dedukti 1.0.0 → 1.0.1
raw patch · 37 files changed
+449/−1469 lines, 37 filesPVP ok
version bump matches the API change (PVP)
API changes (from Hackage documentation)
Files
- dedukti.cabal +13/−2
- t/Coq1univ.eu +0/−70
- t/Logic.eu +0/−290
- t/Logicavecprelude.eu +0/−156
- t/bug.dk +5/−0
- t/bug.eu +0/−5
- t/coc.dk +28/−0
- t/coc.eu +0/−28
- t/conj.eu +0/−3
- t/coq/Datatypes.dk +180/−0
- t/coqlogicprel.eu +0/−156
- t/delta1.dk +2/−0
- t/delta1.eu +0/−2
- t/delta2.dk +7/−0
- t/delta2.eu +0/−7
- t/exemple.dk +9/−0
- t/exemple.eu +0/−9
- t/f.dk +17/−0
- t/f.eu +0/−17
- t/fold/arith.dk +49/−0
- t/gros.eu +0/−360
- t/linearity.eu +0/−3
- t/logic.dk +49/−0
- t/logic.eu +0/−49
- t/loop.eu +0/−4
- t/nat.dk +17/−0
- t/nat.eu +0/−17
- t/peano.dk +66/−0
- t/peano.eu +0/−66
- t/plus.dk +7/−0
- t/plus.eu +0/−7
- t/sigma.eu +0/−41
- t/sigma2.eu +0/−31
- t/stt1.eu +0/−57
- t/test1.eu +0/−8
- t/testcomplet.eu +0/−60
- t/vec.eu +0/−21
dedukti.cabal view
@@ -1,5 +1,5 @@ name: dedukti-version: 1.0.0+version: 1.0.1 author: Mathieu Boespflug maintainer: Mathieu Boespflug <mboes@lix.polytechnique.fr> 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
− t/Coq1univ.eu
@@ -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-
− t/Logic.eu
@@ -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
− t/Logicavecprelude.eu
@@ -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) ) ) ) ) .
+ t/bug.dk view
@@ -0,0 +1,5 @@+nat : Type.++x : nat.++y : x.
− t/bug.eu
@@ -1,5 +0,0 @@-nat : Type.--x : nat.--y : x.
+ t/coc.dk view
@@ -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.
− t/coc.eu
@@ -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.
− t/conj.eu
@@ -1,3 +0,0 @@-o : Type.-conj : o -> o -> Type.-[x : o] conj x x --> conj x x.
+ t/coq/Datatypes.dk view
@@ -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. ++
− t/coqlogicprel.eu
@@ -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) ) ) ) ) .
+ t/delta1.dk view
@@ -0,0 +1,2 @@+delta : a : Type -> (b : Type -> b -> b) -> a -> a.+[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a).
− t/delta1.eu
@@ -1,2 +0,0 @@-delta : a : Type -> (b : Type -> b -> b) -> a -> a.-[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a).
+ t/delta2.dk view
@@ -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.
− t/delta2.eu
@@ -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.
+ t/exemple.dk view
@@ -0,0 +1,9 @@+T : Type.++U : Type.++[] U --> T -> T.++app : f : (T -> T) -> T -> T.++[] app --> f : U => x : T => f x.
− t/exemple.eu
@@ -1,9 +0,0 @@-T : Type.--U : Type.--[] U --> T -> T.--app : f : (T -> T) -> T -> T.--[] app --> f : U => x : T => f x.
+ t/f.dk view
@@ -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.
− t/f.eu
@@ -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.
+ t/fold/arith.dk view
@@ -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.+
− t/gros.eu
@@ -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
− t/linearity.eu
@@ -1,3 +0,0 @@-Nat: Type.-Q : (Nat -> Nat) -> Nat -> Nat -> Type.-[a:Nat, f : Nat -> Nat] Q f a (f a) --> Nat.
+ t/logic.dk view
@@ -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.
− t/logic.eu
@@ -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.
− t/loop.eu
@@ -1,4 +0,0 @@-A : Type.-[] A --> A -> A.-t : A.-[] t --> x : A => x.
+ t/nat.dk view
@@ -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).++
− t/nat.eu
@@ -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).--
+ t/peano.dk view
@@ -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.
− t/peano.eu
@@ -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.
+ t/plus.dk view
@@ -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.
− t/plus.eu
@@ -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.
− t/sigma.eu
@@ -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.
− t/sigma2.eu
@@ -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).-
− t/stt1.eu
@@ -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.
− t/test1.eu
@@ -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).
− t/testcomplet.eu
@@ -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) ) ) .
− t/vec.eu
@@ -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).