diff --git a/dedukti.cabal b/dedukti.cabal --- a/dedukti.cabal +++ b/dedukti.cabal @@ -1,5 +1,5 @@ name: dedukti -version: 1.0.0 +version: 1.0.1 author: Mathieu Boespflug maintainer: Mathieu Boespflug copyright: © 2009 CNRS - École Polytechnique - INRIA @@ -18,7 +18,18 @@ cabal-version: >= 1.6.0 build-type: Simple tested-with: GHC ==6.10 -data-files: t/*.eu +data-files: t/bug.dk + t/coc.dk + t/coq/Datatypes.dk + t/delta1.dk + t/delta2.dk + t/exemple.dk + t/f.dk + t/fold/arith.dk + t/logic.dk + t/nat.dk + t/peano.dk + t/plus.dk executable dedukti diff --git a/t/Coq1univ.eu b/t/Coq1univ.eu deleted file mode 100644 --- a/t/Coq1univ.eu +++ /dev/null @@ -1,70 +0,0 @@ -Uset : Type. -Uprop : Type. -Utype : Type. - -eprop : x : Uprop -> Type. -eset : x : Uset -> Type. -etype : x : Utype -> Type. - -dotset : Utype. -dotprop : Utype. - -; /!\ type : type /!\, should use universes -dottype : Utype. - -; /!\ subtyping in coq, should be unidirectional /!\ -[] Uprop --> Utype. -[] Uset --> Utype. - -dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop. -dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset. -dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype. -dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop. -dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop. -dotpist : x : Uset -> y : (eset x -> Utype) -> Utype. -dotpits : x : Utype -> y : (etype x -> Uset) -> Uset. -dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset. -dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype. - - -[x:Uprop, y : eprop x -> Uprop] - eprop (dotpipp x y) --> w : eprop x -> eprop (y w). - -[x:Uset, y : eset x -> Uprop] - eprop (dotpisp x y) --> w : eset x -> eprop (y w). - -[x:Utype, y : etype x -> Uprop] - eprop (dotpitp x y) --> w : etype x -> eprop (y w). - -; /!\ -[P : Uprop] eprop P --> etype P. - -[x:Uprop, y : eprop x -> Uset] - eset (dotpips x y) --> w : eprop x -> eset (y w). - -[x:Utype, y : etype x -> Uset] - eset (dotpits x y) --> w : etype x -> eset (y w). - -[x:Uset, y : eset x -> Uset] - eset (dotpiss x y) --> w : eset x -> eset (y w). - -; /!\ -[P : Uset] eset P --> etype P. - -[x:Uset, y : eset x -> Utype] - etype (dotpist x y) --> w : eset x -> etype (y w). - -[x:Utype, y : etype x -> Utype] - etype (dotpitt x y) --> w : etype x -> etype (y w). - -[x:Uprop, y : eprop x -> Utype] - etype (dotpipt x y) --> w : eprop x -> etype (y w). - - -[] (etype dotset) --> Uset. -[] (etype dotprop) --> Uprop. -; /!\ -[] (etype dottype) --> Utype. - -; end of Coq1univ - diff --git a/t/Logic.eu b/t/Logic.eu deleted file mode 100644 --- a/t/Logic.eu +++ /dev/null @@ -1,290 +0,0 @@ -True : Uprop. -I : (eprop True) . -case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) . -I_case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (eprop True) ) ) ) . -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( (I_case_0 P) f) t) --> I. -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) ( ( (I_case_0 P) f) t) ) --> f. -True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) . -[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) . -True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) . -[] True_ind --> (P : (etype dotprop) => (True_rect P) ) . -True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) . -[] True_rec --> (P : (etype dotset) => (True_rect P) ) . -False : Uprop. -case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) . -False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) . -[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) . -False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) . -[] False_ind --> (P : (etype dotprop) => (False_rect P) ) . -False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) . -[] False_rec --> (P : (etype dotset) => (False_rect P) ) . -not : (A : Uprop -> Uprop) . -[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) . -and : (A : Uprop -> (B : Uprop -> Uprop) ) . -conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) . -case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) . -conj_case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_11 : (eprop A) -> (_10 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_13 : (eprop A) -> (_12 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Utype, f : (_15 : (eprop A) -> (_14 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) ] ( ( ( ( (conj_case_2 A) B) P) f) a) --> ( (conj A) B) . -[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( ( ( ( (conj_case_2 A) B) P) f) a) var_0) var_1) ) --> ( (f var_0) var_1) . -and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_19 : (eprop A) -> (_18 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) . -[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_17 : (eprop A) => ( (dotpipt B) (_16 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) . -and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_21 : (eprop A) -> (_20 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) . -[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) . -and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_23 : (eprop A) -> (_22 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) . -[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) . -case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_24 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) . -conj_case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_26 : (eprop A) -> (_25 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_3 A) B) H) --> ( (conj A) B) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( ( (conj_case_3 A) B) H) var_2) var_3) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H欧0) ) var_2) var_3) . -proj1 : (A : Uprop -> (B : Uprop -> (_27 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) . -[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) . -case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_28 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) . -conj_case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_30 : (eprop A) -> (_29 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_4 A) B) H) --> ( (conj A) B) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( ( (conj_case_4 A) B) H) var_4) var_5) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) . -proj2 : (A : Uprop -> (B : Uprop -> (_31 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) . -[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) . -or : (A : Uprop -> (B : Uprop -> Uprop) ) . -or_introl : (A : Uprop -> (B : Uprop -> (_32 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) . -or_intror : (A : Uprop -> (B : Uprop -> (_33 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) . -case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_34 : (eprop A) -> (eprop P) ) -> (f欧0 : (_35 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_36 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) . -or_introl_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_39 : (eprop A) -> (eprop P) ) -> (f欧0 : (_40 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_41 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_42 : (eprop A) -> (eprop P) ) , f欧0 : (_43 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) --> ( (or_introl A) B) . -or_intror_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_44 : (eprop A) -> (eprop P) ) -> (f欧0 : (_45 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_46 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_47 : (eprop A) -> (eprop P) ) , f欧0 : (_48 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) --> ( (or_intror A) B) . -[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) var_6) ) --> (f var_6) . -[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) var_7) ) --> (f欧0 var_7) . -or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_51 : (eprop A) -> (eprop P) ) -> (f欧0 : (_52 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) . -[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_50 : (eprop A) => P) ) ) => (f欧0 : (eprop ( (dotpipp B) (_49 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) o) ) ) ) ) ) ) . -iff : (A : Uprop -> (B : Uprop -> Uprop) ) . -[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_53 : (eprop A) => B) ) ) ( (dotpipp B) (_54 : (eprop B) => A) ) ) ) ) . -iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) . -[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_55 : (eprop A) => A) ) ) ( (dotpipp A) (_56 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) . -case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_60 : (eprop ( (and ( (dotpipp A) (_57 : (eprop A) => B) ) ) ( (dotpipp B) (_58 : (eprop B) => A) ) ) ) -> (_59 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) . -conj_case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_68 : (_63 : (eprop A) -> (eprop B) ) -> (_67 : (_64 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_65 : (eprop A) => B) ) ) ( (dotpipp B) (_66 : (eprop B) => A) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) ] ( ( ( (conj_case_6 A) B) C) H) --> ( (conj ( (dotpipp A) (_61 : (eprop A) => B) ) ) ( (dotpipp B) (_62 : (eprop B) => A) ) ) . -case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_71 : (eprop A) -> (eprop B) ) -> (H2 : (_72 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_75 : (eprop ( (and ( (dotpipp B) (_73 : (eprop B) => C) ) ) ( (dotpipp C) (_74 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) . -conj_case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_80 : (eprop A) -> (eprop B) ) -> (H2 : (_81 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_87 : (_82 : (eprop B) -> (eprop C) ) -> (_86 : (_83 : (eprop C) -> (eprop B) ) -> (eprop ( (and ( (dotpipp B) (_84 : (eprop B) => C) ) ) ( (dotpipp C) (_85 : (eprop C) => B) ) ) ) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_88 : (eprop A) -> (eprop B) ) , H2 : (_89 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) ] ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) --> ( (conj ( (dotpipp B) (_78 : (eprop B) => C) ) ) ( (dotpipp C) (_79 : (eprop C) => B) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_76 : (eprop A) -> (eprop B) ) , H2 : (_77 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) , var_10 : (_90 : (eprop B) -> (eprop C) ) , var_11 : (_91 : (eprop C) -> (eprop B) ) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) ( ( ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_95 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_94 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_92 : (eprop A) => C) ) ) ( (dotpipp C) (_93 : (eprop C) => A) ) ) (H欧1 : (eprop A) => (H3 (H1 H欧1) ) ) ) (H欧1 : (eprop C) => (H2 (H1 (H2 (H4 H欧1) ) ) ) ) ) ) ) var_10) var_11) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (_69 : (eprop A) -> (eprop B) ) , var_9 : (_70 : (eprop B) -> (eprop A) ) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( ( ( (conj_case_6 A) B) C) H) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_97 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_96 : (eprop B) => A) ) ) => (H欧0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) H欧0) ) ) ) var_8) var_9) . -iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_99 : (eprop ( (iff A) B) ) -> (_98 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) . -[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) . -case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_102 : (eprop ( (and ( (dotpipp A) (_100 : (eprop A) => B) ) ) ( (dotpipp B) (_101 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) . -conj_case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_110 : (_105 : (eprop A) -> (eprop B) ) -> (_109 : (_106 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_107 : (eprop A) => B) ) ) ( (dotpipp B) (_108 : (eprop B) => A) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_8 A) B) H) --> ( (conj ( (dotpipp A) (_103 : (eprop A) => B) ) ) ( (dotpipp B) (_104 : (eprop B) => A) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (_111 : (eprop A) -> (eprop B) ) , var_13 : (_112 : (eprop B) -> (eprop A) ) ] ( ( ( (case_8 A) B) H) ( ( ( ( (conj_case_8 A) B) H) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_116 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_115 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_113 : (eprop B) => A) ) ) ( (dotpipp A) (_114 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) . -iff_sym : (A : Uprop -> (B : Uprop -> (_117 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) . -[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) . -case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_128 : (eprop ( (and ( (dotpipp A) (_125 : (eprop A) => False) ) ) ( (dotpipp False) (_126 : (eprop False) => A) ) ) ) -> (_127 : (eprop A) -> (eprop False) ) ) ) ) . -conj_case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_136 : (_131 : (eprop A) -> (eprop False) ) -> (_135 : (_132 : (eprop False) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_133 : (eprop A) => False) ) ) ( (dotpipp False) (_134 : (eprop False) => A) ) ) ) ) ) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) ] ( (conj_case_9 A) H) --> ( (conj ( (dotpipp A) (_129 : (eprop A) => False) ) ) ( (dotpipp False) (_130 : (eprop False) => A) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (_137 : (eprop A) -> (eprop False) ) , var_15 : (_138 : (eprop False) -> (eprop A) ) ] ( ( (case_9 A) H) ( ( ( (conj_case_9 A) H) var_14) var_15) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_140 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_139 : (eprop False) => A) ) ) => H欧0) ) var_14) var_15) . -neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) . -[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_118 : (eprop A) => False) ) ) (_119 : (eprop ( (dotpipp A) (_118 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_121 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_120 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_124 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_122 : (eprop A) => False) ) ) ( (dotpipp False) (_123 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) . -and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_168 : (_165 : (eprop B) -> (eprop A) ) -> (_167 : (_166 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_164 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_163 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_141 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_142 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_143 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_144 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_156 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_155 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_154 : (eprop A) => ( (dotpipp B) (_153 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_152 : (eprop A) => ( (dotpipp C) (_151 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp B) (_148 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H欧1 : (eprop A) => (H5 : (eprop C) => ( (H1欧2 : (eprop ( (dotpipp C) (_147 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop A) => ( (H0欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H1欧3 : (eprop A) => (H6 : (eprop B) => H5) ) ) H0欧0) ) (H1欧2 H5) ) ) (H0 H5) ) ) (H2欧0 H4) ) ) ) ) H1欧1) ) (H欧0 H3欧0) ) ) (H1欧0 H4) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp B) (_150 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (dotpipp C) (_149 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H1欧2 : (eprop A) => (H5 : (eprop B) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H0欧1 : (eprop A) => (H6 : (eprop C) => H5) ) ) H欧0) ) (H0欧0 H5) ) ) (H H5) ) ) ) ) H2欧1) ) (H1欧1 H3欧0) ) ) (H2欧0 H4) ) ) (H1欧0 H4) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_157 : (eprop B) => C) ) ) ( (dotpipp C) (_158 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_162 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_161 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_159 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_160 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H1欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2欧0) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj A) B) H3欧0) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_196 : (_193 : (eprop B) -> (eprop A) ) -> (_195 : (_194 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_192 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_191 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_169 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_170 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_171 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_172 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_184 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_183 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_182 : (eprop B) => ( (dotpipp A) (_181 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_180 : (eprop C) => ( (dotpipp A) (_179 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_173 : (eprop B) => C) ) ) ( (dotpipp C) (_174 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_176 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H欧1 : (eprop C) => (H5 : (eprop A) => ( (H1欧2 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_175 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H0欧1 : (eprop B) => (H6 : (eprop A) => H欧1) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H欧1) ) ) (H0 H欧1) ) ) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_178 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H0欧1 : (eprop B) => (H5 : (eprop A) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_177 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H欧1 : (eprop C) => (H6 : (eprop A) => H0欧1) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H0欧1) ) ) (H H0欧1) ) ) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_185 : (eprop B) => C) ) ) ( (dotpipp C) (_186 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_190 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_189 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_187 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_188 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H1欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧0) H2欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj B) A) H0欧0) H3欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_220 : (_217 : (eprop B) -> (eprop (not A) ) ) -> (_219 : (_218 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_216 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_215 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_197 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_198 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_199 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_200 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_208 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_207 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_206 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_205 : (eprop B) => ( (or A) C) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_204 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_203 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_201 : (eprop B) => C) ) ) ( (dotpipp C) (_202 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => H4欧0) ) H0欧0) ) (H5 H4欧0) ) ) (H0 H4欧0) ) ) ) H欧0) ) (H4 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => H5欧0) ) H欧0) ) (H4 H5欧0) ) ) (H H5欧0) ) ) ) H0欧0) ) (H5 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H2欧0 : (eprop A) => (H3 ( ( (or_introl A) C) H2欧0) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H1欧0 : (eprop A) => (H2 ( ( (or_introl A) B) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_209 : (eprop B) => C) ) ) ( (dotpipp C) (_210 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_214 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_213 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_211 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_212 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H1欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_intror A) B) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_244 : (_241 : (eprop B) -> (eprop (not A) ) ) -> (_243 : (_242 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_240 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_239 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_221 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_222 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_223 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_224 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_232 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_231 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_230 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_229 : (eprop A) => ( (or C) A) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_228 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_227 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_225 : (eprop B) => C) ) ) ( (dotpipp C) (_226 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H1欧1 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => H1欧1) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H1欧1) ) ) (H0 H1欧1) ) ) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H2欧1 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => H2欧1) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H2欧1) ) ) (H H2欧1) ) ) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H2欧0 : (eprop C) => (H3 ( ( (or_introl C) A) H2欧0) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H1欧0 : (eprop B) => (H2 ( ( (or_introl B) A) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_233 : (eprop B) => C) ) ) ( (dotpipp C) (_234 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_238 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_237 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_235 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_236 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H1欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_introl B) A) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_251 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) . -[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_245 : (eprop B) => C) ) ) ( (dotpipp C) (_246 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_250 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_249 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_247 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_248 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2) H欧1) ) (H1 H欧1) ) ) (H0 H3) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj A) B) H2) H欧1) ) (H0 H欧1) ) ) (H1 H3) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_258 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) . -[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_252 : (eprop B) => C) ) ) ( (dotpipp C) (_253 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_257 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_256 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_254 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_255 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧1) H3) ) (H1 H欧1) ) ) (H0 H2) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj B) A) H欧1) H3) ) (H0 H欧1) ) ) (H1 H2) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_265 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) . -[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_259 : (eprop B) => C) ) ) ( (dotpipp C) (_260 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_264 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_263 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_261 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_262 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_intror A) B) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_272 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) . -[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_266 : (eprop B) => C) ) ) ( (dotpipp C) (_267 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_271 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_270 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_268 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_269 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H欧0) ) ) (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_introl B) A) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_277 : (eprop ( (and ( (dotpipp A) (_273 : (eprop A) => B) ) ) ( (dotpipp B) (_274 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_275 : (eprop A) => B) ) ) ( (dotpipp B) (_276 : (eprop B) => A) ) ) ) ) ) ) ) . -conj_case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_285 : (_280 : (eprop A) -> (eprop B) ) -> (_284 : (_281 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_282 : (eprop A) => B) ) ) ( (dotpipp B) (_283 : (eprop B) => A) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_10 A) B) H) --> ( (conj ( (dotpipp A) (_278 : (eprop A) => B) ) ) ( (dotpipp B) (_279 : (eprop B) => A) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (_286 : (eprop A) -> (eprop B) ) , var_17 : (_287 : (eprop B) -> (eprop A) ) ] ( ( ( (case_10 A) B) H) ( ( ( ( (conj_case_10 A) B) H) var_16) var_17) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_291 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_290 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_288 : (eprop A) => B) ) ) ( (dotpipp B) (_289 : (eprop B) => A) ) ) H欧0) H0) ) ) var_16) var_17) . -iff_and : (A : Uprop -> (B : Uprop -> (_294 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_292 : (eprop A) => B) ) ) ( (dotpipp B) (_293 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) . -iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_317 : (eprop A) => B) ) ) ( (dotpipp B) (_318 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_297 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_295 : (eprop A) => B) ) ) ( (dotpipp B) (_296 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) (_300 : (eprop ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_301 : (eprop A) => B) ) ) ( (dotpipp B) (_302 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_303 : (eprop A) => B) ) ) ( (dotpipp B) (_304 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_308 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_307 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_305 : (eprop A) => B) ) ) ( (dotpipp B) (_306 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_315 : (eprop A) => B) ) ) ( (dotpipp B) (_316 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_309 : (eprop A) => B) ) ) ( (dotpipp B) (_310 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_314 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_313 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_311 : (eprop A) => B) ) ) ( (dotpipp B) (_312 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) ) ) . -IF_then_else : (P : Uprop -> (Q : Uprop -> (R : Uprop -> Uprop) ) ) . -[] IF_then_else --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => ( (or ( (and P) Q) ) ( (and (not P) ) R) ) ) ) ) . -ex : (A : Utype -> (P : (_319 : (etype A) -> Uprop) -> Uprop) ) . -ex_intro : (A : Utype -> (P : (_320 : (etype A) -> Uprop) -> (x : (etype A) -> (_321 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) . -case_11 : (A : Utype -> (P : (_322 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_323 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (_324 : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) ) . -ex_intro_case_11 : (A : Utype -> (P : (_327 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_328 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (x : (etype A) -> (_329 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_330 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_331 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) ] ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) --> ( (ex_intro A) P) . -[A : Utype, P : (_325 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_326 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) , var_18 : (etype A) , var_19 : (eprop (P var_18) ) ] ( ( ( ( ( (case_11 A) P) P欧0) f) e) ( ( ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) var_18) var_19) ) --> ( (f var_18) var_19) . -ex_ind : (A : Utype -> (P : (_334 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_335 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) . -[] ex_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_333 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_332 : (eprop (P x) ) => P欧0) ) ) ) ) => (e : (eprop ( (ex A) P) ) => ( ( ( ( ( (case_11 A) P) P欧0) f) e) e) ) ) ) ) ) . -ex2 : (A : Utype -> (P : (_336 : (etype A) -> Uprop) -> (Q : (_337 : (etype A) -> Uprop) -> Uprop) ) ) . -ex_intro2 : (A : Utype -> (P : (_338 : (etype A) -> Uprop) -> (Q : (_339 : (etype A) -> Uprop) -> (x : (etype A) -> (_341 : (eprop (P x) ) -> (_340 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) . -case_12 : (A : Utype -> (P : (_342 : (etype A) -> Uprop) -> (Q : (_343 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_345 : (eprop (P x) ) -> (_344 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (_346 : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) ) . -ex_intro2_case_12 : (A : Utype -> (P : (_351 : (etype A) -> Uprop) -> (Q : (_352 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_354 : (eprop (P x) ) -> (_353 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (x : (etype A) -> (_356 : (eprop (P x) ) -> (_355 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_357 : (etype A) -> Uprop) , Q : (_358 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_360 : (eprop (P x) ) -> (_359 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) ] ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) --> ( ( (ex_intro2 A) P) Q) . -[A : Utype, P : (_347 : (etype A) -> Uprop) , Q : (_348 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_350 : (eprop (P x) ) -> (_349 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) , var_20 : (etype A) , var_21 : (eprop (P var_20) ) , var_22 : (eprop (Q var_20) ) ] ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) ( ( ( ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) var_20) var_21) var_22) ) --> ( ( (f var_20) var_21) var_22) . -ex2_ind : (A : Utype -> (P : (_365 : (etype A) -> Uprop) -> (Q : (_366 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_368 : (eprop (P x) ) -> (_367 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) . -[] ex2_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_364 : (etype A) => dotprop) ) ) => (Q : (etype ( (dotpitt A) (_363 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_362 : (eprop (P x) ) => ( (dotpipp (Q x) ) (_361 : (eprop (Q x) ) => P欧0) ) ) ) ) ) ) => (e : (eprop ( ( (ex2 A) P) Q) ) => ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) e) ) ) ) ) ) ) . -all : (A : Utype -> (P : (_370 : (etype A) -> Uprop) -> Uprop) ) . -[] all --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_369 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => (P x) ) ) ) ) . -inst : (A : Utype -> (P : (_372 : (etype A) -> Uprop) -> (x : (etype A) -> (_373 : (eprop ( (all A) (x欧0 : (etype A) => (P x欧0) ) ) ) -> (eprop (P x) ) ) ) ) ) . -[] inst --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_371 : (etype A) => dotprop) ) ) => (x : (etype A) => (H : (eprop ( (dotpitp A) (x欧0 : (etype A) => (P x欧0) ) ) ) => (H x) ) ) ) ) . -gen : (A : Utype -> (P : (_376 : (etype A) -> Uprop) -> (B : Uprop -> (f : (y : (etype A) -> (_377 : (eprop B) -> (eprop (P y) ) ) ) -> (_378 : (eprop B) -> (eprop ( (all A) P) ) ) ) ) ) ) . -[] gen --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_375 : (etype A) => dotprop) ) ) => (B : (etype dotprop) => (f : (eprop ( (dotpitp A) (y : (etype A) => ( (dotpipp B) (_374 : (eprop B) => (P y) ) ) ) ) ) => (H : (eprop B) => (x : (etype A) => ( (f x) H) ) ) ) ) ) ) . -eq : (A : Utype -> (x : (etype A) -> (_379 : (etype A) -> Uprop) ) ) . -refl_equal : (A : Utype -> (x : (etype A) -> (eprop ( ( (eq A) x) x) ) ) ) . -case_13 : (A : Utype -> (x : (etype A) -> (P : (_380 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_381 : (eprop ( ( (eq A) x) y欧0) ) -> (etype (P y欧0) ) ) ) ) ) ) ) ) ) . -refl_equal_case_13 : (A : Utype -> (x : (etype A) -> (P : (_383 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , P : (_384 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) --> ( (refl_equal A) x) . -[A : Utype, x : (etype A) , P : (_382 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) x) ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) ) --> f. -eq_rect : (A : Utype -> (x : (etype A) -> (P : (_386 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (etype (P y) ) ) ) ) ) ) ) . -[] eq_rect --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_385 : (etype A) => dottype) ) ) => (f : (etype (P x) ) => (y : (etype A) => (e : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) y) e) ) ) ) ) ) ) . -eq_ind : (A : Utype -> (x : (etype A) -> (P : (_388 : (etype A) -> Uprop) -> (f : (eprop (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop (P y) ) ) ) ) ) ) ) . -[] eq_ind --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_387 : (etype A) => dotprop) ) ) => ( ( (eq_rect A) x) P) ) ) ) . -eq_rec : (A : Utype -> (x : (etype A) -> (P : (_390 : (etype A) -> Uset) -> (f : (eset (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eset (P y) ) ) ) ) ) ) ) . -[] eq_rec --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_389 : (etype A) => dotset) ) ) => ( ( (eq_rect A) x) P) ) ) ) . -case_14 : (A : Uprop -> (C : Uprop -> (h1 : (eprop A) -> (h2 : (_391 : (eprop A) -> (eprop False) ) -> (f : (eprop False) -> (_392 : (eprop False) -> (eprop C) ) ) ) ) ) ) . -absurd : (A : Uprop -> (C : Uprop -> (_396 : (eprop A) -> (_395 : (eprop (not A) ) -> (eprop C) ) ) ) ) . -[] absurd --> (A : (etype dotprop) => (C : (etype dotprop) => (h1 : (eprop A) => (h2 : (eprop ( (dotpipp A) (_394 : (eprop A) => False) ) ) => ( (f : (eprop False) => ( ( ( ( ( (case_14 A) C) h1) h2) f) f) ) (h2 h1) ) ) ) ) ) . -case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_397 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq A) y欧0) x) ) ) ) ) ) ) ) . -refl_equal_case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( (refl_equal_case_15 A) x) y) H) --> ( (refl_equal A) x) . -[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (case_15 A) x) y) H) x) ( ( ( (refl_equal_case_15 A) x) y) H) ) --> ( (refl_equal A) x) . -sym_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_398 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) . -[] sym_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( (case_15 A) x) y) H) y) H) ) ) ) ) . -case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (y欧0 : (etype A) -> (_399 : (eprop ( ( (eq A) y) y欧0) ) -> (eprop ( ( (eq A) x) y欧0) ) ) ) ) ) ) ) ) ) . -refl_equal_case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) --> ( (refl_equal A) y) . -[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) y) ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) ) --> H. -trans_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_401 : (eprop ( ( (eq A) x) y) ) -> (_400 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) . -[] trans_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => (H0 : (eprop ( ( (eq A) y) z) ) => ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) z) H0) ) ) ) ) ) ) . -case_17 : (A : Utype -> (B : Utype -> (f : (_402 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_403 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq B) (f x) ) (f y欧0) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_17 : (A : Utype -> (B : Utype -> (f : (_405 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) . -[A : Utype, B : Utype, f : (_406 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) --> ( (refl_equal A) x) . -[A : Utype, B : Utype, f : (_404 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) x) ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) ) --> ( (refl_equal B) (f x) ) . -f_equal : (A : Utype -> (B : Utype -> (f : (_408 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (_409 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq B) (f x) ) (f y) ) ) ) ) ) ) ) ) . -[] f_equal --> (A : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A) (_407 : (etype A) => B) ) ) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) y) H) ) ) ) ) ) ) . -case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (y欧0 : (etype A) -> (_410 : (eprop ( ( (eq A) y) y欧0) ) -> (h1欧0 : (eprop (not ( ( (eq A) y欧0) y) ) ) -> (eprop ( ( (eq A) y欧0) y) ) ) ) ) ) ) ) ) ) . -refl_equal_case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) --> ( (refl_equal A) y) . -[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( ( ( (case_18 A) x) y) h1) h2) y) ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) ) --> (h1欧0 : (eprop (not ( ( (eq A) y) y) ) ) => ( (refl_equal A) y) ) . -sym_not_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_411 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) . -[] sym_not_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (h1 : (eprop (not ( ( (eq A) x) y) ) ) => (h2 : (eprop ( ( (eq A) y) x) ) => (h1 ( ( ( ( ( ( ( (case_18 A) x) y) h1) h2) x) h2) h1) ) ) ) ) ) ) . -sym_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_412 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) . -[] sym_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_eq A) x) y) ) ) ) . -sym_not_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_413 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) . -[] sym_not_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_not_eq A) x) y) ) ) ) . -trans_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_415 : (eprop ( ( (eq A) x) y) ) -> (_414 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) . -[] trans_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => ( ( ( (trans_eq A) x) y) z) ) ) ) ) . -eq_ind_r : (A : Utype -> (x : (etype A) -> (P : (_417 : (etype A) -> Uprop) -> (_419 : (eprop (P x) ) -> (y : (etype A) -> (_418 : (eprop ( ( (eq A) y) x) ) -> (eprop (P y) ) ) ) ) ) ) ) . -[] eq_ind_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_416 : (etype A) => dotprop) ) ) => (H : (eprop (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_ind A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -eq_rec_r : (A : Utype -> (x : (etype A) -> (P : (_421 : (etype A) -> Uset) -> (_423 : (eset (P x) ) -> (y : (etype A) -> (_422 : (eprop ( ( (eq A) y) x) ) -> (eset (P y) ) ) ) ) ) ) ) . -[] eq_rec_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_420 : (etype A) => dotset) ) ) => (H : (eset (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rec A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -eq_rect_r : (A : Utype -> (x : (etype A) -> (P : (_425 : (etype A) -> Utype) -> (_427 : (etype (P x) ) -> (y : (etype A) -> (_426 : (eprop ( ( (eq A) y) x) ) -> (etype (P y) ) ) ) ) ) ) ) . -[] eq_rect_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_424 : (etype A) => dottype) ) ) => (H : (etype (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rect A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_429 : (etype A1) -> (_428 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_431 : (eprop ( ( (eq A1) x1) y) ) -> (_430 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y) y2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_435 : (etype A1) -> (_434 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_437 : (etype A1) -> (_436 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) --> ( (refl_equal A1) x1) . -case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_439 : (etype A1) -> (_438 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_440 : (eprop ( ( (eq A2) x2) y) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f x1) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_444 : (etype A1) -> (_443 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_446 : (etype A1) -> (_445 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) --> ( (refl_equal A2) x2) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_442 : (etype A1) -> (_441 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) x2) ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) ) --> ( (refl_equal B) ( (f x1) x2) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_433 : (etype A1) -> (_432 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) x1) ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) y2) H欧0) ) . -f_equal2 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_450 : (etype A1) -> (_449 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (_452 : (eprop ( ( (eq A1) x1) y1) ) -> (_451 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y1) y2) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal2 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_448 : (etype A1) => ( (dotpitt A2) (_447 : (etype A2) => B) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) y1) H) ) ) ) ) ) ) ) ) ) . -case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_455 : (etype A1) -> (_454 : (etype A2) -> (_453 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_458 : (eprop ( ( (eq A1) x1) y) ) -> (_457 : (eprop ( ( (eq A2) x2) y2) ) -> (_456 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_464 : (etype A1) -> (_463 : (etype A2) -> (_462 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_467 : (etype A1) -> (_466 : (etype A2) -> (_465 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) --> ( (refl_equal A1) x1) . -case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_470 : (etype A1) -> (_469 : (etype A2) -> (_468 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_472 : (eprop ( ( (eq A2) x2) y) ) -> (_471 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) y) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_478 : (etype A1) -> (_477 : (etype A2) -> (_476 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_481 : (etype A1) -> (_480 : (etype A2) -> (_479 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) --> ( (refl_equal A2) x2) . -case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_484 : (etype A1) -> (_483 : (etype A2) -> (_482 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_485 : (eprop ( ( (eq A3) x3) y) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) x2) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_491 : (etype A1) -> (_490 : (etype A2) -> (_489 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_494 : (etype A1) -> (_493 : (etype A2) -> (_492 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_488 : (etype A1) -> (_487 : (etype A2) -> (_486 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) ) --> ( (refl_equal B) ( ( (f x1) x2) x3) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_475 : (etype A1) -> (_474 : (etype A2) -> (_473 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_461 : (etype A1) -> (_460 : (etype A2) -> (_459 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) x1) ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) y2) H欧0) ) . -f_equal3 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_500 : (etype A1) -> (_499 : (etype A2) -> (_498 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (_503 : (eprop ( ( (eq A1) x1) y1) ) -> (_502 : (eprop ( ( (eq A2) x2) y2) ) -> (_501 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y1) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal3 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_497 : (etype A1) => ( (dotpitt A2) (_496 : (etype A2) => ( (dotpitt A3) (_495 : (etype A3) => B) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) . -case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_507 : (etype A1) -> (_506 : (etype A2) -> (_505 : (etype A3) -> (_504 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_511 : (eprop ( ( (eq A1) x1) y) ) -> (_510 : (eprop ( ( (eq A2) x2) y2) ) -> (_509 : (eprop ( ( (eq A3) x3) y3) ) -> (_508 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_519 : (etype A1) -> (_518 : (etype A2) -> (_517 : (etype A3) -> (_516 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_523 : (etype A1) -> (_522 : (etype A2) -> (_521 : (etype A3) -> (_520 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) --> ( (refl_equal A1) x1) . -case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_527 : (etype A1) -> (_526 : (etype A2) -> (_525 : (etype A3) -> (_524 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_530 : (eprop ( ( (eq A2) x2) y) ) -> (_529 : (eprop ( ( (eq A3) x3) y3) ) -> (_528 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) y) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_538 : (etype A1) -> (_537 : (etype A2) -> (_536 : (etype A3) -> (_535 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_542 : (etype A1) -> (_541 : (etype A2) -> (_540 : (etype A3) -> (_539 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) --> ( (refl_equal A2) x2) . -case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_546 : (etype A1) -> (_545 : (etype A2) -> (_544 : (etype A3) -> (_543 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_548 : (eprop ( ( (eq A3) x3) y) ) -> (_547 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) y) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_556 : (etype A1) -> (_555 : (etype A2) -> (_554 : (etype A3) -> (_553 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_560 : (etype A1) -> (_559 : (etype A2) -> (_558 : (etype A3) -> (_557 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_564 : (etype A1) -> (_563 : (etype A2) -> (_562 : (etype A3) -> (_561 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_565 : (eprop ( ( (eq A4) x4) y) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) x3) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_573 : (etype A1) -> (_572 : (etype A2) -> (_571 : (etype A3) -> (_570 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_577 : (etype A1) -> (_576 : (etype A2) -> (_575 : (etype A3) -> (_574 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_569 : (etype A1) -> (_568 : (etype A2) -> (_567 : (etype A3) -> (_566 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) ) --> ( (refl_equal B) ( ( ( (f x1) x2) x3) x4) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_552 : (etype A1) -> (_551 : (etype A2) -> (_550 : (etype A3) -> (_549 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) y4) H欧2) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_534 : (etype A1) -> (_533 : (etype A2) -> (_532 : (etype A3) -> (_531 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_515 : (etype A1) -> (_514 : (etype A2) -> (_513 : (etype A3) -> (_512 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) y2) H欧0) ) . -f_equal4 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_585 : (etype A1) -> (_584 : (etype A2) -> (_583 : (etype A3) -> (_582 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (_589 : (eprop ( ( (eq A1) x1) y1) ) -> (_588 : (eprop ( ( (eq A2) x2) y2) ) -> (_587 : (eprop ( ( (eq A3) x3) y3) ) -> (_586 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y1) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal4 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_581 : (etype A1) => ( (dotpitt A2) (_580 : (etype A2) => ( (dotpitt A3) (_579 : (etype A3) => ( (dotpitt A4) (_578 : (etype A4) => B) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_594 : (etype A1) -> (_593 : (etype A2) -> (_592 : (etype A3) -> (_591 : (etype A4) -> (_590 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_599 : (eprop ( ( (eq A1) x1) y) ) -> (_598 : (eprop ( ( (eq A2) x2) y2) ) -> (_597 : (eprop ( ( (eq A3) x3) y3) ) -> (_596 : (eprop ( ( (eq A4) x4) y4) ) -> (_595 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_609 : (etype A1) -> (_608 : (etype A2) -> (_607 : (etype A3) -> (_606 : (etype A4) -> (_605 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_614 : (etype A1) -> (_613 : (etype A2) -> (_612 : (etype A3) -> (_611 : (etype A4) -> (_610 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) --> ( (refl_equal A1) x1) . -case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_619 : (etype A1) -> (_618 : (etype A2) -> (_617 : (etype A3) -> (_616 : (etype A4) -> (_615 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_623 : (eprop ( ( (eq A2) x2) y) ) -> (_622 : (eprop ( ( (eq A3) x3) y3) ) -> (_621 : (eprop ( ( (eq A4) x4) y4) ) -> (_620 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) y) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_633 : (etype A1) -> (_632 : (etype A2) -> (_631 : (etype A3) -> (_630 : (etype A4) -> (_629 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_638 : (etype A1) -> (_637 : (etype A2) -> (_636 : (etype A3) -> (_635 : (etype A4) -> (_634 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) --> ( (refl_equal A2) x2) . -case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_643 : (etype A1) -> (_642 : (etype A2) -> (_641 : (etype A3) -> (_640 : (etype A4) -> (_639 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_646 : (eprop ( ( (eq A3) x3) y) ) -> (_645 : (eprop ( ( (eq A4) x4) y4) ) -> (_644 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) y) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_656 : (etype A1) -> (_655 : (etype A2) -> (_654 : (etype A3) -> (_653 : (etype A4) -> (_652 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_661 : (etype A1) -> (_660 : (etype A2) -> (_659 : (etype A3) -> (_658 : (etype A4) -> (_657 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_666 : (etype A1) -> (_665 : (etype A2) -> (_664 : (etype A3) -> (_663 : (etype A4) -> (_662 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_668 : (eprop ( ( (eq A4) x4) y) ) -> (_667 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) y) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_678 : (etype A1) -> (_677 : (etype A2) -> (_676 : (etype A3) -> (_675 : (etype A4) -> (_674 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_683 : (etype A1) -> (_682 : (etype A2) -> (_681 : (etype A3) -> (_680 : (etype A4) -> (_679 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) . -case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_688 : (etype A1) -> (_687 : (etype A2) -> (_686 : (etype A3) -> (_685 : (etype A4) -> (_684 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (y : (etype A5) -> (_689 : (eprop ( ( (eq A5) x5) y) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) x4) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_699 : (etype A1) -> (_698 : (etype A2) -> (_697 : (etype A3) -> (_696 : (etype A4) -> (_695 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq A5) x5) x5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_704 : (etype A1) -> (_703 : (etype A2) -> (_702 : (etype A3) -> (_701 : (etype A4) -> (_700 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) --> ( (refl_equal A5) x5) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_694 : (etype A1) -> (_693 : (etype A2) -> (_692 : (etype A3) -> (_691 : (etype A4) -> (_690 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) x5) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) ) --> ( (refl_equal B) ( ( ( ( (f x1) x2) x3) x4) x5) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_673 : (etype A1) -> (_672 : (etype A2) -> (_671 : (etype A3) -> (_670 : (etype A4) -> (_669 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) ) --> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) y5) H欧3) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_651 : (etype A1) -> (_650 : (etype A2) -> (_649 : (etype A3) -> (_648 : (etype A4) -> (_647 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) y4) H欧2) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_628 : (etype A1) -> (_627 : (etype A2) -> (_626 : (etype A3) -> (_625 : (etype A4) -> (_624 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_604 : (etype A1) -> (_603 : (etype A2) -> (_602 : (etype A3) -> (_601 : (etype A4) -> (_600 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) y2) H欧0) ) . -f_equal5 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_714 : (etype A1) -> (_713 : (etype A2) -> (_712 : (etype A3) -> (_711 : (etype A4) -> (_710 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (_719 : (eprop ( ( (eq A1) x1) y1) ) -> (_718 : (eprop ( ( (eq A2) x2) y2) ) -> (_717 : (eprop ( ( (eq A3) x3) y3) ) -> (_716 : (eprop ( ( (eq A4) x4) y4) ) -> (_715 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y1) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal5 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (A5 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_709 : (etype A1) => ( (dotpitt A2) (_708 : (etype A2) => ( (dotpitt A3) (_707 : (etype A3) => ( (dotpitt A4) (_706 : (etype A4) => ( (dotpitt A5) (_705 : (etype A5) => B) ) ) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (x5 : (etype A5) => (y5 : (etype A5) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -subrelation : (A : Utype -> (B : Utype -> (R : (_726 : (etype A) -> (_725 : (etype B) -> Uprop) ) -> (R' : (_728 : (etype A) -> (_727 : (etype B) -> Uprop) ) -> Uprop) ) ) ) . -[] subrelation --> (A : (etype dottype) => (B : (etype dottype) => (R : (etype ( (dotpitt A) (_724 : (etype A) => ( (dotpitt B) (_723 : (etype B) => dotprop) ) ) ) ) => (R' : (etype ( (dotpitt A) (_722 : (etype A) => ( (dotpitt B) (_721 : (etype B) => dotprop) ) ) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp B) (y : (etype B) => ( (dotpipp ( (R x) y) ) (_720 : (eprop ( (R x) y) ) => ( (R' x) y) ) ) ) ) ) ) ) ) ) ) . -unique : (A : Utype -> (P : (_731 : (etype A) -> Uprop) -> (x : (etype A) -> Uprop) ) ) . -[] unique --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_730 : (etype A) => dotprop) ) ) => (x : (etype A) => ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_729 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) . -uniqueness : (A : Utype -> (P : (_735 : (etype A) -> Uprop) -> Uprop) ) . -[] uniqueness --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_734 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp A) (y : (etype A) => ( (dotpipp (P x) ) (_733 : (eprop (P x) ) => ( (dotpipp (P y) ) (_732 : (eprop (P y) ) => ( ( (eq A) x) y) ) ) ) ) ) ) ) ) ) ) . -case_33 : (A : Utype -> (P : (_738 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_739 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) . -conj_case_33 : (A : Utype -> (P : (_741 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_743 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_742 : (eprop ( (uniqueness A) P) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) . -[A : Utype, P : (_744 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ] ( ( (conj_case_33 A) P) H) --> ( (conj ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) . -case_34 : (A : Utype -> (P : (_745 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_747 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_746 : (eprop ( (uniqueness A) P) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) ) ) . -ex_intro_case_34 : (A : Utype -> (P : (_749 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_750 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_751 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( ( (ex_intro_case_34 A) P) H) H欧0) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) . -[A : Utype, P : (_748 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_25 : (etype A) , var_26 : (eprop ( (x : (etype A) => (P x) ) var_25) ) ] ( ( ( ( (case_34 A) P) H) H欧0) ( ( ( ( ( (ex_intro_case_34 A) P) H) H欧0) var_25) var_26) ) --> ( ( (x : (etype A) => (Hx : (eprop (P x) ) => (Huni : (eprop ( (uniqueness A) P) ) => ( ( ( (ex_intro A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) x) ( ( ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_752 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) Hx) (x' : (etype A) => (H欧1 : (eprop (P x') ) => ( ( ( (Huni x) x') Hx) H欧1) ) ) ) ) ) ) ) var_25) var_26) . -[A : Utype, P : (_740 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , var_23 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_24 : (eprop ( (uniqueness A) P) ) ] ( ( ( (case_33 A) P) H) ( ( ( ( (conj_case_33 A) P) H) var_23) var_24) ) --> ( ( (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( ( (case_34 A) P) H) H欧0) H欧0) ) var_23) var_24) . -case_35 : (A : Utype -> (P : (_753 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (_754 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) . -ex_intro_case_35 : (A : Utype -> (P : (_756 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (_757 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (eprop ( (ex A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_758 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ] ( ( (ex_intro_case_35 A) P) H) --> ( (ex_intro A) ( (unique A) (x : (etype A) => (P x) ) ) ) . -case_36 : (A : Utype -> (P : (_759 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_761 : (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_760 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) -> (eprop ( (and ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) ) . -conj_case_36 : (A : Utype -> (P : (_764 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_768 : (eprop (P x) ) -> (_767 : (x' : (etype A) -> (_765 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) -> (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_766 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_769 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) ] ( ( ( ( (conj_case_36 A) P) H) x) H欧0) --> ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_763 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) . -[A : Utype, P : (_762 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) , var_29 : (eprop (P x) ) , var_30 : (x' : (etype A) -> (_770 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) ] ( ( ( ( ( (case_36 A) P) H) x) H欧0) ( ( ( ( ( ( (conj_case_36 A) P) H) x) H欧0) var_29) var_30) ) --> ( ( (Hx : (eprop (P x) ) => (Huni : (eprop ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_771 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) => ( ( ( (conj ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ( ( ( (ex_intro A) (x欧0 : (etype A) => (P x欧0) ) ) x) Hx) ) (x' : (etype A) => (x'' : (etype A) => (Hx' : (eprop (P x') ) => (Hx'' : (eprop (P x'') ) => ( ( ( ( ( (trans_eq A) x') x) x'') ( ( ( (sym_eq A) x) x') ( (Huni x') Hx') ) ) ( (Huni x'') Hx'') ) ) ) ) ) ) ) ) var_29) var_30) . -[A : Utype, P : (_755 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , var_27 : (etype A) , var_28 : (eprop ( ( (unique A) (x : (etype A) => (P x) ) ) var_27) ) ] ( ( ( (case_35 A) P) H) ( ( ( ( (ex_intro_case_35 A) P) H) var_27) var_28) ) --> ( ( (x : (etype A) => (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) => ( ( ( ( ( (case_36 A) P) H) x) H欧0) H欧0) ) ) var_27) var_28) . -unique_existence : (A : Utype -> (P : (_773 : (etype A) -> Uprop) -> (eprop ( (iff ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) . -[] unique_existence --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_772 : (etype A) => dotprop) ) ) => ( ( ( (conj ( (dotpipp ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) (_736 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ( (dotpipp ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) (_737 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( ( ( (case_33 A) P) H) H) ) ) (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( ( ( (case_35 A) P) H) H) ) ) ) ) . -inhabited : (A : Utype -> Uprop) . -inhabits : (A : Utype -> (_774 : (etype A) -> (eprop (inhabited A) ) ) ) . -case_37 : (A : Utype -> (P : Uprop -> (f : (_775 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_776 : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) ) . -inhabits_case_37 : (A : Utype -> (P : Uprop -> (f : (_778 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_779 : (etype A) -> (eprop (inhabited A) ) ) ) ) ) ) . -[A : Utype, P : Uprop, f : (_780 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) ] ( ( ( (inhabits_case_37 A) P) f) i) --> (inhabits A) . -[A : Utype, P : Uprop, f : (_777 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) , var_31 : (etype A) ] ( ( ( ( (case_37 A) P) f) i) ( ( ( ( (inhabits_case_37 A) P) f) i) var_31) ) --> (f var_31) . -inhabited_ind : (A : Utype -> (P : Uprop -> (f : (_782 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) . -[] inhabited_ind --> (A : (etype dottype) => (P : (etype dotprop) => (f : (eprop ( (dotpitp A) (_781 : (etype A) => P) ) ) => (i : (eprop (inhabited A) ) => ( ( ( ( (case_37 A) P) f) i) i) ) ) ) ) . -case_38 : (A : Utype -> (P : (_783 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_784 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) ) . -ex_intro_case_38 : (A : Utype -> (P : (_786 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_787 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_788 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( (ex_intro_case_38 A) P) H) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) . -[A : Utype, P : (_785 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_32 : (etype A) , var_33 : (eprop ( (x : (etype A) => (P x) ) var_32) ) ] ( ( ( (case_38 A) P) H) ( ( ( ( (ex_intro_case_38 A) P) H) var_32) var_33) ) --> ( ( (x : (etype A) => (H欧0 : (eprop (P x) ) => ( (inhabits A) x) ) ) var_32) var_33) . -exists_inhabited : (A : Utype -> (P : (_790 : (etype A) -> Uprop) -> (_791 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) . -[] exists_inhabited --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_789 : (etype A) => dotprop) ) ) => (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( (case_38 A) P) H) H) ) ) ) . -eq_stepl : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_793 : (eprop ( ( (eq A) x) y) ) -> (_792 : (eprop ( ( (eq A) x) z) ) -> (eprop ( ( (eq A) z) y) ) ) ) ) ) ) ) . -[] eq_stepl --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H1 : (eprop ( ( (eq A) x) y) ) => (H2 : (eprop ( ( (eq A) x) z) ) => ( ( ( ( ( (eq_ind A) x) (z欧0 : (etype A) => ( ( (eq A) z欧0) y) ) ) H1) z) H2) ) ) ) ) ) ) . -iff_stepl : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_805 : (eprop ( (iff A) B) ) -> (_804 : (eprop ( (iff A) C) ) -> (eprop ( (iff C) B) ) ) ) ) ) ) . -[] iff_stepl --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => (H0 : (eprop ( (iff A) C) ) => ( ( ( ( (and_ind ( (dotpipp A) (_794 : (eprop A) => B) ) ) ( (dotpipp B) (_795 : (eprop B) => A) ) ) ( (iff C) B) ) (H1 : (eprop ( (dotpipp A) (_803 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_802 : (eprop B) => A) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_796 : (eprop A) => C) ) ) ( (dotpipp C) (_797 : (eprop C) => A) ) ) ( (iff C) B) ) (H欧0 : (eprop ( (dotpipp A) (_801 : (eprop A) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_800 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp C) (_798 : (eprop C) => B) ) ) ( (dotpipp B) (_799 : (eprop B) => C) ) ) (H0欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H3欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H3欧0) (H2 H3欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H3 H0欧0) ) ) ) (H0欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H2欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H1欧0) (H3 H1欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H2 H0欧0) ) ) ) ) ) ) H0) ) ) ) H) ) ) ) ) ) . -;Finished module Logic diff --git a/t/Logicavecprelude.eu b/t/Logicavecprelude.eu deleted file mode 100644 --- a/t/Logicavecprelude.eu +++ /dev/null @@ -1,156 +0,0 @@ -Uset : Type. -Uprop : Type. -Utype : Type. - -eprop : x : Uprop -> Type. -eset : x : Uset -> Type. -etype : x : Utype -> Type. - -dotset : Utype. -dotprop : Utype. - -; /!\ type : type /!\, should use universes -dottype : Utype. - -; /!\ subtyping in coq, should be unidirectional /!\ -[] Uprop --> Utype. -[] Uset --> Utype. - -dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop. -dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset. -dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype. -dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop. -dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop. -dotpist : x : Uset -> y : (eset x -> Utype) -> Utype. -dotpits : x : Utype -> y : (etype x -> Uset) -> Uset. -dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset. -dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype. - - -[x:Uprop, y : eprop x -> Uprop] - eprop (dotpipp x y) --> w : eprop x -> eprop (y w). - -[x:Uset, y : eset x -> Uprop] - eprop (dotpisp x y) --> w : eset x -> eprop (y w). - -[x:Utype, y : etype x -> Uprop] - eprop (dotpitp x y) --> w : etype x -> eprop (y w). - -; /!\ -[P : Uprop] eprop P --> etype P. - -[x:Uprop, y : eprop x -> Uset] - eset (dotpips x y) --> w : eprop x -> eset (y w). - -[x:Utype, y : etype x -> Uset] - eset (dotpits x y) --> w : etype x -> eset (y w). - -[x:Uset, y : eset x -> Uset] - eset (dotpiss x y) --> w : eset x -> eset (y w). - -; /!\ -[P : Uset] eset P --> etype P. - -[x:Uset, y : eset x -> Utype] - etype (dotpist x y) --> w : eset x -> etype (y w). - -[x:Utype, y : etype x -> Utype] - etype (dotpitt x y) --> w : etype x -> etype (y w). - -[x:Uprop, y : eprop x -> Utype] - etype (dotpipt x y) --> w : eprop x -> etype (y w). - - -[] (etype dotset) --> Uset. -[] (etype dotprop) --> Uprop. -; /!\ -[] (etype dottype) --> Utype. - -; end of Coq1univ - -True : Uprop. -I : (eprop True) . -case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) . -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) I) --> f. -True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) . -[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) . -True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) . -[] True_ind --> (P : (etype dotprop) => (True_rect P) ) . -True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) . -[] True_rec --> (P : (etype dotset) => (True_rect P) ) . -False : Uprop. -case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) . -False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) . -[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) . -False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) . -[] False_ind --> (P : (etype dotprop) => (False_rect P) ) . -False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) . -[] False_rec --> (P : (etype dotset) => (False_rect P) ) . -not : (A : Uprop -> Uprop) . -[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) . -and : (A : Uprop -> (B : Uprop -> Uprop) ) . -conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) . -case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( (conj A) B) var_0) var_1) ) --> ( (f var_0) var_1) . -and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_13 : (eprop A) -> (_12 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) . -[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_11 : (eprop A) => ( (dotpipt B) (_10 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) . -and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_15 : (eprop A) -> (_14 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) . -[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) . -and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_17 : (eprop A) -> (_16 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) . -[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) . -case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_18 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( (conj A) B) var_2) var_3) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_2) var_3) . -proj1 : (A : Uprop -> (B : Uprop -> (_19 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) . -[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) . -case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_20 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( (conj A) B) var_4) var_5) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) . -proj2 : (A : Uprop -> (B : Uprop -> (_21 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) . -[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) . -or : (A : Uprop -> (B : Uprop -> Uprop) ) . -or_introl : (A : Uprop -> (B : Uprop -> (_22 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) . -or_intror : (A : Uprop -> (B : Uprop -> (_23 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) . -case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_24 : (eprop A) -> (eprop P) ) -> (f0 : (_25 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_26 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_introl A) B) var_6) ) --> (f var_6) . -[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_intror A) B) var_7) ) --> (f0 var_7) . -or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_31 : (eprop A) -> (eprop P) ) -> (f0 : (_32 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) . -[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_30 : (eprop A) => P) ) ) => (f0 : (eprop ( (dotpipp B) (_29 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) o) ) ) ) ) ) ) . -iff : (A : Uprop -> (B : Uprop -> Uprop) ) . -[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_33 : (eprop A) => B) ) ) ( (dotpipp B) (_34 : (eprop B) => A) ) ) ) ) . -iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) . -[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_35 : (eprop A) => A) ) ) ( (dotpipp A) (_36 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) . -case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_40 : (eprop ( (and ( (dotpipp A) (_37 : (eprop A) => B) ) ) ( (dotpipp B) (_38 : (eprop B) => A) ) ) ) -> (_39 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) . -case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_41 : (eprop A) -> (eprop B) ) -> (H2 : (_42 : (eprop B) -> (eprop A) ) -> (H0 : (eprop ( (iff B) C) ) -> (_45 : (eprop ( (and ( (dotpipp B) (_43 : (eprop B) => C) ) ) ( (dotpipp C) (_44 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_46 : (eprop A) -> (eprop B) ) , H2 : (_47 : (eprop B) -> (eprop A) ) , H0 : (eprop ( (iff B) C) ) , var_10 : (eprop B) , var_11 : (eprop C) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) ( ( ( (conj B) C) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_51 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_50 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_48 : (eprop A) => C) ) ) ( (dotpipp C) (_49 : (eprop C) => A) ) ) (H1 : (eprop A) => (H3 (H1 H1) ) ) ) (H1 : (eprop C) => (H2 (H1 (H2 (H4 H1) ) ) ) ) ) ) ) var_10) var_11) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (eprop A) , var_9 : (eprop B) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( (conj A) B) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_53 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_52 : (eprop B) => A) ) ) => (H0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) H0) ) ) ) var_8) var_9) . -iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_55 : (eprop ( (iff A) B) ) -> (_54 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) . -[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) . -case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_58 : (eprop ( (and ( (dotpipp A) (_56 : (eprop A) => B) ) ) ( (dotpipp B) (_57 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (eprop A) , var_13 : (eprop B) ] ( ( ( (case_8 A) B) H) ( ( ( (conj A) B) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_62 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_61 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_59 : (eprop B) => A) ) ) ( (dotpipp A) (_60 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) . -iff_sym : (A : Uprop -> (B : Uprop -> (_63 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) . -[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) . -case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_74 : (eprop ( (and ( (dotpipp A) (_71 : (eprop A) => False) ) ) ( (dotpipp False) (_72 : (eprop False) => A) ) ) ) -> (_73 : (eprop A) -> (eprop False) ) ) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (eprop A) , var_15 : (eprop False) ] ( ( (case_9 A) H) ( ( ( (conj A) False) var_14) var_15) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_76 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_75 : (eprop False) => A) ) ) => H0) ) var_14) var_15) . -neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) . -[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_64 : (eprop A) => False) ) ) (_65 : (eprop ( (dotpipp A) (_64 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_67 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_66 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_70 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_68 : (eprop A) => False) ) ) ( (dotpipp False) (_69 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) . -and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_104 : (_101 : (eprop B) -> (eprop A) ) -> (_103 : (_102 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_100 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_99 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_77 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_78 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_79 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_80 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_92 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_91 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_90 : (eprop A) => ( (dotpipp B) (_89 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_88 : (eprop A) => ( (dotpipp C) (_87 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_81 : (eprop B) => C) ) ) ( (dotpipp C) (_82 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp B) (_84 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H1 : (eprop A) => (H5 : (eprop C) => ( (H12 : (eprop ( (dotpipp C) (_83 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop A) => ( (H00 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H13 : (eprop A) => (H6 : (eprop B) => H5) ) ) H00) ) (H12 H5) ) ) (H0 H5) ) ) (H20 H4) ) ) ) ) H11) ) (H0 H30) ) ) (H10 H4) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp B) (_86 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (dotpipp C) (_85 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H12 : (eprop A) => (H5 : (eprop B) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H01 : (eprop A) => (H6 : (eprop C) => H5) ) ) H0) ) (H00 H5) ) ) (H H5) ) ) ) ) H21) ) (H11 H30) ) ) (H20 H4) ) ) (H10 H4) ) ) (H0 H30) ) ) ) ) (H20 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H20) H4) ) ) ) ) ) (H10 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_93 : (eprop B) => C) ) ) ( (dotpipp C) (_94 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_98 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_97 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_95 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_96 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H10 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H20) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H10) ) ) (H10 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj A) B) H30) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_132 : (_129 : (eprop B) -> (eprop A) ) -> (_131 : (_130 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_128 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_127 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_105 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_106 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_107 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_108 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_120 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_119 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_118 : (eprop B) => ( (dotpipp A) (_117 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_116 : (eprop C) => ( (dotpipp A) (_115 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_109 : (eprop B) => C) ) ) ( (dotpipp C) (_110 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_112 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H1 : (eprop C) => (H5 : (eprop A) => ( (H12 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_111 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H01 : (eprop B) => (H6 : (eprop A) => H1) ) ) H21) ) (H00 H4) ) ) (H20 H1) ) ) (H0 H1) ) ) ) ) H11) ) (H0 H4) ) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_114 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H01 : (eprop B) => (H5 : (eprop A) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_113 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H1 : (eprop C) => (H6 : (eprop A) => H01) ) ) H11) ) (H0 H4) ) ) (H10 H01) ) ) (H H01) ) ) ) ) H21) ) (H00 H4) ) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H20 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H20) H4) ) ) ) ) ) (H10 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_121 : (eprop B) => C) ) ) ( (dotpipp C) (_122 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_126 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_125 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_123 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_124 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H10 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H0) H20) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H10) ) ) (H10 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj B) A) H00) H30) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_156 : (_153 : (eprop B) -> (eprop (not A) ) ) -> (_155 : (_154 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_152 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_151 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_133 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_134 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_135 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_136 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_144 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_143 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_142 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_141 : (eprop B) => ( (or A) C) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_140 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_139 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_137 : (eprop B) => C) ) ) ( (dotpipp C) (_138 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => H40) ) H00) ) (H5 H40) ) ) (H0 H40) ) ) ) H0) ) (H4 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => H50) ) H0) ) (H4 H50) ) ) (H H50) ) ) ) H00) ) (H5 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H20 : (eprop A) => (H3 ( ( (or_introl A) C) H20) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H10 : (eprop A) => (H2 ( ( (or_introl A) B) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_150 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_149 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_147 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_148 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H10 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) H10) ) ) (H10 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_intror A) B) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_180 : (_177 : (eprop B) -> (eprop (not A) ) ) -> (_179 : (_178 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_176 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_175 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_157 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_158 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_159 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_160 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_168 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_167 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_166 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_165 : (eprop A) => ( (or C) A) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_164 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_163 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_161 : (eprop B) => C) ) ) ( (dotpipp C) (_162 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H11 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => H11) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H11) ) ) (H0 H11) ) ) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H21 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => H21) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H21) ) ) (H H21) ) ) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H20 : (eprop C) => (H3 ( ( (or_introl C) A) H20) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H10 : (eprop B) => (H2 ( ( (or_introl B) A) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_169 : (eprop B) => C) ) ) ( (dotpipp C) (_170 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_174 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_173 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_171 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_172 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H10 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H10) ) ) (H10 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_introl B) A) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_187 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) . -[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_181 : (eprop B) => C) ) ) ( (dotpipp C) (_182 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_186 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_185 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_183 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_184 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H2) H1) ) (H1 H1) ) ) (H0 H3) ) ) ) ) H0) ) ) (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj A) B) H2) H1) ) (H0 H1) ) ) (H1 H3) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_194 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) . -[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_188 : (eprop B) => C) ) ) ( (dotpipp C) (_189 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_193 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_192 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_190 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_191 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H1) H3) ) (H1 H1) ) ) (H0 H2) ) ) ) ) H0) ) ) (H0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj B) A) H1) H3) ) (H0 H1) ) ) (H1 H2) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_201 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) . -[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_195 : (eprop B) => C) ) ) ( (dotpipp C) (_196 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_200 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_199 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_197 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_198 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) H0) ) ) (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_intror A) B) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_208 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) . -[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_202 : (eprop B) => C) ) ) ( (dotpipp C) (_203 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_207 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_206 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_204 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_205 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H0) ) ) (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_introl B) A) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_213 : (eprop ( (and ( (dotpipp A) (_209 : (eprop A) => B) ) ) ( (dotpipp B) (_210 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_211 : (eprop A) => B) ) ) ( (dotpipp B) (_212 : (eprop B) => A) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (eprop A) , var_17 : (eprop B) ] ( ( ( (case_10 A) B) H) ( ( ( (conj A) B) var_16) var_17) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_217 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_216 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_214 : (eprop A) => B) ) ) ( (dotpipp B) (_215 : (eprop B) => A) ) ) H0) H0) ) ) var_16) var_17) . -iff_and : (A : Uprop -> (B : Uprop -> (_220 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_218 : (eprop A) => B) ) ) ( (dotpipp B) (_219 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) . -iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_243 : (eprop A) => B) ) ) ( (dotpipp B) (_244 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_223 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_221 : (eprop A) => B) ) ) ( (dotpipp B) (_222 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) (_226 : (eprop ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_227 : (eprop A) => B) ) ) ( (dotpipp B) (_228 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_229 : (eprop A) => B) ) ) ( (dotpipp B) (_230 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_234 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_233 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_231 : (eprop A) => B) ) ) ( (dotpipp B) (_232 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_241 : (eprop A) => B) ) ) ( (dotpipp B) (_242 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_235 : (eprop A) => B) ) ) ( (dotpipp B) (_236 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_240 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_239 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_237 : (eprop A) => B) ) ) ( (dotpipp B) (_238 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) ) ) . diff --git a/t/bug.dk b/t/bug.dk new file mode 100644 --- /dev/null +++ b/t/bug.dk @@ -0,0 +1,5 @@ +nat : Type. + +x : nat. + +y : x. diff --git a/t/bug.eu b/t/bug.eu deleted file mode 100644 --- a/t/bug.eu +++ /dev/null @@ -1,5 +0,0 @@ -nat : Type. - -x : nat. - -y : x. diff --git a/t/coc.dk b/t/coc.dk new file mode 100644 --- /dev/null +++ b/t/coc.dk @@ -0,0 +1,28 @@ +Utype : Type. + +Ukind : Type. + +etype : Utype -> Type. + +ekind : Ukind -> Type. + +dottype : Ukind. + +dotpi1 : x : Utype -> y : (etype x -> Utype) -> Utype. +dotpi2 : x : Utype -> y : (etype x -> Ukind) -> Ukind. +dotpi3 : x : Ukind -> y : (ekind x -> Utype) -> Utype. +dotpi4 : x : Ukind -> y : (ekind x -> Ukind) -> Ukind. + +[x:Utype, y : etype x -> Utype] + etype (dotpi1 x y) --> w : etype x -> etype (y w). +[x:Ukind, y : ekind x -> Utype] + etype (dotpi3 x y) --> w : ekind x -> etype (y w). + +[] ekind dottype --> Utype. +[x:Utype, y : etype x -> Ukind] + ekind (dotpi2 x y) --> w : etype x -> ekind (y w). +[x:Ukind, y : ekind x -> Ukind] + ekind (dotpi4 x y) --> w : ekind x -> ekind (y w). + +a : x : Utype -> y : etype x -> etype x. +[] a --> x : Utype => y : etype x => y. diff --git a/t/coc.eu b/t/coc.eu deleted file mode 100644 --- a/t/coc.eu +++ /dev/null @@ -1,28 +0,0 @@ -Utype : Type. - -Ukind : Type. - -etype : Utype -> Type. - -ekind : Ukind -> Type. - -dottype : Ukind. - -dotpi1 : x : Utype -> y : (etype x -> Utype) -> Utype. -dotpi2 : x : Utype -> y : (etype x -> Ukind) -> Ukind. -dotpi3 : x : Ukind -> y : (ekind x -> Utype) -> Utype. -dotpi4 : x : Ukind -> y : (ekind x -> Ukind) -> Ukind. - -[x:Utype, y : etype x -> Utype] - etype (dotpi1 x y) --> w : etype x -> etype (y w). -[x:Ukind, y : ekind x -> Utype] - etype (dotpi3 x y) --> w : ekind x -> etype (y w). - -[] ekind dottype --> Utype. -[x:Utype, y : etype x -> Ukind] - ekind (dotpi2 x y) --> w : etype x -> ekind (y w). -[x:Ukind, y : ekind x -> Ukind] - ekind (dotpi4 x y) --> w : ekind x -> ekind (y w). - -a : x : Utype -> y : etype x -> etype x. -[] a --> x : Utype => y : etype x => y. diff --git a/t/conj.eu b/t/conj.eu deleted file mode 100644 --- a/t/conj.eu +++ /dev/null @@ -1,3 +0,0 @@ -o : Type. -conj : o -> o -> Type. -[x : o] conj x x --> conj x x. diff --git a/t/coq/Datatypes.dk b/t/coq/Datatypes.dk new file mode 100644 --- /dev/null +++ b/t/coq/Datatypes.dk @@ -0,0 +1,180 @@ +Set : Type. +eps : Set -> Type. + +Prop : Type. +eps' : Prop -> Type. + +;; logic + +True : Prop. +I : eps' True. +True_rec : P : Set -> eps P -> eps' True -> eps P. +True_ind : P : Prop -> eps' P -> eps' True -> eps' P. + +[P:Set,x:eps P] True_rec P x I --> x. +[P:Prop,x:eps' P] True_ind P x I --> x. + +False : Prop. +False_rec : P : Set -> eps' False -> eps P. +False_ind : P : Prop -> eps' False -> eps' P. + +not : Prop -> Prop. +[A:Prop] not A --> implies A False. + +and : Prop -> Prop -> Prop. +conj : A:Prop -> B:Prop -> x:eps' A -> y:eps' B -> eps' (and A B). +conj_rec : A:Prop -> B:Prop -> P : Set -> (eps' A -> eps' B -> eps P) -> eps' (and A B) -> eps P. +conj_ind : A:Prop -> B:Prop -> P : Prop -> (eps' A -> eps' B -> eps' P) -> eps' (and A B) -> eps' P. + +[A:Prop, B:Prop, P : Set, f: eps' A -> eps' B -> eps P, x:eps' A, y:eps' B] + conj_rec A B P f (conj A B x y) --> f x y. + +[A:Prop, B:Prop, P : Prop, f: eps' A -> eps' B -> eps' P, x:eps' A, y:eps' B] + conj_ind A B P f (conj A B x y) --> f x y. + +proj1 : A:Prop -> B:Prop -> eps' (and A B) -> eps' A. +[A:Prop, B:Prop, x:eps' A, y:eps' B] proj1 A B (conj A B x y) --> x. + +proj2 : A:Prop -> B:Prop -> eps' (and A B) -> eps' B. +[A:Prop, B:Prop, x:eps' A, y:eps' B] proj2 A B (conj A B x y) --> y. + +or : Prop -> Prop -> Prop. +or_introl : A:Prop -> B:Prop -> eps' A -> eps' (or A B). +or_intror : A:Prop -> B:Prop -> eps' B -> eps' (or A B). + +or_ind : A:Prop -> B:Prop -> P:Prop -> (eps' A -> eps' P) -> (eps' B -> eps' P) -> eps' (or A B) -> eps' P. +[A:Prop, B:Prop, P:Prop, f:(eps' A -> eps' P), g:(eps' B -> eps' P), x: eps' A] or_ind A B P f g (or_introl A B x) --> f x. +[A:Prop, B:Prop, P:Prop, f:(eps' A -> eps' P), g:(eps' B -> eps' P), x: eps' B] or_ind A B P f g (or_intror A B x) --> g x. + +;; --- not in Coq --- + +pi_spp : A:Set -> (eps A -> Prop) -> Prop. +pi_ppp : A:Prop -> (eps' A -> Prop) -> Prop. +implies : A:Prop -> B:Prop -> Prop. + +[A:Set, f:eps A -> Prop] eps' (pi_spp A f) --> x:eps A -> eps' (f x). +[A:Prop, f:eps' A -> Prop] eps' (pi_ppp A f) --> x:eps' A -> eps' (f x). +[A:Prop, B:Prop] implies A B --> pi_ppp A (_:eps' A => B). + +;; ------------------ + +iff : Prop -> Prop -> Prop. +[A:Prop,B:Prop] iff A B --> and (implies A B) (implies B A). + +iff_refl : A:Prop -> eps' (iff A A). +[A:Prop] iff_refl A --> conj (implies A A) (implies A A) (H:eps' A => H) (H:eps' A => H). + +iff_trans : A:Prop -> B:Prop -> C:Prop -> eps' (iff A B) -> eps' (iff B C) -> eps' (iff A C). +[A: Prop, + B: Prop, + C: Prop, + H1: eps' (implies A B), + H2: eps' (implies B A), + H3: eps' (implies B C), + H4: eps' (implies C B) +] iff_trans A B C (conj (implies A B) (implies B A) H1 H2) (conj (implies B C) (implies C B) H3 H4) + --> conj (implies A C) (implies C A) (H5:eps' A => H3 (H1 H5)) (H5:eps' C => (H2 (H4 H5))). + +iff_sym : A:Prop -> B:Prop -> eps' (iff A B) -> eps' (iff B A). +[A:Prop, B:Prop, H1:eps' (implies A B), H2:eps' (implies B A)] + iff_sym A B (conj (implies A B) (implies B A) H1 H2) --> conj (implies B A) (implies A B) H2 H1. + +neg_false : A : Prop -> eps' (iff (not A) (iff A False)). +[A:Prop] neg_false A --> + conj (implies (not A) (iff A False)) + (implies (iff A False) (not A)) + (H : eps' (not A) => conj (implies A False) (implies False A) H (H1 : eps' False => False_ind A H1)) + (H : eps' (iff A False) => match1 A H). + +match1 : A:Prop -> H:eps' (iff A False) -> eps' (implies A False). +[A:Prop, H0:eps' (implies A False), _:eps' (implies False A)] + match1 A (conj (implies A False) (implies False A) H0 _) --> H0. + +and_cancel_l : A:Prop -> B:Prop -> C:Prop -> eps' (implies (implies B A) (implies (implies C A) (iff (iff (and A B) (and A C)) (iff B C)))). +; TODO PROOF +and_cancel_r : A:Prop -> B:Prop -> C:Prop -> eps' (implies (implies B A) (implies (implies C A) (iff (iff (and B A) (and C A)) (iff B C)))). +; TODO PROOF +or_cancel_l : A:Prop -> B:Prop -> C:Prop -> eps' (implies (implies B (not A)) (implies (implies C (not A)) (iff (iff (or A B) (or A C)) (iff B C)))). +; TODO PROOF +or_cancel_r : A:Prop -> B:Prop -> C:Prop -> eps' (implies (implies B (not A)) (implies (implies C (not A)) (iff (iff (or B A) (or C A)) (iff B C)))). +; TODO PROOF + +and_iff_compat_l : A:Prop -> B:Prop -> C:Prop -> eps' (implies (iff B C) (iff (and A B) (and A C))). +; TODO PROOF +and_iff_compat_r : A:Prop -> B:Prop -> C:Prop -> eps' (implies (iff B C) (iff (and B A) (and C A))). +; TODO PROOF +or_iff_compat_l : A:Prop -> B:Prop -> C:Prop -> eps' (implies (iff B C) (iff (or A B) (or A C))). +; TODO PROOF +or_iff_compat_r : A:Prop -> B:Prop -> C:Prop -> eps' (implies (iff B C) (iff (or B A) (or C A))). +; TODO PROOF + +iff_and : A:Prop -> B:Prop -> eps' (implies (iff A B) (and (implies A B) (implies B A))). +; TODO PROOF +iff_to_and : A:Prop -> B:Prop -> eps' (iff (iff A B) (and (implies A B) (implies B A))). +; TODO PROOF + +ex : A:Set -> (eps A -> Prop) -> Prop. +ex_intro : A:Set -> P:(eps A -> Prop) -> x:(eps A) -> eps' (P x) -> eps' (ex A P). +ex_ind : A:Set -> P:(eps A -> Prop) -> P0:Prop -> (x:eps A -> eps' (P x) -> eps' P0) -> eps' (ex A P) -> eps' P0. + +[A:Set, P:eps A -> Prop, P0:Prop, f:(x:(eps A) -> eps' (P x) -> eps' P0), x:(eps A), x0:eps' (P x)] + ex_ind A P P0 f (ex_intro A P x x0) --> f x x0. + +ex2 : A:Set -> P:(eps A -> Prop) -> Q:(eps A -> Prop) -> Prop. +ex2_intro : A:Set -> P:(eps A -> Prop) -> Q:(eps A -> Prop) -> x:(eps A) -> eps' (P x) -> eps' (Q x) -> eps' (ex2 A P Q). +ex2_ind : A:Set -> P:(eps A -> Prop) -> Q:(eps A -> Prop) -> P0:Prop -> (x:eps A -> eps' (P x) -> eps' (Q x) -> eps' P0) -> eps' (ex2 A P Q) -> eps' P0. +[A:Set, P:(eps A -> Prop), Q:(eps A -> Prop), P0:Prop, f:(x:eps A -> eps' (P x) -> eps' (Q x) -> eps' P0),x:(eps A), x0:eps' (P x), x1:eps' (Q x)] + ex2_ind A P Q P0 f (ex2_intro A P Q x x0 x1) --> f x x0 x1. + +; all == pi_Xpp + +;;inst_spp : A:Set -> P:(eps A -> Prop) -> x:(eps A) -> eps' (pi_spp (x0:(eps A) => P x0)) -> eps' (P x). + +;; datatypes + +unit : Set. +tt : eps unit. +unit_rec : P : (eps unit -> Set) -> eps (P tt) -> u : eps unit -> eps (P u). +unit_ind : P : (eps unit -> Prop) -> eps' (P tt) -> u : eps unit -> eps' (P u). + +[f:eps unit -> Set, a: eps (f tt)] unit_rec f a tt --> a. +[P:eps unit -> Prop, a: eps' (P tt)] unit_ind P a tt --> a. + +bool : Set. +true : eps bool. +false : eps bool. + +bool_rec : P : (eps bool -> Set) -> eps (P true) -> eps (P false) -> b : eps bool -> eps (P b). +bool_ind : P : (eps bool -> Prop) -> eps' (P true) -> eps' (P false) -> b : eps bool -> eps' (P b). + +[P:eps bool -> Set, a:eps (P true), b:eps (P false)] bool_rec P a b true --> a. +[P:eps bool -> Set, a:eps (P true), b:eps (P false)] bool_rec P a b false --> b. +[P:eps bool -> Prop, a:eps' (P true), b:eps' (P false)] bool_ind P a b true --> a. +[P:eps bool -> Prop, a:eps' (P true), b:eps' (P false)] bool_ind P a b false --> b. + +andb : eps bool -> eps bool -> eps bool. +[] andb true true --> true. +[_:eps bool] andb _ false --> false. +[_:eps bool] andb false _ --> false. + +orb : eps bool -> eps bool -> eps bool. +[_:eps bool] orb true _ --> true. +[_:eps bool] orb _ true --> true. +[] orb false false --> false. + +implb : eps bool -> eps bool -> eps bool. +[] implb true false --> false. +[] implb true true --> true. +[_:eps bool] implb false _ --> true. + +xorb : eps bool -> eps bool -> eps bool. +[] xorb true true --> false. +[] xorb false false --> false. +[] xorb true false --> true. +[] xorb false true --> true. + +negb : eps bool -> eps bool. +[] negb true --> false. +[] negb false --> true. + + diff --git a/t/coqlogicprel.eu b/t/coqlogicprel.eu deleted file mode 100644 --- a/t/coqlogicprel.eu +++ /dev/null @@ -1,156 +0,0 @@ -Uset : Type. -Uprop : Type. -Utype : Type. - -eprop : x : Uprop -> Type. -eset : x : Uset -> Type. -etype : x : Utype -> Type. - -dotset : Utype. -dotprop : Utype. - -; /!\ type : type /!\, should use universes -dottype : Utype. - -; /!\ subtyping in coq, should be unidirectional /!\ -[] Uprop --> Utype. -[] Uset --> Utype. - -dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop. -dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset. -dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype. -dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop. -dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop. -dotpist : x : Uset -> y : (eset x -> Utype) -> Utype. -dotpits : x : Utype -> y : (etype x -> Uset) -> Uset. -dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset. -dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype. - - -[x:Uprop, y : eprop x -> Uprop] - eprop (dotpipp x y) --> w : eprop x -> eprop (y w). - -[x:Uset, y : eset x -> Uprop] - eprop (dotpisp x y) --> w : eset x -> eprop (y w). - -[x:Utype, y : etype x -> Uprop] - eprop (dotpitp x y) --> w : etype x -> eprop (y w). - -; /!\ -[P : Uprop] eprop P --> etype P. - -[x:Uprop, y : eprop x -> Uset] - eset (dotpips x y) --> w : eprop x -> eset (y w). - -[x:Utype, y : etype x -> Uset] - eset (dotpits x y) --> w : etype x -> eset (y w). - -[x:Uset, y : eset x -> Uset] - eset (dotpiss x y) --> w : eset x -> eset (y w). - -; /!\ -[P : Uset] eset P --> etype P. - -[x:Uset, y : eset x -> Utype] - etype (dotpist x y) --> w : eset x -> etype (y w). - -[x:Utype, y : etype x -> Utype] - etype (dotpitt x y) --> w : etype x -> etype (y w). - -[x:Uprop, y : eprop x -> Utype] - etype (dotpipt x y) --> w : eprop x -> etype (y w). - - -[] (etype dotset) --> Uset. -[] (etype dotprop) --> Uprop. -; /!\ -[] (etype dottype) --> Utype. - -; end of Coq1univ - -True : Uprop. -I : (eprop True) . -case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) . -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) I) --> f. -True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) . -[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) . -True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) . -[] True_ind --> (P : (etype dotprop) => (True_rect P) ) . -True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) . -[] True_rec --> (P : (etype dotset) => (True_rect P) ) . -False : Uprop. -case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) . -False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) . -[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) . -False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) . -[] False_ind --> (P : (etype dotprop) => (False_rect P) ) . -False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) . -[] False_rec --> (P : (etype dotset) => (False_rect P) ) . -not : (A : Uprop -> Uprop) . -[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) . -and : (A : Uprop -> (B : Uprop -> Uprop) ) . -conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) . -case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( (conj A) B) var_0) var_1) ) --> ( (f var_0) var_1) . -and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_13 : (eprop A) -> (_12 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) . -[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_11 : (eprop A) => ( (dotpipt B) (_10 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) . -and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_15 : (eprop A) -> (_14 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) . -[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) . -and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_17 : (eprop A) -> (_16 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) . -[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) . -case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_18 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( (conj A) B) var_2) var_3) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_2) var_3) . -proj1 : (A : Uprop -> (B : Uprop -> (_19 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) . -[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) . -case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_20 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( (conj A) B) var_4) var_5) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) . -proj2 : (A : Uprop -> (B : Uprop -> (_21 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) . -[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) . -or : (A : Uprop -> (B : Uprop -> Uprop) ) . -or_introl : (A : Uprop -> (B : Uprop -> (_22 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) . -or_intror : (A : Uprop -> (B : Uprop -> (_23 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) . -case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_24 : (eprop A) -> (eprop P) ) -> (f0 : (_25 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_26 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_introl A) B) var_6) ) --> (f var_6) . -[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_intror A) B) var_7) ) --> (f0 var_7) . -or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_31 : (eprop A) -> (eprop P) ) -> (f0 : (_32 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) . -[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_30 : (eprop A) => P) ) ) => (f0 : (eprop ( (dotpipp B) (_29 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) o) ) ) ) ) ) ) . -iff : (A : Uprop -> (B : Uprop -> Uprop) ) . -[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_33 : (eprop A) => B) ) ) ( (dotpipp B) (_34 : (eprop B) => A) ) ) ) ) . -iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) . -[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_35 : (eprop A) => A) ) ) ( (dotpipp A) (_36 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) . -case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_40 : (eprop ( (and ( (dotpipp A) (_37 : (eprop A) => B) ) ) ( (dotpipp B) (_38 : (eprop B) => A) ) ) ) -> (_39 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) . -case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_41 : (eprop A) -> (eprop B) ) -> (H2 : (_42 : (eprop B) -> (eprop A) ) -> (H0 : (eprop ( (iff B) C) ) -> (_45 : (eprop ( (and ( (dotpipp B) (_43 : (eprop B) => C) ) ) ( (dotpipp C) (_44 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_46 : (eprop A) -> (eprop B) ) , H2 : (_47 : (eprop B) -> (eprop A) ) , H0 : (eprop ( (iff B) C) ) , var_10 : (eprop B) , var_11 : (eprop C) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) ( ( ( (conj B) C) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_51 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_50 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_48 : (eprop A) => C) ) ) ( (dotpipp C) (_49 : (eprop C) => A) ) ) (H1 : (eprop A) => (H3 (H1 H1) ) ) ) (H1 : (eprop C) => (H2 (H1 (H2 (H4 H1) ) ) ) ) ) ) ) var_10) var_11) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (eprop A) , var_9 : (eprop B) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( (conj A) B) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_53 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_52 : (eprop B) => A) ) ) => (H0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) H0) ) ) ) var_8) var_9) . -iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_55 : (eprop ( (iff A) B) ) -> (_54 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) . -[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) . -case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_58 : (eprop ( (and ( (dotpipp A) (_56 : (eprop A) => B) ) ) ( (dotpipp B) (_57 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (eprop A) , var_13 : (eprop B) ] ( ( ( (case_8 A) B) H) ( ( ( (conj A) B) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_62 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_61 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_59 : (eprop B) => A) ) ) ( (dotpipp A) (_60 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) . -iff_sym : (A : Uprop -> (B : Uprop -> (_63 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) . -[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) . -case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_74 : (eprop ( (and ( (dotpipp A) (_71 : (eprop A) => False) ) ) ( (dotpipp False) (_72 : (eprop False) => A) ) ) ) -> (_73 : (eprop A) -> (eprop False) ) ) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (eprop A) , var_15 : (eprop False) ] ( ( (case_9 A) H) ( ( ( (conj A) False) var_14) var_15) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_76 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_75 : (eprop False) => A) ) ) => H0) ) var_14) var_15) . -neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) . -[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_64 : (eprop A) => False) ) ) (_65 : (eprop ( (dotpipp A) (_64 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_67 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_66 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_70 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_68 : (eprop A) => False) ) ) ( (dotpipp False) (_69 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) . -and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_104 : (_101 : (eprop B) -> (eprop A) ) -> (_103 : (_102 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_100 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_99 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_77 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_78 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_79 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_80 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_92 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_91 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_90 : (eprop A) => ( (dotpipp B) (_89 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_88 : (eprop A) => ( (dotpipp C) (_87 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_81 : (eprop B) => C) ) ) ( (dotpipp C) (_82 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp B) (_84 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H1 : (eprop A) => (H5 : (eprop C) => ( (H12 : (eprop ( (dotpipp C) (_83 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop A) => ( (H00 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H13 : (eprop A) => (H6 : (eprop B) => H5) ) ) H00) ) (H12 H5) ) ) (H0 H5) ) ) (H20 H4) ) ) ) ) H11) ) (H0 H30) ) ) (H10 H4) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp B) (_86 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (dotpipp C) (_85 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H12 : (eprop A) => (H5 : (eprop B) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H01 : (eprop A) => (H6 : (eprop C) => H5) ) ) H0) ) (H00 H5) ) ) (H H5) ) ) ) ) H21) ) (H11 H30) ) ) (H20 H4) ) ) (H10 H4) ) ) (H0 H30) ) ) ) ) (H20 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H20) H4) ) ) ) ) ) (H10 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_93 : (eprop B) => C) ) ) ( (dotpipp C) (_94 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_98 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_97 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_95 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_96 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H10 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H20) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H10) ) ) (H10 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj A) B) H30) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_132 : (_129 : (eprop B) -> (eprop A) ) -> (_131 : (_130 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_128 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_127 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_105 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_106 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_107 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_108 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_120 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_119 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_118 : (eprop B) => ( (dotpipp A) (_117 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_116 : (eprop C) => ( (dotpipp A) (_115 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_109 : (eprop B) => C) ) ) ( (dotpipp C) (_110 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_112 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H1 : (eprop C) => (H5 : (eprop A) => ( (H12 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_111 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H01 : (eprop B) => (H6 : (eprop A) => H1) ) ) H21) ) (H00 H4) ) ) (H20 H1) ) ) (H0 H1) ) ) ) ) H11) ) (H0 H4) ) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_114 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H01 : (eprop B) => (H5 : (eprop A) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_113 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H1 : (eprop C) => (H6 : (eprop A) => H01) ) ) H11) ) (H0 H4) ) ) (H10 H01) ) ) (H H01) ) ) ) ) H21) ) (H00 H4) ) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H20 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H20) H4) ) ) ) ) ) (H10 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_121 : (eprop B) => C) ) ) ( (dotpipp C) (_122 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_126 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_125 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_123 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_124 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H10 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H0) H20) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H10) ) ) (H10 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj B) A) H00) H30) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_156 : (_153 : (eprop B) -> (eprop (not A) ) ) -> (_155 : (_154 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_152 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_151 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_133 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_134 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_135 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_136 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_144 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_143 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_142 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_141 : (eprop B) => ( (or A) C) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_140 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_139 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_137 : (eprop B) => C) ) ) ( (dotpipp C) (_138 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => H40) ) H00) ) (H5 H40) ) ) (H0 H40) ) ) ) H0) ) (H4 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => H50) ) H0) ) (H4 H50) ) ) (H H50) ) ) ) H00) ) (H5 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H20 : (eprop A) => (H3 ( ( (or_introl A) C) H20) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H10 : (eprop A) => (H2 ( ( (or_introl A) B) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_150 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_149 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_147 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_148 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H10 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) H10) ) ) (H10 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_intror A) B) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_180 : (_177 : (eprop B) -> (eprop (not A) ) ) -> (_179 : (_178 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_176 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_175 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_157 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_158 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_159 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_160 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_168 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_167 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_166 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_165 : (eprop A) => ( (or C) A) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_164 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_163 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_161 : (eprop B) => C) ) ) ( (dotpipp C) (_162 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H11 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => H11) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H11) ) ) (H0 H11) ) ) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H21 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => H21) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H21) ) ) (H H21) ) ) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H20 : (eprop C) => (H3 ( ( (or_introl C) A) H20) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H10 : (eprop B) => (H2 ( ( (or_introl B) A) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_169 : (eprop B) => C) ) ) ( (dotpipp C) (_170 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_174 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_173 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_171 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_172 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H10 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H10) ) ) (H10 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_introl B) A) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_187 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) . -[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_181 : (eprop B) => C) ) ) ( (dotpipp C) (_182 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_186 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_185 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_183 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_184 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H2) H1) ) (H1 H1) ) ) (H0 H3) ) ) ) ) H0) ) ) (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj A) B) H2) H1) ) (H0 H1) ) ) (H1 H3) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_194 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) . -[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_188 : (eprop B) => C) ) ) ( (dotpipp C) (_189 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_193 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_192 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_190 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_191 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H1) H3) ) (H1 H1) ) ) (H0 H2) ) ) ) ) H0) ) ) (H0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj B) A) H1) H3) ) (H0 H1) ) ) (H1 H2) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_201 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) . -[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_195 : (eprop B) => C) ) ) ( (dotpipp C) (_196 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_200 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_199 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_197 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_198 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) H0) ) ) (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_intror A) B) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_208 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) . -[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_202 : (eprop B) => C) ) ) ( (dotpipp C) (_203 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_207 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_206 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_204 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_205 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H0) ) ) (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_introl B) A) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H0) ) ) ) ) ) H) ) ) ) ) . -case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_213 : (eprop ( (and ( (dotpipp A) (_209 : (eprop A) => B) ) ) ( (dotpipp B) (_210 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_211 : (eprop A) => B) ) ) ( (dotpipp B) (_212 : (eprop B) => A) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (eprop A) , var_17 : (eprop B) ] ( ( ( (case_10 A) B) H) ( ( ( (conj A) B) var_16) var_17) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_217 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_216 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_214 : (eprop A) => B) ) ) ( (dotpipp B) (_215 : (eprop B) => A) ) ) H0) H0) ) ) var_16) var_17) . -iff_and : (A : Uprop -> (B : Uprop -> (_220 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_218 : (eprop A) => B) ) ) ( (dotpipp B) (_219 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) . -iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_243 : (eprop A) => B) ) ) ( (dotpipp B) (_244 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_223 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_221 : (eprop A) => B) ) ) ( (dotpipp B) (_222 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) (_226 : (eprop ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_227 : (eprop A) => B) ) ) ( (dotpipp B) (_228 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_229 : (eprop A) => B) ) ) ( (dotpipp B) (_230 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_234 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_233 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_231 : (eprop A) => B) ) ) ( (dotpipp B) (_232 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_241 : (eprop A) => B) ) ) ( (dotpipp B) (_242 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_235 : (eprop A) => B) ) ) ( (dotpipp B) (_236 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_240 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_239 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_237 : (eprop A) => B) ) ) ( (dotpipp B) (_238 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) ) ) . diff --git a/t/delta1.dk b/t/delta1.dk new file mode 100644 --- /dev/null +++ b/t/delta1.dk @@ -0,0 +1,2 @@ +delta : a : Type -> (b : Type -> b -> b) -> a -> a. +[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a). diff --git a/t/delta1.eu b/t/delta1.eu deleted file mode 100644 --- a/t/delta1.eu +++ /dev/null @@ -1,2 +0,0 @@ -delta : a : Type -> (b : Type -> b -> b) -> a -> a. -[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a). diff --git a/t/delta2.dk b/t/delta2.dk new file mode 100644 --- /dev/null +++ b/t/delta2.dk @@ -0,0 +1,7 @@ +;; Same as delta1.eu but with d2 declared of type 'delta delta', which of +;; course is ill-typed. + +delta : a : Type -> (b : Type -> b -> b) -> a -> a. +[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a). + +d2 : delta delta. diff --git a/t/delta2.eu b/t/delta2.eu deleted file mode 100644 --- a/t/delta2.eu +++ /dev/null @@ -1,7 +0,0 @@ -;; Same as delta1.eu but with d2 declared of type 'delta delta', which of -;; course is ill-typed. - -delta : a : Type -> (b : Type -> b -> b) -> a -> a. -[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a). - -d2 : delta delta. diff --git a/t/exemple.dk b/t/exemple.dk new file mode 100644 --- /dev/null +++ b/t/exemple.dk @@ -0,0 +1,9 @@ +T : Type. + +U : Type. + +[] U --> T -> T. + +app : f : (T -> T) -> T -> T. + +[] app --> f : U => x : T => f x. diff --git a/t/exemple.eu b/t/exemple.eu deleted file mode 100644 --- a/t/exemple.eu +++ /dev/null @@ -1,9 +0,0 @@ -T : Type. - -U : Type. - -[] U --> T -> T. - -app : f : (T -> T) -> T -> T. - -[] app --> f : U => x : T => f x. diff --git a/t/f.dk b/t/f.dk new file mode 100644 --- /dev/null +++ b/t/f.dk @@ -0,0 +1,17 @@ +Utype : Type. +Ukind : Type. +etype : Utype -> Type. +ekind : Ukind -> Type. +dottype : Ukind. +dotpi1 : x : Utype -> (etype x -> Utype) -> Utype. +dotpi3 : x : Ukind -> (ekind x -> Utype) -> Utype. +[] ekind dottype --> Utype. +[x:Utype, y : etype x -> Utype] + etype ((dotpi1 x) y) --> w : etype x -> etype (y w). + +[x:Ukind, y : ekind x -> Utype] + etype ((dotpi3 x) y) --> w : ekind x -> etype (y w). + +a : x : Utype -> etype x -> etype x. + +[] a --> x : Utype => y : etype x => y. diff --git a/t/f.eu b/t/f.eu deleted file mode 100644 --- a/t/f.eu +++ /dev/null @@ -1,17 +0,0 @@ -Utype : Type. -Ukind : Type. -etype : Utype -> Type. -ekind : Ukind -> Type. -dottype : Ukind. -dotpi1 : x : Utype -> (etype x -> Utype) -> Utype. -dotpi3 : x : Ukind -> (ekind x -> Utype) -> Utype. -[] ekind dottype --> Utype. -[x:Utype, y : etype x -> Utype] - etype ((dotpi1 x) y) --> w : etype x -> etype (y w). - -[x:Ukind, y : ekind x -> Utype] - etype ((dotpi3 x) y) --> w : ekind x -> etype (y w). - -a : x : Utype -> etype x -> etype x. - -[] a --> x : Utype => y : etype x => y. diff --git a/t/fold/arith.dk b/t/fold/arith.dk new file mode 100644 --- /dev/null +++ b/t/fold/arith.dk @@ -0,0 +1,49 @@ +prop : Type. +eps : prop -> Type. + +implies : prop -> prop -> prop. + +nat : Type. +nat_ : prop. + +bool : Type. +bool_ : prop. + +true : bool. +false : bool. + +isTrue : bool -> Type. +trueisTrue : isTrue true. + + +[] eps nat_ --> nat. +[] eps bool_ --> bool. +[a:prop,b:prop] eps (implies a b) --> eps a -> eps b. + +unfold : nat -> p:prop -> eps p -> (nat -> eps p -> eps p) -> eps p. +fold : (p:prop -> eps p -> (nat -> eps p -> eps p) -> eps p) -> nat. + +[pi:p:prop -> eps p -> (nat -> eps p -> eps p) -> eps p] unfold (fold pi) --> pi. + +0 : nat. +S : nat -> nat. + +[] 0 --> fold (p:prop => u:eps p => v:(nat -> eps p -> eps p) => u). +[n:nat] S n --> fold (p:prop => u:eps p => v:(nat -> eps p -> eps p) => v n (unfold n p u v)). + +pred : nat -> nat. +[n:nat] pred n --> unfold n nat_ 0 (m:nat => _:nat => m). + +iszero : nat -> bool. +[n:nat] iszero n --> unfold n bool_ true (_:nat => _:bool => false). + +eq : nat -> nat -> bool. +[n:nat] eq n --> unfold n (implies nat_ bool_) iszero (_:nat => f:(nat -> bool) => m:nat => unfold m bool_ false (p:nat => _:bool => f p)). + + +test1 : nat. +[] test1 --> S (S (S (S (S 0)))). + +test2 : isTrue (eq test1 test1). +[] test2 --> trueisTrue. + diff --git a/t/gros.eu b/t/gros.eu deleted file mode 100644 --- a/t/gros.eu +++ /dev/null @@ -1,360 +0,0 @@ -Uset : Type. -Uprop : Type. -Utype : Type. - -eprop : x : Uprop -> Type. -eset : x : Uset -> Type. -etype : x : Utype -> Type. - -dotset : Utype. -dotprop : Utype. - -; /!\ type : type /!\, should use universes -dottype : Utype. - -; /!\ subtyping in coq, should be unidirectional /!\ -[] Uprop --> Utype. -[] Uset --> Utype. - -dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop. -dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset. -dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype. -dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop. -dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop. -dotpist : x : Uset -> y : (eset x -> Utype) -> Utype. -dotpits : x : Utype -> y : (etype x -> Uset) -> Uset. -dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset. -dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype. - - -[x:Uprop, y : eprop x -> Uprop] - eprop (dotpipp x y) --> w : eprop x -> eprop (y w). - -[x:Uset, y : eset x -> Uprop] - eprop (dotpisp x y) --> w : eset x -> eprop (y w). - -[x:Utype, y : etype x -> Uprop] - eprop (dotpitp x y) --> w : etype x -> eprop (y w). - -; /!\ -[P : Uprop] eprop P --> etype P. - -[x:Uprop, y : eprop x -> Uset] - eset (dotpips x y) --> w : eprop x -> eset (y w). - -[x:Utype, y : etype x -> Uset] - eset (dotpits x y) --> w : etype x -> eset (y w). - -[x:Uset, y : eset x -> Uset] - eset (dotpiss x y) --> w : eset x -> eset (y w). - -; /!\ -[P : Uset] eset P --> etype P. - -[x:Uset, y : eset x -> Utype] - etype (dotpist x y) --> w : eset x -> etype (y w). - -[x:Utype, y : etype x -> Utype] - etype (dotpitt x y) --> w : etype x -> etype (y w). - -[x:Uprop, y : eprop x -> Utype] - etype (dotpipt x y) --> w : eprop x -> etype (y w). - - -[] (etype dotset) --> Uset. -[] (etype dotprop) --> Uprop. -; /!\ -[] (etype dottype) --> Utype. - -; end of Coq1univ - -True : Uprop. -I : (eprop True) . -case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) . -I_case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (eprop True) ) ) ) . -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( (I_case_0 P) f) t) --> I. -[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) ( ( (I_case_0 P) f) t) ) --> f. -True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) . -[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) . -True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) . -[] True_ind --> (P : (etype dotprop) => (True_rect P) ) . -True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) . -[] True_rec --> (P : (etype dotset) => (True_rect P) ) . -False : Uprop. -case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) . -False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) . -[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) . -False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) . -[] False_ind --> (P : (etype dotprop) => (False_rect P) ) . -False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) . -[] False_rec --> (P : (etype dotset) => (False_rect P) ) . -not : (A : Uprop -> Uprop) . -[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) . -and : (A : Uprop -> (B : Uprop -> Uprop) ) . -conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) . -case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) . -conj_case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_11 : (eprop A) -> (_10 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_13 : (eprop A) -> (_12 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Utype, f : (_15 : (eprop A) -> (_14 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) ] ( ( ( ( (conj_case_2 A) B) P) f) a) --> ( (conj A) B) . -[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( ( ( ( (conj_case_2 A) B) P) f) a) var_0) var_1) ) --> ( (f var_0) var_1) . -and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_19 : (eprop A) -> (_18 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) . -[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_17 : (eprop A) => ( (dotpipt B) (_16 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) . -and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_21 : (eprop A) -> (_20 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) . -[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) . -and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_23 : (eprop A) -> (_22 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) . -[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) . -case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_24 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) . -conj_case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_26 : (eprop A) -> (_25 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_3 A) B) H) --> ( (conj A) B) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( ( (conj_case_3 A) B) H) var_2) var_3) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H欧0) ) var_2) var_3) . -proj1 : (A : Uprop -> (B : Uprop -> (_27 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) . -[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) . -case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_28 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) . -conj_case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_30 : (eprop A) -> (_29 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_4 A) B) H) --> ( (conj A) B) . -[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( ( (conj_case_4 A) B) H) var_4) var_5) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) . -proj2 : (A : Uprop -> (B : Uprop -> (_31 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) . -[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) . -or : (A : Uprop -> (B : Uprop -> Uprop) ) . -or_introl : (A : Uprop -> (B : Uprop -> (_32 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) . -or_intror : (A : Uprop -> (B : Uprop -> (_33 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) . -case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_34 : (eprop A) -> (eprop P) ) -> (f欧0 : (_35 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_36 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) . -or_introl_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_39 : (eprop A) -> (eprop P) ) -> (f欧0 : (_40 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_41 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_42 : (eprop A) -> (eprop P) ) , f欧0 : (_43 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) --> ( (or_introl A) B) . -or_intror_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_44 : (eprop A) -> (eprop P) ) -> (f欧0 : (_45 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_46 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, P : Uprop, f : (_47 : (eprop A) -> (eprop P) ) , f欧0 : (_48 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) --> ( (or_intror A) B) . -[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) var_6) ) --> (f var_6) . -[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) var_7) ) --> (f欧0 var_7) . -or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_51 : (eprop A) -> (eprop P) ) -> (f欧0 : (_52 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) . -[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_50 : (eprop A) => P) ) ) => (f欧0 : (eprop ( (dotpipp B) (_49 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) o) ) ) ) ) ) ) . -iff : (A : Uprop -> (B : Uprop -> Uprop) ) . -[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_53 : (eprop A) => B) ) ) ( (dotpipp B) (_54 : (eprop B) => A) ) ) ) ) . -iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) . -[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_55 : (eprop A) => A) ) ) ( (dotpipp A) (_56 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) . -case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_60 : (eprop ( (and ( (dotpipp A) (_57 : (eprop A) => B) ) ) ( (dotpipp B) (_58 : (eprop B) => A) ) ) ) -> (_59 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) . -conj_case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_68 : (_63 : (eprop A) -> (eprop B) ) -> (_67 : (_64 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_65 : (eprop A) => B) ) ) ( (dotpipp B) (_66 : (eprop B) => A) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) ] ( ( ( (conj_case_6 A) B) C) H) --> ( (conj ( (dotpipp A) (_61 : (eprop A) => B) ) ) ( (dotpipp B) (_62 : (eprop B) => A) ) ) . -case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_71 : (eprop A) -> (eprop B) ) -> (H2 : (_72 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_75 : (eprop ( (and ( (dotpipp B) (_73 : (eprop B) => C) ) ) ( (dotpipp C) (_74 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) . -conj_case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_80 : (eprop A) -> (eprop B) ) -> (H2 : (_81 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_87 : (_82 : (eprop B) -> (eprop C) ) -> (_86 : (_83 : (eprop C) -> (eprop B) ) -> (eprop ( (and ( (dotpipp B) (_84 : (eprop B) => C) ) ) ( (dotpipp C) (_85 : (eprop C) => B) ) ) ) ) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_88 : (eprop A) -> (eprop B) ) , H2 : (_89 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) ] ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) --> ( (conj ( (dotpipp B) (_78 : (eprop B) => C) ) ) ( (dotpipp C) (_79 : (eprop C) => B) ) ) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_76 : (eprop A) -> (eprop B) ) , H2 : (_77 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) , var_10 : (_90 : (eprop B) -> (eprop C) ) , var_11 : (_91 : (eprop C) -> (eprop B) ) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) ( ( ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_95 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_94 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_92 : (eprop A) => C) ) ) ( (dotpipp C) (_93 : (eprop C) => A) ) ) (H欧1 : (eprop A) => (H3 (H1 H欧1) ) ) ) (H欧1 : (eprop C) => (H2 (H1 (H2 (H4 H欧1) ) ) ) ) ) ) ) var_10) var_11) . -[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (_69 : (eprop A) -> (eprop B) ) , var_9 : (_70 : (eprop B) -> (eprop A) ) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( ( ( (conj_case_6 A) B) C) H) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_97 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_96 : (eprop B) => A) ) ) => (H欧0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) H欧0) ) ) ) var_8) var_9) . -iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_99 : (eprop ( (iff A) B) ) -> (_98 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) . -[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) . -case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_102 : (eprop ( (and ( (dotpipp A) (_100 : (eprop A) => B) ) ) ( (dotpipp B) (_101 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) . -conj_case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_110 : (_105 : (eprop A) -> (eprop B) ) -> (_109 : (_106 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_107 : (eprop A) => B) ) ) ( (dotpipp B) (_108 : (eprop B) => A) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_8 A) B) H) --> ( (conj ( (dotpipp A) (_103 : (eprop A) => B) ) ) ( (dotpipp B) (_104 : (eprop B) => A) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (_111 : (eprop A) -> (eprop B) ) , var_13 : (_112 : (eprop B) -> (eprop A) ) ] ( ( ( (case_8 A) B) H) ( ( ( ( (conj_case_8 A) B) H) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_116 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_115 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_113 : (eprop B) => A) ) ) ( (dotpipp A) (_114 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) . -iff_sym : (A : Uprop -> (B : Uprop -> (_117 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) . -[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) . -case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_128 : (eprop ( (and ( (dotpipp A) (_125 : (eprop A) => False) ) ) ( (dotpipp False) (_126 : (eprop False) => A) ) ) ) -> (_127 : (eprop A) -> (eprop False) ) ) ) ) . -conj_case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_136 : (_131 : (eprop A) -> (eprop False) ) -> (_135 : (_132 : (eprop False) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_133 : (eprop A) => False) ) ) ( (dotpipp False) (_134 : (eprop False) => A) ) ) ) ) ) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) ] ( (conj_case_9 A) H) --> ( (conj ( (dotpipp A) (_129 : (eprop A) => False) ) ) ( (dotpipp False) (_130 : (eprop False) => A) ) ) . -[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (_137 : (eprop A) -> (eprop False) ) , var_15 : (_138 : (eprop False) -> (eprop A) ) ] ( ( (case_9 A) H) ( ( ( (conj_case_9 A) H) var_14) var_15) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_140 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_139 : (eprop False) => A) ) ) => H欧0) ) var_14) var_15) . -neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) . -[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_118 : (eprop A) => False) ) ) (_119 : (eprop ( (dotpipp A) (_118 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_121 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_120 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_124 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_122 : (eprop A) => False) ) ) ( (dotpipp False) (_123 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) . -and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_168 : (_165 : (eprop B) -> (eprop A) ) -> (_167 : (_166 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_164 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_163 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_141 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_142 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_143 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_144 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_156 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_155 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_154 : (eprop A) => ( (dotpipp B) (_153 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_152 : (eprop A) => ( (dotpipp C) (_151 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp B) (_148 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H欧1 : (eprop A) => (H5 : (eprop C) => ( (H1欧2 : (eprop ( (dotpipp C) (_147 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop A) => ( (H0欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H1欧3 : (eprop A) => (H6 : (eprop B) => H5) ) ) H0欧0) ) (H1欧2 H5) ) ) (H0 H5) ) ) (H2欧0 H4) ) ) ) ) H1欧1) ) (H欧0 H3欧0) ) ) (H1欧0 H4) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp B) (_150 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (dotpipp C) (_149 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H1欧2 : (eprop A) => (H5 : (eprop B) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H0欧1 : (eprop A) => (H6 : (eprop C) => H5) ) ) H欧0) ) (H0欧0 H5) ) ) (H H5) ) ) ) ) H2欧1) ) (H1欧1 H3欧0) ) ) (H2欧0 H4) ) ) (H1欧0 H4) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_157 : (eprop B) => C) ) ) ( (dotpipp C) (_158 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_162 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_161 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_159 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_160 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H1欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2欧0) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj A) B) H3欧0) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_196 : (_193 : (eprop B) -> (eprop A) ) -> (_195 : (_194 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_192 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_191 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_169 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_170 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_171 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_172 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_184 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_183 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_182 : (eprop B) => ( (dotpipp A) (_181 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_180 : (eprop C) => ( (dotpipp A) (_179 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_173 : (eprop B) => C) ) ) ( (dotpipp C) (_174 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_176 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H欧1 : (eprop C) => (H5 : (eprop A) => ( (H1欧2 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_175 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H0欧1 : (eprop B) => (H6 : (eprop A) => H欧1) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H欧1) ) ) (H0 H欧1) ) ) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_178 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H0欧1 : (eprop B) => (H5 : (eprop A) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_177 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H欧1 : (eprop C) => (H6 : (eprop A) => H0欧1) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H0欧1) ) ) (H H0欧1) ) ) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_185 : (eprop B) => C) ) ) ( (dotpipp C) (_186 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_190 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_189 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_187 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_188 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H1欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧0) H2欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj B) A) H0欧0) H3欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_220 : (_217 : (eprop B) -> (eprop (not A) ) ) -> (_219 : (_218 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_216 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_215 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_197 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_198 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_199 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_200 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_208 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_207 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_206 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_205 : (eprop B) => ( (or A) C) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_204 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_203 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_201 : (eprop B) => C) ) ) ( (dotpipp C) (_202 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => H4欧0) ) H0欧0) ) (H5 H4欧0) ) ) (H0 H4欧0) ) ) ) H欧0) ) (H4 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => H5欧0) ) H欧0) ) (H4 H5欧0) ) ) (H H5欧0) ) ) ) H0欧0) ) (H5 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H2欧0 : (eprop A) => (H3 ( ( (or_introl A) C) H2欧0) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H1欧0 : (eprop A) => (H2 ( ( (or_introl A) B) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_209 : (eprop B) => C) ) ) ( (dotpipp C) (_210 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_214 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_213 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_211 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_212 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H1欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_intror A) B) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_244 : (_241 : (eprop B) -> (eprop (not A) ) ) -> (_243 : (_242 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) . -[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_240 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_239 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_221 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_222 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_223 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_224 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_232 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_231 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_230 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_229 : (eprop A) => ( (or C) A) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_228 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_227 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_225 : (eprop B) => C) ) ) ( (dotpipp C) (_226 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H1欧1 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => H1欧1) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H1欧1) ) ) (H0 H1欧1) ) ) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H2欧1 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => H2欧1) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H2欧1) ) ) (H H2欧1) ) ) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H2欧0 : (eprop C) => (H3 ( ( (or_introl C) A) H2欧0) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H1欧0 : (eprop B) => (H2 ( ( (or_introl B) A) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_233 : (eprop B) => C) ) ) ( (dotpipp C) (_234 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_238 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_237 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_235 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_236 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H1欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_introl B) A) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) . -and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_251 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) . -[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_245 : (eprop B) => C) ) ) ( (dotpipp C) (_246 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_250 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_249 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_247 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_248 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2) H欧1) ) (H1 H欧1) ) ) (H0 H3) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj A) B) H2) H欧1) ) (H0 H欧1) ) ) (H1 H3) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_258 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) . -[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_252 : (eprop B) => C) ) ) ( (dotpipp C) (_253 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_257 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_256 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_254 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_255 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧1) H3) ) (H1 H欧1) ) ) (H0 H2) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj B) A) H欧1) H3) ) (H0 H欧1) ) ) (H1 H2) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_265 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) . -[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_259 : (eprop B) => C) ) ) ( (dotpipp C) (_260 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_264 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_263 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_261 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_262 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_intror A) B) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_272 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) . -[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_266 : (eprop B) => C) ) ) ( (dotpipp C) (_267 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_271 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_270 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_268 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_269 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H欧0) ) ) (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_introl B) A) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) . -case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_277 : (eprop ( (and ( (dotpipp A) (_273 : (eprop A) => B) ) ) ( (dotpipp B) (_274 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_275 : (eprop A) => B) ) ) ( (dotpipp B) (_276 : (eprop B) => A) ) ) ) ) ) ) ) . -conj_case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_285 : (_280 : (eprop A) -> (eprop B) ) -> (_284 : (_281 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_282 : (eprop A) => B) ) ) ( (dotpipp B) (_283 : (eprop B) => A) ) ) ) ) ) ) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_10 A) B) H) --> ( (conj ( (dotpipp A) (_278 : (eprop A) => B) ) ) ( (dotpipp B) (_279 : (eprop B) => A) ) ) . -[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (_286 : (eprop A) -> (eprop B) ) , var_17 : (_287 : (eprop B) -> (eprop A) ) ] ( ( ( (case_10 A) B) H) ( ( ( ( (conj_case_10 A) B) H) var_16) var_17) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_291 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_290 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_288 : (eprop A) => B) ) ) ( (dotpipp B) (_289 : (eprop B) => A) ) ) H欧0) H0) ) ) var_16) var_17) . -iff_and : (A : Uprop -> (B : Uprop -> (_294 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_292 : (eprop A) => B) ) ) ( (dotpipp B) (_293 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) . -iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_317 : (eprop A) => B) ) ) ( (dotpipp B) (_318 : (eprop B) => A) ) ) ) ) ) ) . -[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_297 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_295 : (eprop A) => B) ) ) ( (dotpipp B) (_296 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) (_300 : (eprop ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_301 : (eprop A) => B) ) ) ( (dotpipp B) (_302 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_303 : (eprop A) => B) ) ) ( (dotpipp B) (_304 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_308 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_307 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_305 : (eprop A) => B) ) ) ( (dotpipp B) (_306 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_315 : (eprop A) => B) ) ) ( (dotpipp B) (_316 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_309 : (eprop A) => B) ) ) ( (dotpipp B) (_310 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_314 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_313 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_311 : (eprop A) => B) ) ) ( (dotpipp B) (_312 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) ) ) . -IF_then_else : (P : Uprop -> (Q : Uprop -> (R : Uprop -> Uprop) ) ) . -[] IF_then_else --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => ( (or ( (and P) Q) ) ( (and (not P) ) R) ) ) ) ) . -ex : (A : Utype -> (P : (_319 : (etype A) -> Uprop) -> Uprop) ) . -ex_intro : (A : Utype -> (P : (_320 : (etype A) -> Uprop) -> (x : (etype A) -> (_321 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) . -case_11 : (A : Utype -> (P : (_322 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_323 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (_324 : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) ) . -ex_intro_case_11 : (A : Utype -> (P : (_327 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_328 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (x : (etype A) -> (_329 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_330 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_331 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) ] ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) --> ( (ex_intro A) P) . -[A : Utype, P : (_325 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_326 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) , var_18 : (etype A) , var_19 : (eprop (P var_18) ) ] ( ( ( ( ( (case_11 A) P) P欧0) f) e) ( ( ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) var_18) var_19) ) --> ( (f var_18) var_19) . -ex_ind : (A : Utype -> (P : (_334 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_335 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) . -[] ex_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_333 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_332 : (eprop (P x) ) => P欧0) ) ) ) ) => (e : (eprop ( (ex A) P) ) => ( ( ( ( ( (case_11 A) P) P欧0) f) e) e) ) ) ) ) ) . -ex2 : (A : Utype -> (P : (_336 : (etype A) -> Uprop) -> (Q : (_337 : (etype A) -> Uprop) -> Uprop) ) ) . -ex_intro2 : (A : Utype -> (P : (_338 : (etype A) -> Uprop) -> (Q : (_339 : (etype A) -> Uprop) -> (x : (etype A) -> (_341 : (eprop (P x) ) -> (_340 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) . -case_12 : (A : Utype -> (P : (_342 : (etype A) -> Uprop) -> (Q : (_343 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_345 : (eprop (P x) ) -> (_344 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (_346 : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) ) . -ex_intro2_case_12 : (A : Utype -> (P : (_351 : (etype A) -> Uprop) -> (Q : (_352 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_354 : (eprop (P x) ) -> (_353 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (x : (etype A) -> (_356 : (eprop (P x) ) -> (_355 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_357 : (etype A) -> Uprop) , Q : (_358 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_360 : (eprop (P x) ) -> (_359 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) ] ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) --> ( ( (ex_intro2 A) P) Q) . -[A : Utype, P : (_347 : (etype A) -> Uprop) , Q : (_348 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_350 : (eprop (P x) ) -> (_349 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) , var_20 : (etype A) , var_21 : (eprop (P var_20) ) , var_22 : (eprop (Q var_20) ) ] ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) ( ( ( ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) var_20) var_21) var_22) ) --> ( ( (f var_20) var_21) var_22) . -ex2_ind : (A : Utype -> (P : (_365 : (etype A) -> Uprop) -> (Q : (_366 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_368 : (eprop (P x) ) -> (_367 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) . -[] ex2_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_364 : (etype A) => dotprop) ) ) => (Q : (etype ( (dotpitt A) (_363 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_362 : (eprop (P x) ) => ( (dotpipp (Q x) ) (_361 : (eprop (Q x) ) => P欧0) ) ) ) ) ) ) => (e : (eprop ( ( (ex2 A) P) Q) ) => ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) e) ) ) ) ) ) ) . -all : (A : Utype -> (P : (_370 : (etype A) -> Uprop) -> Uprop) ) . -[] all --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_369 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => (P x) ) ) ) ) . -inst : (A : Utype -> (P : (_372 : (etype A) -> Uprop) -> (x : (etype A) -> (_373 : (eprop ( (all A) (x欧0 : (etype A) => (P x欧0) ) ) ) -> (eprop (P x) ) ) ) ) ) . -[] inst --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_371 : (etype A) => dotprop) ) ) => (x : (etype A) => (H : (eprop ( (dotpitp A) (x欧0 : (etype A) => (P x欧0) ) ) ) => (H x) ) ) ) ) . -gen : (A : Utype -> (P : (_376 : (etype A) -> Uprop) -> (B : Uprop -> (f : (y : (etype A) -> (_377 : (eprop B) -> (eprop (P y) ) ) ) -> (_378 : (eprop B) -> (eprop ( (all A) P) ) ) ) ) ) ) . -[] gen --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_375 : (etype A) => dotprop) ) ) => (B : (etype dotprop) => (f : (eprop ( (dotpitp A) (y : (etype A) => ( (dotpipp B) (_374 : (eprop B) => (P y) ) ) ) ) ) => (H : (eprop B) => (x : (etype A) => ( (f x) H) ) ) ) ) ) ) . -eq : (A : Utype -> (x : (etype A) -> (_379 : (etype A) -> Uprop) ) ) . -refl_equal : (A : Utype -> (x : (etype A) -> (eprop ( ( (eq A) x) x) ) ) ) . -case_13 : (A : Utype -> (x : (etype A) -> (P : (_380 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_381 : (eprop ( ( (eq A) x) y欧0) ) -> (etype (P y欧0) ) ) ) ) ) ) ) ) ) . -refl_equal_case_13 : (A : Utype -> (x : (etype A) -> (P : (_383 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , P : (_384 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) --> ( (refl_equal A) x) . -[A : Utype, x : (etype A) , P : (_382 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) x) ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) ) --> f. -eq_rect : (A : Utype -> (x : (etype A) -> (P : (_386 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (etype (P y) ) ) ) ) ) ) ) . -[] eq_rect --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_385 : (etype A) => dottype) ) ) => (f : (etype (P x) ) => (y : (etype A) => (e : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) y) e) ) ) ) ) ) ) . -eq_ind : (A : Utype -> (x : (etype A) -> (P : (_388 : (etype A) -> Uprop) -> (f : (eprop (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop (P y) ) ) ) ) ) ) ) . -[] eq_ind --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_387 : (etype A) => dotprop) ) ) => ( ( (eq_rect A) x) P) ) ) ) . -eq_rec : (A : Utype -> (x : (etype A) -> (P : (_390 : (etype A) -> Uset) -> (f : (eset (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eset (P y) ) ) ) ) ) ) ) . -[] eq_rec --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_389 : (etype A) => dotset) ) ) => ( ( (eq_rect A) x) P) ) ) ) . -case_14 : (A : Uprop -> (C : Uprop -> (h1 : (eprop A) -> (h2 : (_391 : (eprop A) -> (eprop False) ) -> (f : (eprop False) -> (_392 : (eprop False) -> (eprop C) ) ) ) ) ) ) . -absurd : (A : Uprop -> (C : Uprop -> (_396 : (eprop A) -> (_395 : (eprop (not A) ) -> (eprop C) ) ) ) ) . -[] absurd --> (A : (etype dotprop) => (C : (etype dotprop) => (h1 : (eprop A) => (h2 : (eprop ( (dotpipp A) (_394 : (eprop A) => False) ) ) => ( (f : (eprop False) => ( ( ( ( ( (case_14 A) C) h1) h2) f) f) ) (h2 h1) ) ) ) ) ) . -case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_397 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq A) y欧0) x) ) ) ) ) ) ) ) . -refl_equal_case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( (refl_equal_case_15 A) x) y) H) --> ( (refl_equal A) x) . -[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (case_15 A) x) y) H) x) ( ( ( (refl_equal_case_15 A) x) y) H) ) --> ( (refl_equal A) x) . -sym_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_398 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) . -[] sym_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( (case_15 A) x) y) H) y) H) ) ) ) ) . -case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (y欧0 : (etype A) -> (_399 : (eprop ( ( (eq A) y) y欧0) ) -> (eprop ( ( (eq A) x) y欧0) ) ) ) ) ) ) ) ) ) . -refl_equal_case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) --> ( (refl_equal A) y) . -[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) y) ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) ) --> H. -trans_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_401 : (eprop ( ( (eq A) x) y) ) -> (_400 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) . -[] trans_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => (H0 : (eprop ( ( (eq A) y) z) ) => ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) z) H0) ) ) ) ) ) ) . -case_17 : (A : Utype -> (B : Utype -> (f : (_402 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_403 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq B) (f x) ) (f y欧0) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_17 : (A : Utype -> (B : Utype -> (f : (_405 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) . -[A : Utype, B : Utype, f : (_406 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) --> ( (refl_equal A) x) . -[A : Utype, B : Utype, f : (_404 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) x) ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) ) --> ( (refl_equal B) (f x) ) . -f_equal : (A : Utype -> (B : Utype -> (f : (_408 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (_409 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq B) (f x) ) (f y) ) ) ) ) ) ) ) ) . -[] f_equal --> (A : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A) (_407 : (etype A) => B) ) ) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) y) H) ) ) ) ) ) ) . -case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (y欧0 : (etype A) -> (_410 : (eprop ( ( (eq A) y) y欧0) ) -> (h1欧0 : (eprop (not ( ( (eq A) y欧0) y) ) ) -> (eprop ( ( (eq A) y欧0) y) ) ) ) ) ) ) ) ) ) . -refl_equal_case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) . -[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) --> ( (refl_equal A) y) . -[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( ( ( (case_18 A) x) y) h1) h2) y) ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) ) --> (h1欧0 : (eprop (not ( ( (eq A) y) y) ) ) => ( (refl_equal A) y) ) . -sym_not_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_411 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) . -[] sym_not_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (h1 : (eprop (not ( ( (eq A) x) y) ) ) => (h2 : (eprop ( ( (eq A) y) x) ) => (h1 ( ( ( ( ( ( ( (case_18 A) x) y) h1) h2) x) h2) h1) ) ) ) ) ) ) . -sym_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_412 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) . -[] sym_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_eq A) x) y) ) ) ) . -sym_not_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_413 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) . -[] sym_not_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_not_eq A) x) y) ) ) ) . -trans_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_415 : (eprop ( ( (eq A) x) y) ) -> (_414 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) . -[] trans_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => ( ( ( (trans_eq A) x) y) z) ) ) ) ) . -eq_ind_r : (A : Utype -> (x : (etype A) -> (P : (_417 : (etype A) -> Uprop) -> (_419 : (eprop (P x) ) -> (y : (etype A) -> (_418 : (eprop ( ( (eq A) y) x) ) -> (eprop (P y) ) ) ) ) ) ) ) . -[] eq_ind_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_416 : (etype A) => dotprop) ) ) => (H : (eprop (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_ind A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -eq_rec_r : (A : Utype -> (x : (etype A) -> (P : (_421 : (etype A) -> Uset) -> (_423 : (eset (P x) ) -> (y : (etype A) -> (_422 : (eprop ( ( (eq A) y) x) ) -> (eset (P y) ) ) ) ) ) ) ) . -[] eq_rec_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_420 : (etype A) => dotset) ) ) => (H : (eset (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rec A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -eq_rect_r : (A : Utype -> (x : (etype A) -> (P : (_425 : (etype A) -> Utype) -> (_427 : (etype (P x) ) -> (y : (etype A) -> (_426 : (eprop ( ( (eq A) y) x) ) -> (etype (P y) ) ) ) ) ) ) ) . -[] eq_rect_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_424 : (etype A) => dottype) ) ) => (H : (etype (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rect A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) . -case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_429 : (etype A1) -> (_428 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_431 : (eprop ( ( (eq A1) x1) y) ) -> (_430 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y) y2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_435 : (etype A1) -> (_434 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_437 : (etype A1) -> (_436 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) --> ( (refl_equal A1) x1) . -case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_439 : (etype A1) -> (_438 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_440 : (eprop ( ( (eq A2) x2) y) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f x1) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_444 : (etype A1) -> (_443 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_446 : (etype A1) -> (_445 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) --> ( (refl_equal A2) x2) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_442 : (etype A1) -> (_441 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) x2) ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) ) --> ( (refl_equal B) ( (f x1) x2) ) . -[A1 : Utype, A2 : Utype, B : Utype, f : (_433 : (etype A1) -> (_432 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) x1) ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) y2) H欧0) ) . -f_equal2 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_450 : (etype A1) -> (_449 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (_452 : (eprop ( ( (eq A1) x1) y1) ) -> (_451 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y1) y2) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal2 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_448 : (etype A1) => ( (dotpitt A2) (_447 : (etype A2) => B) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) y1) H) ) ) ) ) ) ) ) ) ) . -case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_455 : (etype A1) -> (_454 : (etype A2) -> (_453 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_458 : (eprop ( ( (eq A1) x1) y) ) -> (_457 : (eprop ( ( (eq A2) x2) y2) ) -> (_456 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_464 : (etype A1) -> (_463 : (etype A2) -> (_462 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_467 : (etype A1) -> (_466 : (etype A2) -> (_465 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) --> ( (refl_equal A1) x1) . -case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_470 : (etype A1) -> (_469 : (etype A2) -> (_468 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_472 : (eprop ( ( (eq A2) x2) y) ) -> (_471 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) y) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_478 : (etype A1) -> (_477 : (etype A2) -> (_476 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_481 : (etype A1) -> (_480 : (etype A2) -> (_479 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) --> ( (refl_equal A2) x2) . -case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_484 : (etype A1) -> (_483 : (etype A2) -> (_482 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_485 : (eprop ( ( (eq A3) x3) y) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) x2) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_491 : (etype A1) -> (_490 : (etype A2) -> (_489 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_494 : (etype A1) -> (_493 : (etype A2) -> (_492 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_488 : (etype A1) -> (_487 : (etype A2) -> (_486 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) ) --> ( (refl_equal B) ( ( (f x1) x2) x3) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_475 : (etype A1) -> (_474 : (etype A2) -> (_473 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_461 : (etype A1) -> (_460 : (etype A2) -> (_459 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) x1) ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) y2) H欧0) ) . -f_equal3 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_500 : (etype A1) -> (_499 : (etype A2) -> (_498 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (_503 : (eprop ( ( (eq A1) x1) y1) ) -> (_502 : (eprop ( ( (eq A2) x2) y2) ) -> (_501 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y1) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal3 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_497 : (etype A1) => ( (dotpitt A2) (_496 : (etype A2) => ( (dotpitt A3) (_495 : (etype A3) => B) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) . -case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_507 : (etype A1) -> (_506 : (etype A2) -> (_505 : (etype A3) -> (_504 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_511 : (eprop ( ( (eq A1) x1) y) ) -> (_510 : (eprop ( ( (eq A2) x2) y2) ) -> (_509 : (eprop ( ( (eq A3) x3) y3) ) -> (_508 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_519 : (etype A1) -> (_518 : (etype A2) -> (_517 : (etype A3) -> (_516 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_523 : (etype A1) -> (_522 : (etype A2) -> (_521 : (etype A3) -> (_520 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) --> ( (refl_equal A1) x1) . -case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_527 : (etype A1) -> (_526 : (etype A2) -> (_525 : (etype A3) -> (_524 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_530 : (eprop ( ( (eq A2) x2) y) ) -> (_529 : (eprop ( ( (eq A3) x3) y3) ) -> (_528 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) y) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_538 : (etype A1) -> (_537 : (etype A2) -> (_536 : (etype A3) -> (_535 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_542 : (etype A1) -> (_541 : (etype A2) -> (_540 : (etype A3) -> (_539 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) --> ( (refl_equal A2) x2) . -case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_546 : (etype A1) -> (_545 : (etype A2) -> (_544 : (etype A3) -> (_543 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_548 : (eprop ( ( (eq A3) x3) y) ) -> (_547 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) y) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_556 : (etype A1) -> (_555 : (etype A2) -> (_554 : (etype A3) -> (_553 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_560 : (etype A1) -> (_559 : (etype A2) -> (_558 : (etype A3) -> (_557 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_564 : (etype A1) -> (_563 : (etype A2) -> (_562 : (etype A3) -> (_561 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_565 : (eprop ( ( (eq A4) x4) y) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) x3) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_573 : (etype A1) -> (_572 : (etype A2) -> (_571 : (etype A3) -> (_570 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_577 : (etype A1) -> (_576 : (etype A2) -> (_575 : (etype A3) -> (_574 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_569 : (etype A1) -> (_568 : (etype A2) -> (_567 : (etype A3) -> (_566 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) ) --> ( (refl_equal B) ( ( ( (f x1) x2) x3) x4) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_552 : (etype A1) -> (_551 : (etype A2) -> (_550 : (etype A3) -> (_549 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) y4) H欧2) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_534 : (etype A1) -> (_533 : (etype A2) -> (_532 : (etype A3) -> (_531 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_515 : (etype A1) -> (_514 : (etype A2) -> (_513 : (etype A3) -> (_512 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) y2) H欧0) ) . -f_equal4 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_585 : (etype A1) -> (_584 : (etype A2) -> (_583 : (etype A3) -> (_582 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (_589 : (eprop ( ( (eq A1) x1) y1) ) -> (_588 : (eprop ( ( (eq A2) x2) y2) ) -> (_587 : (eprop ( ( (eq A3) x3) y3) ) -> (_586 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y1) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal4 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_581 : (etype A1) => ( (dotpitt A2) (_580 : (etype A2) => ( (dotpitt A3) (_579 : (etype A3) => ( (dotpitt A4) (_578 : (etype A4) => B) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_594 : (etype A1) -> (_593 : (etype A2) -> (_592 : (etype A3) -> (_591 : (etype A4) -> (_590 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_599 : (eprop ( ( (eq A1) x1) y) ) -> (_598 : (eprop ( ( (eq A2) x2) y2) ) -> (_597 : (eprop ( ( (eq A3) x3) y3) ) -> (_596 : (eprop ( ( (eq A4) x4) y4) ) -> (_595 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_609 : (etype A1) -> (_608 : (etype A2) -> (_607 : (etype A3) -> (_606 : (etype A4) -> (_605 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_614 : (etype A1) -> (_613 : (etype A2) -> (_612 : (etype A3) -> (_611 : (etype A4) -> (_610 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) --> ( (refl_equal A1) x1) . -case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_619 : (etype A1) -> (_618 : (etype A2) -> (_617 : (etype A3) -> (_616 : (etype A4) -> (_615 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_623 : (eprop ( ( (eq A2) x2) y) ) -> (_622 : (eprop ( ( (eq A3) x3) y3) ) -> (_621 : (eprop ( ( (eq A4) x4) y4) ) -> (_620 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) y) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_633 : (etype A1) -> (_632 : (etype A2) -> (_631 : (etype A3) -> (_630 : (etype A4) -> (_629 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_638 : (etype A1) -> (_637 : (etype A2) -> (_636 : (etype A3) -> (_635 : (etype A4) -> (_634 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) --> ( (refl_equal A2) x2) . -case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_643 : (etype A1) -> (_642 : (etype A2) -> (_641 : (etype A3) -> (_640 : (etype A4) -> (_639 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_646 : (eprop ( ( (eq A3) x3) y) ) -> (_645 : (eprop ( ( (eq A4) x4) y4) ) -> (_644 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) y) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_656 : (etype A1) -> (_655 : (etype A2) -> (_654 : (etype A3) -> (_653 : (etype A4) -> (_652 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_661 : (etype A1) -> (_660 : (etype A2) -> (_659 : (etype A3) -> (_658 : (etype A4) -> (_657 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) --> ( (refl_equal A3) x3) . -case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_666 : (etype A1) -> (_665 : (etype A2) -> (_664 : (etype A3) -> (_663 : (etype A4) -> (_662 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_668 : (eprop ( ( (eq A4) x4) y) ) -> (_667 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) y) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_678 : (etype A1) -> (_677 : (etype A2) -> (_676 : (etype A3) -> (_675 : (etype A4) -> (_674 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_683 : (etype A1) -> (_682 : (etype A2) -> (_681 : (etype A3) -> (_680 : (etype A4) -> (_679 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) . -case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_688 : (etype A1) -> (_687 : (etype A2) -> (_686 : (etype A3) -> (_685 : (etype A4) -> (_684 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (y : (etype A5) -> (_689 : (eprop ( ( (eq A5) x5) y) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) x4) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -refl_equal_case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_699 : (etype A1) -> (_698 : (etype A2) -> (_697 : (etype A3) -> (_696 : (etype A4) -> (_695 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq A5) x5) x5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_704 : (etype A1) -> (_703 : (etype A2) -> (_702 : (etype A3) -> (_701 : (etype A4) -> (_700 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) --> ( (refl_equal A5) x5) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_694 : (etype A1) -> (_693 : (etype A2) -> (_692 : (etype A3) -> (_691 : (etype A4) -> (_690 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) x5) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) ) --> ( (refl_equal B) ( ( ( ( (f x1) x2) x3) x4) x5) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_673 : (etype A1) -> (_672 : (etype A2) -> (_671 : (etype A3) -> (_670 : (etype A4) -> (_669 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) ) --> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) y5) H欧3) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_651 : (etype A1) -> (_650 : (etype A2) -> (_649 : (etype A3) -> (_648 : (etype A4) -> (_647 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) y4) H欧2) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_628 : (etype A1) -> (_627 : (etype A2) -> (_626 : (etype A3) -> (_625 : (etype A4) -> (_624 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) y3) H欧1) ) . -[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_604 : (etype A1) -> (_603 : (etype A2) -> (_602 : (etype A3) -> (_601 : (etype A4) -> (_600 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) y2) H欧0) ) . -f_equal5 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_714 : (etype A1) -> (_713 : (etype A2) -> (_712 : (etype A3) -> (_711 : (etype A4) -> (_710 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (_719 : (eprop ( ( (eq A1) x1) y1) ) -> (_718 : (eprop ( ( (eq A2) x2) y2) ) -> (_717 : (eprop ( ( (eq A3) x3) y3) ) -> (_716 : (eprop ( ( (eq A4) x4) y4) ) -> (_715 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y1) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -[] f_equal5 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (A5 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_709 : (etype A1) => ( (dotpitt A2) (_708 : (etype A2) => ( (dotpitt A3) (_707 : (etype A3) => ( (dotpitt A4) (_706 : (etype A4) => ( (dotpitt A5) (_705 : (etype A5) => B) ) ) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (x5 : (etype A5) => (y5 : (etype A5) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . -subrelation : (A : Utype -> (B : Utype -> (R : (_726 : (etype A) -> (_725 : (etype B) -> Uprop) ) -> (R' : (_728 : (etype A) -> (_727 : (etype B) -> Uprop) ) -> Uprop) ) ) ) . -[] subrelation --> (A : (etype dottype) => (B : (etype dottype) => (R : (etype ( (dotpitt A) (_724 : (etype A) => ( (dotpitt B) (_723 : (etype B) => dotprop) ) ) ) ) => (R' : (etype ( (dotpitt A) (_722 : (etype A) => ( (dotpitt B) (_721 : (etype B) => dotprop) ) ) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp B) (y : (etype B) => ( (dotpipp ( (R x) y) ) (_720 : (eprop ( (R x) y) ) => ( (R' x) y) ) ) ) ) ) ) ) ) ) ) . -unique : (A : Utype -> (P : (_731 : (etype A) -> Uprop) -> (x : (etype A) -> Uprop) ) ) . -[] unique --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_730 : (etype A) => dotprop) ) ) => (x : (etype A) => ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_729 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) . -uniqueness : (A : Utype -> (P : (_735 : (etype A) -> Uprop) -> Uprop) ) . -[] uniqueness --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_734 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp A) (y : (etype A) => ( (dotpipp (P x) ) (_733 : (eprop (P x) ) => ( (dotpipp (P y) ) (_732 : (eprop (P y) ) => ( ( (eq A) x) y) ) ) ) ) ) ) ) ) ) ) . -case_33 : (A : Utype -> (P : (_738 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_739 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) . -conj_case_33 : (A : Utype -> (P : (_741 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_743 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_742 : (eprop ( (uniqueness A) P) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) . -[A : Utype, P : (_744 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ] ( ( (conj_case_33 A) P) H) --> ( (conj ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) . -case_34 : (A : Utype -> (P : (_745 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_747 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_746 : (eprop ( (uniqueness A) P) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) ) ) . -ex_intro_case_34 : (A : Utype -> (P : (_749 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_750 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_751 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( ( (ex_intro_case_34 A) P) H) H欧0) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) . -[A : Utype, P : (_748 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_25 : (etype A) , var_26 : (eprop ( (x : (etype A) => (P x) ) var_25) ) ] ( ( ( ( (case_34 A) P) H) H欧0) ( ( ( ( ( (ex_intro_case_34 A) P) H) H欧0) var_25) var_26) ) --> ( ( (x : (etype A) => (Hx : (eprop (P x) ) => (Huni : (eprop ( (uniqueness A) P) ) => ( ( ( (ex_intro A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) x) ( ( ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_752 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) Hx) (x' : (etype A) => (H欧1 : (eprop (P x') ) => ( ( ( (Huni x) x') Hx) H欧1) ) ) ) ) ) ) ) var_25) var_26) . -[A : Utype, P : (_740 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , var_23 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_24 : (eprop ( (uniqueness A) P) ) ] ( ( ( (case_33 A) P) H) ( ( ( ( (conj_case_33 A) P) H) var_23) var_24) ) --> ( ( (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( ( (case_34 A) P) H) H欧0) H欧0) ) var_23) var_24) . -case_35 : (A : Utype -> (P : (_753 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (_754 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) . -ex_intro_case_35 : (A : Utype -> (P : (_756 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (_757 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (eprop ( (ex A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_758 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ] ( ( (ex_intro_case_35 A) P) H) --> ( (ex_intro A) ( (unique A) (x : (etype A) => (P x) ) ) ) . -case_36 : (A : Utype -> (P : (_759 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_761 : (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_760 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) -> (eprop ( (and ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) ) . -conj_case_36 : (A : Utype -> (P : (_764 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_768 : (eprop (P x) ) -> (_767 : (x' : (etype A) -> (_765 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) -> (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_766 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_769 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) ] ( ( ( ( (conj_case_36 A) P) H) x) H欧0) --> ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_763 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) . -[A : Utype, P : (_762 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) , var_29 : (eprop (P x) ) , var_30 : (x' : (etype A) -> (_770 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) ] ( ( ( ( ( (case_36 A) P) H) x) H欧0) ( ( ( ( ( ( (conj_case_36 A) P) H) x) H欧0) var_29) var_30) ) --> ( ( (Hx : (eprop (P x) ) => (Huni : (eprop ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_771 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) => ( ( ( (conj ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ( ( ( (ex_intro A) (x欧0 : (etype A) => (P x欧0) ) ) x) Hx) ) (x' : (etype A) => (x'' : (etype A) => (Hx' : (eprop (P x') ) => (Hx'' : (eprop (P x'') ) => ( ( ( ( ( (trans_eq A) x') x) x'') ( ( ( (sym_eq A) x) x') ( (Huni x') Hx') ) ) ( (Huni x'') Hx'') ) ) ) ) ) ) ) ) var_29) var_30) . -[A : Utype, P : (_755 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , var_27 : (etype A) , var_28 : (eprop ( ( (unique A) (x : (etype A) => (P x) ) ) var_27) ) ] ( ( ( (case_35 A) P) H) ( ( ( ( (ex_intro_case_35 A) P) H) var_27) var_28) ) --> ( ( (x : (etype A) => (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) => ( ( ( ( ( (case_36 A) P) H) x) H欧0) H欧0) ) ) var_27) var_28) . -unique_existence : (A : Utype -> (P : (_773 : (etype A) -> Uprop) -> (eprop ( (iff ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) . -[] unique_existence --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_772 : (etype A) => dotprop) ) ) => ( ( ( (conj ( (dotpipp ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) (_736 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ( (dotpipp ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) (_737 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( ( ( (case_33 A) P) H) H) ) ) (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( ( ( (case_35 A) P) H) H) ) ) ) ) . -inhabited : (A : Utype -> Uprop) . -inhabits : (A : Utype -> (_774 : (etype A) -> (eprop (inhabited A) ) ) ) . -case_37 : (A : Utype -> (P : Uprop -> (f : (_775 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_776 : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) ) . -inhabits_case_37 : (A : Utype -> (P : Uprop -> (f : (_778 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_779 : (etype A) -> (eprop (inhabited A) ) ) ) ) ) ) . -[A : Utype, P : Uprop, f : (_780 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) ] ( ( ( (inhabits_case_37 A) P) f) i) --> (inhabits A) . -[A : Utype, P : Uprop, f : (_777 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) , var_31 : (etype A) ] ( ( ( ( (case_37 A) P) f) i) ( ( ( ( (inhabits_case_37 A) P) f) i) var_31) ) --> (f var_31) . -inhabited_ind : (A : Utype -> (P : Uprop -> (f : (_782 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) . -[] inhabited_ind --> (A : (etype dottype) => (P : (etype dotprop) => (f : (eprop ( (dotpitp A) (_781 : (etype A) => P) ) ) => (i : (eprop (inhabited A) ) => ( ( ( ( (case_37 A) P) f) i) i) ) ) ) ) . -case_38 : (A : Utype -> (P : (_783 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_784 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) ) . -ex_intro_case_38 : (A : Utype -> (P : (_786 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_787 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) . -[A : Utype, P : (_788 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( (ex_intro_case_38 A) P) H) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) . -[A : Utype, P : (_785 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_32 : (etype A) , var_33 : (eprop ( (x : (etype A) => (P x) ) var_32) ) ] ( ( ( (case_38 A) P) H) ( ( ( ( (ex_intro_case_38 A) P) H) var_32) var_33) ) --> ( ( (x : (etype A) => (H欧0 : (eprop (P x) ) => ( (inhabits A) x) ) ) var_32) var_33) . -exists_inhabited : (A : Utype -> (P : (_790 : (etype A) -> Uprop) -> (_791 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) . -[] exists_inhabited --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_789 : (etype A) => dotprop) ) ) => (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( (case_38 A) P) H) H) ) ) ) . -eq_stepl : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_793 : (eprop ( ( (eq A) x) y) ) -> (_792 : (eprop ( ( (eq A) x) z) ) -> (eprop ( ( (eq A) z) y) ) ) ) ) ) ) ) . -[] eq_stepl --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H1 : (eprop ( ( (eq A) x) y) ) => (H2 : (eprop ( ( (eq A) x) z) ) => ( ( ( ( ( (eq_ind A) x) (z欧0 : (etype A) => ( ( (eq A) z欧0) y) ) ) H1) z) H2) ) ) ) ) ) ) . -iff_stepl : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_805 : (eprop ( (iff A) B) ) -> (_804 : (eprop ( (iff A) C) ) -> (eprop ( (iff C) B) ) ) ) ) ) ) . -[] iff_stepl --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => (H0 : (eprop ( (iff A) C) ) => ( ( ( ( (and_ind ( (dotpipp A) (_794 : (eprop A) => B) ) ) ( (dotpipp B) (_795 : (eprop B) => A) ) ) ( (iff C) B) ) (H1 : (eprop ( (dotpipp A) (_803 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_802 : (eprop B) => A) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_796 : (eprop A) => C) ) ) ( (dotpipp C) (_797 : (eprop C) => A) ) ) ( (iff C) B) ) (H欧0 : (eprop ( (dotpipp A) (_801 : (eprop A) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_800 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp C) (_798 : (eprop C) => B) ) ) ( (dotpipp B) (_799 : (eprop B) => C) ) ) (H0欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H3欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H3欧0) (H2 H3欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H3 H0欧0) ) ) ) (H0欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H2欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H1欧0) (H3 H1欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H2 H0欧0) ) ) ) ) ) ) H0) ) ) ) H) ) ) ) ) ) . -;Finished module Logic diff --git a/t/linearity.eu b/t/linearity.eu deleted file mode 100644 --- a/t/linearity.eu +++ /dev/null @@ -1,3 +0,0 @@ -Nat: Type. -Q : (Nat -> Nat) -> Nat -> Nat -> Type. -[a:Nat, f : Nat -> Nat] Q f a (f a) --> Nat. diff --git a/t/logic.dk b/t/logic.dk new file mode 100644 --- /dev/null +++ b/t/logic.dk @@ -0,0 +1,49 @@ +False : coc.Utype. + +True : coc.Utype. + +I : coc.etype True. + +eq : t : coc.Utype -> coc.etype t -> coc.etype t -> Type. + +eq_ : t : coc.Utype -> coc.etype t -> coc.etype t -> coc.Utype. + +[ t : coc.Utype +, x : coc.etype t +, y : coc.etype t ] +eq t x y --> coc.etype (eq_ t x y). + + +refl_equal : t : coc.Utype -> x : coc.etype t -> eq t x x. + +eq_rec : t : coc.Utype + -> x : coc.etype t + -> p : (coc.etype t -> coc.Utype) + -> g : coc.etype (p x) + -> y : coc.etype t + -> h : eq t x y + -> coc.etype (p y). + +[ t : coc.Utype +, x : coc.etype t +, p : coc.etype t -> coc.Utype +, f : coc.etype (p x) ] +eq_rec t x p f x (refl_equal t x) --> f. + +f_equal + : A : coc.Utype + -> B : coc.Utype + -> f : (coc.etype A -> coc.etype B) + -> x : coc.etype A + -> y : coc.etype A + -> H : eq A x y + -> eq B (f x) (f y). + +[] f_equal --> + A : coc.Utype + => B : coc.Utype + => f : (coc.etype A -> coc.etype B) + => x : coc.etype A + => y : coc.etype A + => H : eq A x y + => eq_rec A x (z : coc.etype A => eq_ B (f x) (f z)) (refl_equal B (f x)) y H. diff --git a/t/logic.eu b/t/logic.eu deleted file mode 100644 --- a/t/logic.eu +++ /dev/null @@ -1,49 +0,0 @@ -False : coc.Utype. - -True : coc.Utype. - -I : coc.etype True. - -eq : t : coc.Utype -> coc.etype t -> coc.etype t -> Type. - -eq_ : t : coc.Utype -> coc.etype t -> coc.etype t -> coc.Utype. - -[ t : coc.Utype -, x : coc.etype t -, y : coc.etype t ] -eq t x y --> coc.etype (eq_ t x y). - - -refl_equal : t : coc.Utype -> x : coc.etype t -> eq t x x. - -eq_rec : t : coc.Utype - -> x : coc.etype t - -> p : (coc.etype t -> coc.Utype) - -> g : coc.etype (p x) - -> y : coc.etype t - -> h : eq t x y - -> coc.etype (p y). - -[ t : coc.Utype -, x : coc.etype t -, p : coc.etype t -> coc.Utype -, f : coc.etype (p x) ] -eq_rec t x p f x (refl_equal t x) --> f. - -f_equal - : A : coc.Utype - -> B : coc.Utype - -> f : (coc.etype A -> coc.etype B) - -> x : coc.etype A - -> y : coc.etype A - -> H : eq A x y - -> eq B (f x) (f y). - -[] f_equal --> - A : coc.Utype - => B : coc.Utype - => f : (coc.etype A -> coc.etype B) - => x : coc.etype A - => y : coc.etype A - => H : eq A x y - => eq_rec A x (z : coc.etype A => eq_ B (f x) (f z)) (refl_equal B (f x)) y H. diff --git a/t/loop.eu b/t/loop.eu deleted file mode 100644 --- a/t/loop.eu +++ /dev/null @@ -1,4 +0,0 @@ -A : Type. -[] A --> A -> A. -t : A. -[] t --> x : A => x. diff --git a/t/nat.dk b/t/nat.dk new file mode 100644 --- /dev/null +++ b/t/nat.dk @@ -0,0 +1,17 @@ +nat : Type. + +0 : nat. + +S : nat -> nat. + +1 : nat. + +[] 1 --> (S 0). + +plus : nat -> nat -> nat. +[x : nat] plus x 0 --> x. +[x : nat] plus 0 x --> x. +[x : nat, y : nat] plus x (S y) --> S (plus x y). +[x : nat, y : nat] plus (S x) y --> S (plus x y). + + diff --git a/t/nat.eu b/t/nat.eu deleted file mode 100644 --- a/t/nat.eu +++ /dev/null @@ -1,17 +0,0 @@ -nat : Type. - -0 : nat. - -S : nat -> nat. - -1 : nat. - -[] 1 --> (S 0). - -plus : nat -> nat -> nat. -[x : nat] plus x 0 --> x. -[x : nat] plus 0 x --> x. -[x : nat, y : nat] plus x (S y) --> S (plus x y). -[x : nat, y : nat] plus (S x) y --> S (plus x y). - - diff --git a/t/peano.dk b/t/peano.dk new file mode 100644 --- /dev/null +++ b/t/peano.dk @@ -0,0 +1,66 @@ +nat : Type. + +nat_ : coc.Utype. + +[] nat --> coc.etype nat_. + +0 : nat. + +S : nat -> nat. + +nat_rec : t : coc.Utype + -> coc.etype t + -> (nat -> coc.etype t -> coc.etype t) + -> nat + -> coc.etype t. + +[ t : coc.Utype +, a : coc.etype t +, f : nat -> coc.etype t -> coc.etype t +] nat_rec t a f 0 --> a. + +[ t : coc.Utype +, a : coc.etype t +, f : nat -> coc.etype t -> coc.etype t +, n : nat +] nat_rec t a f (S n) --> f n (nat_rec t a f (S n)). + +plus : nat -> nat -> nat. + +[] plus --> x : nat => y : nat => nat_rec nat_ 0 (x : nat => y : nat => y) x. + +plus2 : nat -> nat -> nat. + +[x : nat] plus2 x 0 --> x. +[x : nat] plus2 0 x --> x. +[x : nat, y : nat] plus2 x (S y) --> S (plus2 x y). +[x : nat, y : nat] plus2 (S x) y --> S (plus2 x y). + +eq_S : x : nat + -> y : nat + -> logic.eq nat_ x y + -> logic.eq nat_ (S x) (S y). + +[] eq_S --> logic.f_equal nat_ nat_ S. + +eq_S2 : coc.etype (coc.dotpi1 nat_ (x : nat + => coc.dotpi1 nat_ (y : nat + => coc.dotpi1 (logic.eq_ nat_ x y) (h : logic.eq nat_ x y + => logic.eq_ nat_ (S x) (S y))))). + +[] eq_S2 --> eq_S. + + +pred : nat -> nat. + +[] pred --> nat_rec nat_ 0 (x:nat => nat => x). + +pred2 : nat -> nat. + +[] pred2 0 --> 0. + +[x : nat] pred2 (S x) --> x. + +pred_Sn : n : nat -> logic.eq nat_ n (pred (S n)). + +[] pred_Sn --> n : nat => logic.refl_equal nat_ n. diff --git a/t/peano.eu b/t/peano.eu deleted file mode 100644 --- a/t/peano.eu +++ /dev/null @@ -1,66 +0,0 @@ -nat : Type. - -nat_ : coc.Utype. - -[] nat --> coc.etype nat_. - -0 : nat. - -S : nat -> nat. - -nat_rec : t : coc.Utype - -> coc.etype t - -> (nat -> coc.etype t -> coc.etype t) - -> nat - -> coc.etype t. - -[ t : coc.Utype -, a : coc.etype t -, f : nat -> coc.etype t -> coc.etype t -] nat_rec t a f 0 --> a. - -[ t : coc.Utype -, a : coc.etype t -, f : nat -> coc.etype t -> coc.etype t -, n : nat -] nat_rec t a f (S n) --> f n (nat_rec t a f (S n)). - -plus : nat -> nat -> nat. - -[] plus --> x : nat => y : nat => nat_rec nat_ 0 (x : nat => y : nat => y) x. - -plus2 : nat -> nat -> nat. - -[x : nat] plus2 x 0 --> x. -[x : nat] plus2 0 x --> x. -[x : nat, y : nat] plus2 x (S y) --> S (plus2 x y). -[x : nat, y : nat] plus2 (S x) y --> S (plus2 x y). - -eq_S : x : nat - -> y : nat - -> logic.eq nat_ x y - -> logic.eq nat_ (S x) (S y). - -[] eq_S --> logic.f_equal nat_ nat_ S. - -eq_S2 : coc.etype (coc.dotpi1 nat_ (x : nat - => coc.dotpi1 nat_ (y : nat - => coc.dotpi1 (logic.eq_ nat_ x y) (h : logic.eq nat_ x y - => logic.eq_ nat_ (S x) (S y))))). - -[] eq_S2 --> eq_S. - - -pred : nat -> nat. - -[] pred --> nat_rec nat_ 0 (x:nat => nat => x). - -pred2 : nat -> nat. - -[] pred2 0 --> 0. - -[x : nat] pred2 (S x) --> x. - -pred_Sn : n : nat -> logic.eq nat_ n (pred (S n)). - -[] pred_Sn --> n : nat => logic.refl_equal nat_ n. diff --git a/t/plus.dk b/t/plus.dk new file mode 100644 --- /dev/null +++ b/t/plus.dk @@ -0,0 +1,7 @@ +P : nat.nat -> Type. + +y : P (nat.S nat.0). + +w : P (nat.S nat.0). + +[] w --> (x : P (nat.plus nat.0 (nat.S nat.0)) => x) y. diff --git a/t/plus.eu b/t/plus.eu deleted file mode 100644 --- a/t/plus.eu +++ /dev/null @@ -1,7 +0,0 @@ -P : nat.nat -> Type. - -y : P (nat.S nat.0). - -w : P (nat.S nat.0). - -[] w --> (x : P (nat.plus nat.0 (nat.S nat.0)) => x) y. diff --git a/t/sigma.eu b/t/sigma.eu deleted file mode 100644 --- a/t/sigma.eu +++ /dev/null @@ -1,41 +0,0 @@ -o : Type. -eps : o -> Type. - -sigma_ : A : o -> (eps A -> o) -> o. -exist_ : A : o -> P : (eps A -> o) -> x : eps A -> eps (P x) -> eps (sigma_ A P). - -fst : A : o -> P : (eps A -> o) -> eps (sigma_ A P) -> eps A. -[A : o, P : eps A -> o, w : eps A, pi : P w] -fst _ _ (eps (exist_ _ _ w pi)) --> w. - -snd : A : o -> P : (eps A -> o) -> s : eps (sigma_ A P) -> eps (P (fst A P s)). -[A : o, P : eps A -> o, w : eps A, pi : P w] -snd _ _ (eps (exist_ _ _ w pi)) --> pi. - - -;; test - -nat : Type. -nat_ : o. - -O : nat. -S : nat -> nat. - -plus : nat -> nat -> nat. -[n:nat,m:nat] plus (S n) m --> S (plus n m). -[n:nat,m:nat] plus O m --> m. - -eq : nat -> nat -> Type. -[n:nat,m:nat] eq (S n) (S m) --> eq n m. -ax: eq O O. - -eq_ : nat -> nat -> o. - -[x:nat,y:nat] eps (eq_ x y) --> eq x y. -[] eps nat_ --> nat. - -thm : n:nat -> eps (sigma_ nat_ (m:nat => eq_ (plus (S O) n) m)). -[] thm --> n:nat => exist_ nat_ (m:nat => eq_ (plus (S O) n) m) (S n) ax. - -verif : eq (fst nat_ (m:nat => eq_ (plus (S O) O) m) (thm O)) (S O). -[] verif --> ax. diff --git a/t/sigma2.eu b/t/sigma2.eu deleted file mode 100644 --- a/t/sigma2.eu +++ /dev/null @@ -1,31 +0,0 @@ -nat : Type. -0 : nat. -S : nat -> nat. - -plus:nat -> nat -> nat. -[x:nat] plus x 0 --> x. -[x:nat] (plus 0) x --> x. -[x:nat, y:nat] plus x (S y) --> S (plus x y). -[x:nat, y:nat] (plus (S x)) y --> S (plus x y). - -eqnat : nat -> nat -> Type. -ax: eqnat 0 0. -[n:nat, m:nat] eqnat (S n) (S m) --> eqnat n m. - -o : Type. -eps : o -> Type. - -_nat : o. -[] (eps _nat) --> nat. - -_eqnat : nat -> nat -> o. -[n:nat,m:nat] eps (_eqnat n m) --> eqnat n m. - -[x:o] eps x --> (eps x). - -sigma : a:o -> (eps a -> o) -> Type. - - -th : Type. -[] th --> sigma _nat (n:nat => _nat). - diff --git a/t/stt1.eu b/t/stt1.eu deleted file mode 100644 --- a/t/stt1.eu +++ /dev/null @@ -1,57 +0,0 @@ -; Simple Type Theory as a theory in predicate logic. -; -; Use reverse polish notation for names, eg: -; * i -> i becomes iia, -; * i -> i -> i becomes iiiaa -; * (i -> i) -> i becomes iiaia - -o : Type. -eps : o -> Type. - -i : Type. -iia : Type. -iiiaa : Type. -iiiiaaa : Type. -iiaia : Type. -iiiaaiiaiiaaa : Type. -iiaiiaa : Type. - -ooa : Type. -oooaa : Type. -ioaoa : Type. -iiaoaoa : Type. -iiiaaoaoa : Type. - -O : i. -S : iia. - -ap_iia : iia -> i -> i. -ap_iiiaa : iiiaa -> i -> iia. -ap_iiiaaiiaiiaaa : iiiaaiiaiiaaa -> iiiaa -> iiaiiaa. -ap_iiaiiaa : iiaiiaa -> iia -> iia. - -ap_ooa : ooa -> o -> o. -ap_oooaa : oooaa -> o -> ooa. - -one : i. -[] one --> ap_iia S O. - -imp : oooaa. -forall_i : ioaoa. -forall_iia : iiaoaoa. -forall_iiiaa : iiiaaoaoa. - -; S and K combinators. -S_iiiaaiiaiiaaa : iiiaaiiaiiaaa. -K_iiiaa : iiiaa. - -[ x : iiiaa -, y : iia -, z : i ] -ap_iia (ap_iiaiiaa (ap_iiiaaiiaiiaaa S_iiiaaiiaiiaaa x) y) z --> ap_iia (ap_iiiaa x z) (ap_iia y z). -[ x : i -, y : i ] -ap_iia (ap_iiiaa K_iiiaa x) y --> x. -[ x : o -, y : o ] -eps (ap_ooa (ap_oooaa imp x) y) --> eps x -> eps y. diff --git a/t/test1.eu b/t/test1.eu deleted file mode 100644 --- a/t/test1.eu +++ /dev/null @@ -1,8 +0,0 @@ -nat : Type. -0 : nat. -S : nat -> nat. -a : Type. -vec : nat -> Type. -vec' : n : nat -> vec n. -nil : vec 0. -cons : n : nat -> a -> vec n -> vec (S n). diff --git a/t/testcomplet.eu b/t/testcomplet.eu deleted file mode 100644 --- a/t/testcomplet.eu +++ /dev/null @@ -1,60 +0,0 @@ -Uset : Type. -Uprop : Type. -Utype : Type. - -eprop : x : Uprop -> Type. -eset : x : Uset -> Type. -etype : x : Utype -> Type. - -dotset : Utype. -dotprop : Utype. - -dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop. -dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset. -dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype. -dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop. -dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop. -dotpist : x : Uset -> y : (eset x -> Utype) -> Utype. -dotpits : x : Utype -> y : (etype x -> Uset) -> Uset. -dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset. -dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype. - - -[x:Uprop, y : eprop x -> Uprop] - eprop (dotpipp x y) --> w : eprop x -> eprop (y w). - -[x:Uset, y : eset x -> Uprop] - eprop (dotpisp x y) --> w : eset x -> eprop (y w). - -[x:Utype, y : etype x -> Uprop] - eprop (dotpitp x y) --> w : etype x -> eprop (y w). - -[x:Uprop, y : eprop x -> Uset] - eset (dotpips x y) --> w : eprop x -> eset (y w). - -[x:Utype, y : etype x -> Uset] - eset (dotpits x y) --> w : etype x -> eset (y w). - -[x:Uset, y : eset x -> Uset] - eset (dotpiss x y) --> w : eset x -> eset (y w). - -[x:Uset, y : eset x -> Utype] - etype (dotpist x y) --> w : eset x -> etype (y w). - -[x:Utype, y : etype x -> Utype] - etype (dotpitt x y) --> w : etype x -> etype (y w). - -[x:Uprop, y : eprop x -> Utype] - etype (dotpipt x y) --> w : eprop x -> etype (y w). - - -[] (etype dotset) --> Uset. -[] (etype dotprop) --> Uprop. -simple : (P : Uprop -> (_ : (eprop P) -> (eprop P) ) ) . -[] simple --> (P : (etype dotprop) => (H : (eprop P) => H) ) . -K : (P : Uprop -> (Q : Uprop -> (_ : (eprop P) -> (_ : (eprop Q) -> (eprop P) ) ) ) ) . -[] K --> (P : (etype dotprop) => (Q : (etype dotprop) => (H : (eprop P) => (H0 : (eprop Q) => ( (simple P) H) ) ) ) ) . -S : (P : Uprop -> (Q : Uprop -> (R : Uprop -> (_ : (_ : (eprop P) -> (_ : (eprop Q) -> (eprop R) ) ) -> (_ : (_ : (eprop P) -> (eprop Q) ) -> (_ : (eprop P) -> (eprop R) ) ) ) ) ) ) . -[] S --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => (H : (eprop ( (dotpipp P) (_ : (eprop P) => ( (dotpipp Q) (_ : (eprop Q) => R) ) ) ) ) => (H0 : (eprop ( (dotpipp P) (_ : (eprop P) => Q) ) ) => (H1 : (eprop P) => ( (H H1) (H0 H1) ) ) ) ) ) ) ) . -I : (P : Uprop -> (_ : (eprop P) -> (eprop P) ) ) . -[] I --> (P : (etype dotprop) => ( ( ( ( (S P) ( (dotpipp P) (_ : (eprop P) => P) ) ) P) ( (K P) ( (dotpipp P) (_ : (eprop P) => P) ) ) ) ( (K P) P) ) ) . diff --git a/t/vec.eu b/t/vec.eu deleted file mode 100644 --- a/t/vec.eu +++ /dev/null @@ -1,21 +0,0 @@ -nat : Type. -O : nat. -S : nat -> nat. - -o : Type. - -nat_ : nat -> o. - -eps : o -> Type. -[n : nat] eps (nat_ n) --> Nat n. - -Nat : nat -> Type. -[n : nat] Nat n --> P : (nat -> o) -> eps (P O) - -> (m : nat -> (Nat m) -> eps (P m) -> eps (P (S m))) - -> eps (P n). - -one : Nat (S O). -[p : nat -> o, z : eps (p O), s : m : nat -> (Nat m) -> eps (P m) -> eps (P (S m))] -one p z s --> s O z. - -;; suc : n : nat -> Nat n -> Nat (S n).