1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1A/wf3/ty3.ma".
19 include "basic_1A/app/defs.ma".
22 \forall (g: G).(\forall (c1: C).(ex C (\lambda (c2: C).(wf3 g c1 c2))))
24 \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(ex C (\lambda (c2:
25 C).(wf3 g c c2)))) (\lambda (n: nat).(ex_intro C (\lambda (c2: C).(wf3 g
26 (CSort n) c2)) (CSort n) (wf3_sort g n))) (\lambda (c: C).(\lambda (H: (ex C
27 (\lambda (c2: C).(wf3 g c c2)))).(\lambda (k: K).(\lambda (t: T).(let H0 \def
28 H in (ex_ind C (\lambda (c2: C).(wf3 g c c2)) (ex C (\lambda (c2: C).(wf3 g
29 (CHead c k t) c2))) (\lambda (x: C).(\lambda (H1: (wf3 g c x)).(K_ind
30 (\lambda (k0: K).(ex C (\lambda (c2: C).(wf3 g (CHead c k0 t) c2)))) (\lambda
31 (b: B).(let H_x \def (ty3_inference g c t) in (let H2 \def H_x in (or_ind (ex
32 T (\lambda (t2: T).(ty3 g c t t2))) (\forall (t2: T).((ty3 g c t t2) \to
33 False)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
34 (H3: (ex T (\lambda (t2: T).(ty3 g c t t2)))).(ex_ind T (\lambda (t2: T).(ty3
35 g c t t2)) (ex C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))) (\lambda
36 (x0: T).(\lambda (H4: (ty3 g c t x0)).(ex_intro C (\lambda (c2: C).(wf3 g
37 (CHead c (Bind b) t) c2)) (CHead x (Bind b) t) (wf3_bind g c x H1 t x0 H4
38 b)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c t t2) \to
39 False)))).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Bind b) t) c2))
40 (CHead x (Bind Void) (TSort O)) (wf3_void g c x H1 t H3 b))) H2)))) (\lambda
41 (f: F).(ex_intro C (\lambda (c2: C).(wf3 g (CHead c (Flat f) t) c2)) x
42 (wf3_flat g c x H1 t f))) k))) H0)))))) c1)).
45 \forall (g: G).(\forall (c: C).((wf3 g c c) \to (\forall (t1: T).(\forall
46 (t2: T).((ty3 g c t1 t2) \to (ty3 g (CSort (cbk c)) (app1 c t1) (app1 c
49 \lambda (g: G).(\lambda (c: C).(\lambda (H: (wf3 g c c)).(insert_eq C c
50 (\lambda (c0: C).(wf3 g c0 c)) (\lambda (c0: C).(\forall (t1: T).(\forall
51 (t2: T).((ty3 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0
52 t2)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y c)).(wf3_ind g (\lambda (c0:
53 C).(\lambda (c1: C).((eq C c0 c1) \to (\forall (t1: T).(\forall (t2: T).((ty3
54 g c0 t1 t2) \to (ty3 g (CSort (cbk c0)) (app1 c0 t1) (app1 c0 t2))))))))
55 (\lambda (m: nat).(\lambda (_: (eq C (CSort m) (CSort m))).(\lambda (t1:
56 T).(\lambda (t2: T).(\lambda (H2: (ty3 g (CSort m) t1 t2)).H2))))) (\lambda
57 (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C
58 c1 c2) \to (\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g
59 (CSort (cbk c1)) (app1 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda
60 (t: T).(\lambda (H3: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C
61 (CHead c1 (Bind b) u) (CHead c2 (Bind b) u))).(\lambda (t1: T).(\lambda (t2:
62 T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def (f_equal C
63 C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
64 \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind b) u) H4) in (let H7
65 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3:
66 T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1
67 t3) (app1 c1 t4))))))) H2 c1 H6) in (let H8 \def (eq_ind_r C c2 (\lambda (c0:
68 C).(wf3 g c1 c0)) H1 c1 H6) in (ex_ind T (\lambda (t0: T).(ty3 g (CHead c1
69 (Bind b) u) t2 t0)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind b) u t1))
70 (app1 c1 (THead (Bind b) u t2))) (\lambda (x: T).(\lambda (_: (ty3 g (CHead
71 c1 (Bind b) u) t2 x)).(H7 (refl_equal C c1) (THead (Bind b) u t1) (THead
72 (Bind b) u t2) (ty3_bind g c1 u t H3 b t1 t2 H5)))) (ty3_correct g (CHead c1
73 (Bind b) u) t1 t2 H5))))))))))))))))) (\lambda (c1: C).(\lambda (c2:
74 C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall
75 (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1
76 c1 t1) (app1 c1 t2)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t:
77 T).((ty3 g c1 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead
78 c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)))).(\lambda (t1: T).(\lambda
79 (t2: T).(\lambda (H5: (ty3 g (CHead c1 (Bind b) u) t1 t2)).(let H6 \def
80 (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow c1 | (CHead
81 c0 _ _) \Rightarrow c0])) (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort
82 O)) H4) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e with [(CSort
83 _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k with [(Bind b0)
84 \Rightarrow b0 | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2
85 (Bind Void) (TSort O)) H4) in ((let H8 \def (f_equal C T (\lambda (e:
86 C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
87 (CHead c1 (Bind b) u) (CHead c2 (Bind Void) (TSort O)) H4) in (\lambda (H9:
88 (eq B b Void)).(\lambda (H10: (eq C c1 c2)).(let H11 \def (eq_ind B b
89 (\lambda (b0: B).(ty3 g (CHead c1 (Bind b0) u) t1 t2)) H5 Void H9) in
90 (eq_ind_r B Void (\lambda (b0: B).(ty3 g (CSort (cbk (CHead c1 (Bind b0) u)))
91 (app1 (CHead c1 (Bind b0) u) t1) (app1 (CHead c1 (Bind b0) u) t2))) (let H12
92 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c1 (Bind Void) t) t1 t2)) H11
93 (TSort O) H8) in (let H13 \def (eq_ind T u (\lambda (t: T).(\forall (t0:
94 T).((ty3 g c1 t t0) \to False))) H3 (TSort O) H8) in (eq_ind_r T (TSort O)
95 (\lambda (t: T).(ty3 g (CSort (cbk (CHead c1 (Bind Void) t))) (app1 (CHead c1
96 (Bind Void) t) t1) (app1 (CHead c1 (Bind Void) t) t2))) (let H14 \def
97 (eq_ind_r C c2 (\lambda (c0: C).((eq C c1 c0) \to (\forall (t3: T).(\forall
98 (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort (cbk c1)) (app1 c1 t3) (app1 c1
99 t4))))))) H2 c1 H10) in (let H15 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g
100 c1 c0)) H1 c1 H10) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c1 (Bind Void)
101 (TSort O)) t2 t)) (ty3 g (CSort (cbk c1)) (app1 c1 (THead (Bind Void) (TSort
102 O) t1)) (app1 c1 (THead (Bind Void) (TSort O) t2))) (\lambda (x: T).(\lambda
103 (_: (ty3 g (CHead c1 (Bind Void) (TSort O)) t2 x)).(H14 (refl_equal C c1)
104 (THead (Bind Void) (TSort O) t1) (THead (Bind Void) (TSort O) t2) (ty3_bind g
105 c1 (TSort O) (TSort (next g O)) (ty3_sort g c1 O) Void t1 t2 H12))))
106 (ty3_correct g (CHead c1 (Bind Void) (TSort O)) t1 t2 H12)))) u H8))) b
107 H9))))) H7)) H6))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
108 (wf3 g c1 c2)).(\lambda (H2: (((eq C c1 c2) \to (\forall (t1: T).(\forall
109 (t2: T).((ty3 g c1 t1 t2) \to (ty3 g (CSort (cbk c1)) (app1 c1 t1) (app1 c1
110 t2)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1
111 (Flat f) u) c2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead
112 c1 (Flat f) u) t1 t2)).(let H5 \def (f_equal C C (\lambda (e: C).e) (CHead c1
113 (Flat f) u) c2 H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c0: C).((eq C c1
114 c0) \to (\forall (t3: T).(\forall (t4: T).((ty3 g c1 t3 t4) \to (ty3 g (CSort
115 (cbk c1)) (app1 c1 t3) (app1 c1 t4))))))) H2 (CHead c1 (Flat f) u) H5) in
116 (let H7 \def (eq_ind_r C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 (CHead c1
117 (Flat f) u) H5) in (let H_x \def (wf3_gen_head2 g c1 c1 u (Flat f) H7) in
118 (let H8 \def H_x in (ex_ind B (\lambda (b: B).(eq K (Flat f) (Bind b))) (ty3
119 g (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
120 t2))) (\lambda (x: B).(\lambda (H9: (eq K (Flat f) (Bind x))).(let H10 \def
121 (eq_ind K (Flat f) (\lambda (ee: K).(match ee with [(Bind _) \Rightarrow
122 False | (Flat _) \Rightarrow True])) I (Bind x) H9) in (False_ind (ty3 g
123 (CSort (cbk c1)) (app1 c1 (THead (Flat f) u t1)) (app1 c1 (THead (Flat f) u
124 t2))) H10)))) H8)))))))))))))))) y c H0))) H))).
127 \forall (g: G).(\forall (c1: C).(\forall (c2: C).((wf3 g c1 c2) \to (wf3 g
130 \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wf3 g c1
131 c2)).(wf3_ind g (\lambda (_: C).(\lambda (c0: C).(wf3 g c0 c0))) (\lambda (m:
132 nat).(wf3_sort g m)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wf3 g
133 c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (t: T).(\lambda
134 (H2: (ty3 g c3 u t)).(\lambda (b: B).(wf3_bind g c4 c4 H1 u t (wf3_ty3_conf g
135 c3 u t H2 c4 H0) b))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
136 (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4 c4)).(\lambda (u: T).(\lambda (_:
137 ((\forall (t: T).((ty3 g c3 u t) \to False)))).(\lambda (_: B).(wf3_bind g c4
138 c4 H1 (TSort O) (TSort (next g O)) (ty3_sort g c4 O) Void)))))))) (\lambda
139 (c3: C).(\lambda (c4: C).(\lambda (_: (wf3 g c3 c4)).(\lambda (H1: (wf3 g c4
140 c4)).(\lambda (_: T).(\lambda (_: F).H1)))))) c1 c2 H)))).
143 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (u: T).((ty3 g c1 t
144 u) \to (ex2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
147 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
148 (ty3 g c1 t u)).(let H_x \def (wf3_total g c1) in (let H0 \def H_x in (ex_ind
149 C (\lambda (c2: C).(wf3 g c1 c2)) (ex2 C (\lambda (c2: C).(wf3 g c1 c2))
150 (\lambda (c2: C).(ty3 g c2 t u))) (\lambda (x: C).(\lambda (H1: (wf3 g c1
151 x)).(ex_intro2 C (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (c2: C).(ty3 g c2 t
152 u)) x H1 (wf3_ty3_conf g c1 t u H x H1)))) H0))))))).