include "basic_1/app/defs.ma".
-theorem wf3_total:
+lemma wf3_total:
\forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
\def
- \lambda (g: G).(\lambda (c1: C).(let TMP_2 \def (\lambda (c: C).(let TMP_1
-\def (\lambda (c2: C).(wf3 g c c2)) in (ex C TMP_1))) in (let TMP_7 \def
-(\lambda (n: nat).(let TMP_4 \def (\lambda (c2: C).(let TMP_3 \def (CSort n)
-in (wf3 g TMP_3 c2))) in (let TMP_5 \def (CSort n) in (let TMP_6 \def
-(wf3_sort g n) in (ex_intro C TMP_4 TMP_5 TMP_6))))) in (let TMP_50 \def
-(\lambda (c: C).(\lambda (H: (ex C (\lambda (c2: C).(wf3 g c c2)))).(\lambda
-(k: K).(\lambda (t: T).(let H0 \def H in (let TMP_8 \def (\lambda (c2:
-C).(wf3 g c c2)) in (let TMP_10 \def (\lambda (c2: C).(let TMP_9 \def (CHead
-c k t) in (wf3 g TMP_9 c2))) in (let TMP_11 \def (ex C TMP_10) in (let TMP_49
-\def (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(let TMP_14 \def (\lambda
-(k0: K).(let TMP_13 \def (\lambda (c2: C).(let TMP_12 \def (CHead c k0 t) in
-(wf3 g TMP_12 c2))) in (ex C TMP_13))) in (let TMP_43 \def (\lambda (b:
-B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (let TMP_15
-\def (\lambda (t2: T).(ty3 g c t t2)) in (let TMP_16 \def (ex T TMP_15) in
-(let TMP_17 \def (\forall (t2: T).((ty3 g c t t2) \to False)) in (let TMP_20
-\def (\lambda (c2: C).(let TMP_18 \def (Bind b) in (let TMP_19 \def (CHead c
-TMP_18 t) in (wf3 g TMP_19 c2)))) in (let TMP_21 \def (ex C TMP_20) in (let
-TMP_34 \def (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(let
-TMP_22 \def (\lambda (t2: T).(ty3 g c t t2)) in (let TMP_25 \def (\lambda
-(c2: C).(let TMP_23 \def (Bind b) in (let TMP_24 \def (CHead c TMP_23 t) in
-(wf3 g TMP_24 c2)))) in (let TMP_26 \def (ex C TMP_25) in (let TMP_33 \def
-(\lambda (x0: T).(\lambda (H4: (ty3 g c t x0)).(let TMP_29 \def (\lambda (c2:
-C).(let TMP_27 \def (Bind b) in (let TMP_28 \def (CHead c TMP_27 t) in (wf3 g
-TMP_28 c2)))) in (let TMP_30 \def (Bind b) in (let TMP_31 \def (CHead x
-TMP_30 t) in (let TMP_32 \def (wf3_bind g c x H1 t x0 H4 b) in (ex_intro C
-TMP_29 TMP_31 TMP_32))))))) in (ex_ind T TMP_22 TMP_26 TMP_33 H3)))))) in
-(let TMP_42 \def (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
-False)))).(let TMP_37 \def (\lambda (c2: C).(let TMP_35 \def (Bind b) in (let
-TMP_36 \def (CHead c TMP_35 t) in (wf3 g TMP_36 c2)))) in (let TMP_38 \def
-(Bind Void) in (let TMP_39 \def (TSort O) in (let TMP_40 \def (CHead x TMP_38
-TMP_39) in (let TMP_41 \def (wf3_void g c x H1 t H3 b) in (ex_intro C TMP_37
-TMP_40 TMP_41))))))) in (or_ind TMP_16 TMP_17 TMP_21 TMP_34 TMP_42
-H2))))))))))) in (let TMP_48 \def (\lambda (f: F).(let TMP_46 \def (\lambda
-(c2: C).(let TMP_44 \def (Flat f) in (let TMP_45 \def (CHead c TMP_44 t) in
-(wf3 g TMP_45 c2)))) in (let TMP_47 \def (wf3_flat g c x H1 t f) in (ex_intro
-C TMP_46 x TMP_47)))) in (K_ind TMP_14 TMP_43 TMP_48 k)))))) in (ex_ind C
-TMP_8 TMP_11 TMP_49 H0)))))))))) in (C_ind TMP_2 TMP_7 TMP_50 c1))))).
+ \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
+C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
+(CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
+(\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
+H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
+(CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
+(\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
+(b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
+T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
+False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
+g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
+(x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
+(CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
+b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
+False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
+(CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
+(f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
+(wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
-theorem ty3_shift1:
+lemma ty3_shift1:
\forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
(t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
t2)))))))
\def
- \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(let TMP_1 \def
-(\lambda (c0: C).(wf3 g c0 c)) in (let TMP_6 \def (\lambda (c0: C).(\forall
-(t1: T).(\forall (t2: T).((ty3 g c0 t1 t2) \to (let TMP_2 \def (cbk c0) in
-(let TMP_3 \def (CSort TMP_2) in (let TMP_4 \def (app1 c0 t1) in (let TMP_5
-\def (app1 c0 t2) in (ty3 g TMP_3 TMP_4 TMP_5))))))))) in (let TMP_180 \def
-(\lambda (y: C).(\lambda (H0: (wf3 g y c)).(let TMP_11 \def (\lambda (c0:
+ \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
+(\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
+(t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
+t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
-g c0 t1 t2) \to (let TMP_7 \def (cbk c0) in (let TMP_8 \def (CSort TMP_7) in
-(let TMP_9 \def (app1 c0 t1) in (let TMP_10 \def (app1 c0 t2) in (ty3 g TMP_8
-TMP_9 TMP_10))))))))))) in (let TMP_12 \def (\lambda (m: nat).(\lambda (_:
-(eq C (CSort m) (CSort m))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2:
-(ty3 g (CSort m) t1 t2)).H2))))) in (let TMP_46 \def (\lambda (c1:
-C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2)
-\to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort
-(cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda (t:
-T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead
-c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let TMP_13 \def
-(\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) in (let TMP_14 \def (Bind b) in (let TMP_15 \def (CHead c1
-TMP_14 u) in (let TMP_16 \def (Bind b) in (let TMP_17 \def (CHead c2 TMP_16
-u) in (let H6 \def (f_equal C C TMP_13 TMP_15 TMP_17 H4) in (let TMP_22 \def
-(\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g
-c1 t3 t4) \to (let TMP_18 \def (cbk c1) in (let TMP_19 \def (CSort TMP_18) in
-(let TMP_20 \def (app1 c1 t3) in (let TMP_21 \def (app1 c1 t4) in (ty3 g
-TMP_19 TMP_20 TMP_21)))))))))) in (let H7 \def (eq_ind_r C c2 TMP_22 H2 c1
-H6) in (let TMP_23 \def (\lambda (c0: C).(wf3 g c1 c0)) in (let H8 \def
-(eq_ind_r C c2 TMP_23 H1 c1 H6) in (let TMP_26 \def (\lambda (t0: T).(let
-TMP_24 \def (Bind b) in (let TMP_25 \def (CHead c1 TMP_24 u) in (ty3 g TMP_25
-t2 t0)))) in (let TMP_27 \def (cbk c1) in (let TMP_28 \def (CSort TMP_27) in
-(let TMP_29 \def (Bind b) in (let TMP_30 \def (THead TMP_29 u t1) in (let
-TMP_31 \def (app1 c1 TMP_30) in (let TMP_32 \def (Bind b) in (let TMP_33 \def
-(THead TMP_32 u t2) in (let TMP_34 \def (app1 c1 TMP_33) in (let TMP_35 \def
-(ty3 g TMP_28 TMP_31 TMP_34) in (let TMP_42 \def (\lambda (x: T).(\lambda (_:
-(ty3 g (CHead c1 (Bind b) u) t2 x)).(let TMP_36 \def (refl_equal C c1) in
-(let TMP_37 \def (Bind b) in (let TMP_38 \def (THead TMP_37 u t1) in (let
-TMP_39 \def (Bind b) in (let TMP_40 \def (THead TMP_39 u t2) in (let TMP_41
-\def (ty3_bind g c1 u t H3 b t1 t2 H5) in (H7 TMP_36 TMP_38 TMP_40
-TMP_41))))))))) in (let TMP_43 \def (Bind b) in (let TMP_44 \def (CHead c1
-TMP_43 u) in (let TMP_45 \def (ty3_correct g TMP_44 t1 t2 H5) in (ex_ind T
-TMP_26 TMP_35 TMP_42 TMP_45))))))))))))))))))))))))))))))))))))) in (let
-TMP_139 \def (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
-c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3
-g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
-t2)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g c1 u t) \to
-False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CHead c2
-(Bind Void) (TSort O)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H5: (ty3
-g (CHead c1 (Bind b) u) t1 t2)).(let TMP_47 \def (\lambda (e: C).(match e
-with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) in (let
-TMP_48 \def (Bind b) in (let TMP_49 \def (CHead c1 TMP_48 u) in (let TMP_50
-\def (Bind Void) in (let TMP_51 \def (TSort O) in (let TMP_52 \def (CHead c2
-TMP_50 TMP_51) in (let H6 \def (f_equal C C TMP_47 TMP_49 TMP_52 H4) in (let
-TMP_53 \def (\lambda (e: C).(match e with [(CSort _) \Rightarrow b | (CHead _
-k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow b])])) in (let TMP_54 \def (Bind b) in (let TMP_55 \def (CHead c1
-TMP_54 u) in (let TMP_56 \def (Bind Void) in (let TMP_57 \def (TSort O) in
-(let TMP_58 \def (CHead c2 TMP_56 TMP_57) in (let H7 \def (f_equal C B TMP_53
-TMP_55 TMP_58 H4) in (let TMP_59 \def (\lambda (e: C).(match e with [(CSort
-_) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) in (let TMP_60 \def (Bind
-b) in (let TMP_61 \def (CHead c1 TMP_60 u) in (let TMP_62 \def (Bind Void) in
-(let TMP_63 \def (TSort O) in (let TMP_64 \def (CHead c2 TMP_62 TMP_63) in
-(let H8 \def (f_equal C T TMP_59 TMP_61 TMP_64 H4) in (let TMP_137 \def
-(\lambda (H9: (eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let TMP_67 \def
-(\lambda (b0: B).(let TMP_65 \def (Bind b0) in (let TMP_66 \def (CHead c1
-TMP_65 u) in (ty3 g TMP_66 t1 t2)))) in (let H11 \def (eq_ind B b TMP_67 H5
-Void H9) in (let TMP_78 \def (\lambda (b0: B).(let TMP_68 \def (Bind b0) in
-(let TMP_69 \def (CHead c1 TMP_68 u) in (let TMP_70 \def (cbk TMP_69) in (let
-TMP_71 \def (CSort TMP_70) in (let TMP_72 \def (Bind b0) in (let TMP_73 \def
-(CHead c1 TMP_72 u) in (let TMP_74 \def (app1 TMP_73 t1) in (let TMP_75 \def
-(Bind b0) in (let TMP_76 \def (CHead c1 TMP_75 u) in (let TMP_77 \def (app1
-TMP_76 t2) in (ty3 g TMP_71 TMP_74 TMP_77)))))))))))) in (let TMP_81 \def
-(\lambda (t: T).(let TMP_79 \def (Bind Void) in (let TMP_80 \def (CHead c1
-TMP_79 t) in (ty3 g TMP_80 t1 t2)))) in (let TMP_82 \def (TSort O) in (let
-H12 \def (eq_ind T u TMP_81 H11 TMP_82 H8) in (let TMP_83 \def (\lambda (t:
-T).(\forall (t0: T).((ty3 g c1 t t0) \to False))) in (let TMP_84 \def (TSort
-O) in (let H13 \def (eq_ind T u TMP_83 H3 TMP_84 H8) in (let TMP_85 \def
-(TSort O) in (let TMP_96 \def (\lambda (t: T).(let TMP_86 \def (Bind Void) in
-(let TMP_87 \def (CHead c1 TMP_86 t) in (let TMP_88 \def (cbk TMP_87) in (let
-TMP_89 \def (CSort TMP_88) in (let TMP_90 \def (Bind Void) in (let TMP_91
-\def (CHead c1 TMP_90 t) in (let TMP_92 \def (app1 TMP_91 t1) in (let TMP_93
-\def (Bind Void) in (let TMP_94 \def (CHead c1 TMP_93 t) in (let TMP_95 \def
-(app1 TMP_94 t2) in (ty3 g TMP_89 TMP_92 TMP_95)))))))))))) in (let TMP_101
-\def (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4:
-T).((ty3 g c1 t3 t4) \to (let TMP_97 \def (cbk c1) in (let TMP_98 \def (CSort
-TMP_97) in (let TMP_99 \def (app1 c1 t3) in (let TMP_100 \def (app1 c1 t4) in
-(ty3 g TMP_98 TMP_99 TMP_100)))))))))) in (let H14 \def (eq_ind_r C c2
-TMP_101 H2 c1 H10) in (let TMP_102 \def (\lambda (c0: C).(wf3 g c1 c0)) in
-(let H15 \def (eq_ind_r C c2 TMP_102 H1 c1 H10) in (let TMP_106 \def (\lambda
-(t: T).(let TMP_103 \def (Bind Void) in (let TMP_104 \def (TSort O) in (let
-TMP_105 \def (CHead c1 TMP_103 TMP_104) in (ty3 g TMP_105 t2 t))))) in (let
-TMP_107 \def (cbk c1) in (let TMP_108 \def (CSort TMP_107) in (let TMP_109
-\def (Bind Void) in (let TMP_110 \def (TSort O) in (let TMP_111 \def (THead
-TMP_109 TMP_110 t1) in (let TMP_112 \def (app1 c1 TMP_111) in (let TMP_113
-\def (Bind Void) in (let TMP_114 \def (TSort O) in (let TMP_115 \def (THead
-TMP_113 TMP_114 t2) in (let TMP_116 \def (app1 c1 TMP_115) in (let TMP_117
-\def (ty3 g TMP_108 TMP_112 TMP_116) in (let TMP_130 \def (\lambda (x:
-T).(\lambda (_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(let TMP_118
-\def (refl_equal C c1) in (let TMP_119 \def (Bind Void) in (let TMP_120 \def
-(TSort O) in (let TMP_121 \def (THead TMP_119 TMP_120 t1) in (let TMP_122
-\def (Bind Void) in (let TMP_123 \def (TSort O) in (let TMP_124 \def (THead
-TMP_122 TMP_123 t2) in (let TMP_125 \def (TSort O) in (let TMP_126 \def (next
-g O) in (let TMP_127 \def (TSort TMP_126) in (let TMP_128 \def (ty3_sort g c1
-O) in (let TMP_129 \def (ty3_bind g c1 TMP_125 TMP_127 TMP_128 Void t1 t2
-H12) in (H14 TMP_118 TMP_121 TMP_124 TMP_129))))))))))))))) in (let TMP_131
-\def (Bind Void) in (let TMP_132 \def (TSort O) in (let TMP_133 \def (CHead
-c1 TMP_131 TMP_132) in (let TMP_134 \def (ty3_correct g TMP_133 t1 t2 H12) in
-(let TMP_135 \def (ex_ind T TMP_106 TMP_117 TMP_130 TMP_134) in (let TMP_136
-\def (eq_ind_r T TMP_85 TMP_96 TMP_135 u H8) in (eq_ind_r B Void TMP_78
-TMP_136 b H9))))))))))))))))))))))))))))))))))))) in (let TMP_138 \def
-(TMP_137 H7) in (TMP_138 H6))))))))))))))))))))))))))))))))))) in (let
-TMP_179 \def (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
-c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3
-g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
+g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
+(\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
+c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
+(CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
+(t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
+(CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
+\Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind b) u) H4) in (let H7
+\def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3:
+T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1
+t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 \def (eq_ind_r C c2 (\lambda (c0:
+C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T (\lambda (t0: T).(ty3 g (CHead c1
+(Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind b) u t1))
+(app1 c1 (THead (Bind b) u t2))) (\lambda (x: T).(\lambda (_: (ty3 g (CHead
+c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) (THead (Bind b) u t1) (THead
+(Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 H5)))) (ty3_correct g (CHead c1
+(Bind b) u) t1 t2 H5))))))))))))))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall
+(t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1
+c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t:
+T).((ty3 g c1 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead
+c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead
+c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
+O)) H4) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort
+_) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2
+(Bind Void) (TSort O)) H4) in ((let H8 \def (f_equal C T (\lambda (e:
+C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
+(eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
+(\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
+(eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
+(app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
+\def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
+(TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
+T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
+(\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
+(Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
+(eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
+(t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
+t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
+c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
+(TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
+O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
+(_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
+(THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
+c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
+(ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
+H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
+(wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
+(t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
(Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
-c1 (Flat f) u) t1 t2)).(let TMP_140 \def (\lambda (e: C).e) in (let TMP_141
-\def (Flat f) in (let TMP_142 \def (CHead c1 TMP_141 u) in (let H5 \def
-(f_equal C C TMP_140 TMP_142 c2 H3) in (let TMP_147 \def (\lambda (c0:
-C).((eq C c1 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to
-(let TMP_143 \def (cbk c1) in (let TMP_144 \def (CSort TMP_143) in (let
-TMP_145 \def (app1 c1 t3) in (let TMP_146 \def (app1 c1 t4) in (ty3 g TMP_144
-TMP_145 TMP_146)))))))))) in (let TMP_148 \def (Flat f) in (let TMP_149 \def
-(CHead c1 TMP_148 u) in (let H6 \def (eq_ind_r C c2 TMP_147 H2 TMP_149 H5) in
-(let TMP_150 \def (\lambda (c0: C).(wf3 g c1 c0)) in (let TMP_151 \def (Flat
-f) in (let TMP_152 \def (CHead c1 TMP_151 u) in (let H7 \def (eq_ind_r C c2
-TMP_150 H1 TMP_152 H5) in (let TMP_153 \def (Flat f) in (let H_x \def
-(wf3_gen_head2 g c1 c1 u TMP_153 H7) in (let H8 \def H_x in (let TMP_156 \def
-(\lambda (b: B).(let TMP_154 \def (Flat f) in (let TMP_155 \def (Bind b) in
-(eq K TMP_154 TMP_155)))) in (let TMP_157 \def (cbk c1) in (let TMP_158 \def
-(CSort TMP_157) in (let TMP_159 \def (Flat f) in (let TMP_160 \def (THead
-TMP_159 u t1) in (let TMP_161 \def (app1 c1 TMP_160) in (let TMP_162 \def
-(Flat f) in (let TMP_163 \def (THead TMP_162 u t2) in (let TMP_164 \def (app1
-c1 TMP_163) in (let TMP_165 \def (ty3 g TMP_158 TMP_161 TMP_164) in (let
-TMP_178 \def (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let
-TMP_166 \def (Flat f) in (let TMP_167 \def (\lambda (ee: K).(match ee with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) in (let TMP_168
-\def (Bind x) in (let H10 \def (eq_ind K TMP_166 TMP_167 I TMP_168 H9) in
-(let TMP_169 \def (cbk c1) in (let TMP_170 \def (CSort TMP_169) in (let
-TMP_171 \def (Flat f) in (let TMP_172 \def (THead TMP_171 u t1) in (let
-TMP_173 \def (app1 c1 TMP_172) in (let TMP_174 \def (Flat f) in (let TMP_175
-\def (THead TMP_174 u t2) in (let TMP_176 \def (app1 c1 TMP_175) in (let
-TMP_177 \def (ty3 g TMP_170 TMP_173 TMP_176) in (False_ind TMP_177
-H10)))))))))))))))) in (ex_ind B TMP_156 TMP_165 TMP_178
-H8))))))))))))))))))))))))))))))))))))) in (wf3_ind g TMP_11 TMP_12 TMP_46
-TMP_139 TMP_179 y c H0)))))))) in (insert_eq C c TMP_1 TMP_6 TMP_180 H)))))).
+c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
+(Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
+c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
+(cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
+(let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
+(Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
+(let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
+g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
+t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
+(eq_ind K (Flat f) (\lambda (ee: K).(match ee with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])) I (Bind x) H9) in (False_ind (ty3 g
+(CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
+t2))) H10)))) H8)))))))))))))))) y c H0))) H))).
-theorem wf3_idem:
+lemma wf3_idem:
\forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
c2 c2))))
\def
\lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
-c2)).(let TMP_1 \def (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) in (let
-TMP_2 \def (\lambda (m: nat).(wf3_sort g m)) in (let TMP_4 \def (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (H0: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
-c4)).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c3 u t)).(\lambda
-(b: B).(let TMP_3 \def (wf3_ty3_conf g c3 u t H2 c4 H0) in (wf3_bind g c4 c4
-H1 u t TMP_3 b)))))))))) in (let TMP_9 \def (\lambda (c3: C).(\lambda (c4:
-C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u:
-T).(\lambda (_: ((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_:
-B).(let TMP_5 \def (TSort O) in (let TMP_6 \def (next g O) in (let TMP_7 \def
-(TSort TMP_6) in (let TMP_8 \def (ty3_sort g c4 O) in (wf3_bind g c4 c4 H1
-TMP_5 TMP_7 TMP_8 Void)))))))))))) in (let TMP_10 \def (\lambda (c3:
-C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
-c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) in (wf3_ind g TMP_1 TMP_2 TMP_4
-TMP_9 TMP_10 c1 c2 H))))))))).
+c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
+nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
+c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
+(H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
+c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
+(wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
+((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
+c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
+(c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
+c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
-theorem wf3_ty3:
+lemma wf3_ty3:
\forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
u)))))))
\def
\lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
-(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (let
-TMP_1 \def (\lambda (c2: C).(wf3 g c1 c2)) in (let TMP_2 \def (\lambda (c2:
-C).(wf3 g c1 c2)) in (let TMP_3 \def (\lambda (c2: C).(ty3 g c2 t u)) in (let
-TMP_4 \def (ex2 C TMP_2 TMP_3) in (let TMP_8 \def (\lambda (x: C).(\lambda
-(H1: (wf3 g c1 x)).(let TMP_5 \def (\lambda (c2: C).(wf3 g c1 c2)) in (let
-TMP_6 \def (\lambda (c2: C).(ty3 g c2 t u)) in (let TMP_7 \def (wf3_ty3_conf
-g c1 t u H x H1) in (ex_intro2 C TMP_5 TMP_6 x H1 TMP_7)))))) in (ex_ind C
-TMP_1 TMP_4 TMP_8 H0)))))))))))).
+(ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
+C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
+(\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
+x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
+u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).
projects / helm.git / blobdiff