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 "LambdaDelta-1/aprem/defs.ma".
19 include "LambdaDelta-1/leq/defs.ma".
22 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
23 (i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g
24 b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
26 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
27 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
28 (b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
29 (b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda
30 (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g
31 (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2:
32 A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H2 \def (match H1 in aprem
33 return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem
34 n a a0)).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2
35 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1)
36 b1)))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H2: (eq nat O
37 i)).(\lambda (H3: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H4: (eq A a0
38 b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to
39 ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
40 A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H5: (eq A (AHead a0 a3) (ASort
41 h2 n2))).(let H6 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e in A
42 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
43 _) \Rightarrow True])) I (ASort h2 n2) H5) in (False_ind ((eq A a0 b2) \to
44 (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1
45 n1) b1)))) H6))) i H2 H3 H4)))) | (aprem_succ a0 a i0 H2 a3) \Rightarrow
46 (\lambda (H3: (eq nat (S i0) i)).(\lambda (H4: (eq A (AHead a3 a0) (ASort h2
47 n2))).(\lambda (H5: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A
48 (AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A
49 (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1)
50 b1))))))) (\lambda (H6: (eq A (AHead a3 a0) (ASort h2 n2))).(let H7 \def
51 (eq_ind A (AHead a3 a0) (\lambda (e: A).(match e in A return (\lambda (_:
52 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
53 True])) I (ASort h2 n2) H6) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a)
54 \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0)
55 (ASort h1 n1) b1))))) H7))) i H3 H4 H5 H2))))]) in (H2 (refl_equal nat i)
56 (refl_equal A (ASort h2 n2)) (refl_equal A b2)))))))))))) (\lambda (a0:
57 A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall (i:
58 nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1
59 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda (a5:
60 A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall (b2:
61 A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
62 A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H4:
63 (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead a3
64 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n
65 (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H6 \def
66 (match H5 in aprem return (\lambda (n: nat).(\lambda (a: A).(\lambda (a6:
67 A).(\lambda (_: (aprem n a a6)).((eq nat n O) \to ((eq A a (AHead a3 a5)) \to
68 ((eq A a6 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
69 A).(aprem O (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow
70 (\lambda (_: (eq nat O O)).(\lambda (H7: (eq A (AHead a6 a7) (AHead a3
71 a5))).(\lambda (H8: (eq A a6 b2)).((let H9 \def (f_equal A A (\lambda (e:
72 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 |
73 (AHead _ a) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H7) in ((let H10
74 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
75 with [(ASort _ _) \Rightarrow a6 | (AHead a _) \Rightarrow a])) (AHead a6 a7)
76 (AHead a3 a5) H7) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a5) \to ((eq A a
77 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O
78 (AHead a0 a4) b1)))))) (\lambda (H11: (eq A a7 a5)).(eq_ind A a5 (\lambda (_:
79 A).((eq A a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
80 A).(aprem O (AHead a0 a4) b1))))) (\lambda (H12: (eq A a3 b2)).(eq_ind A b2
81 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
82 A).(aprem O (AHead a0 a4) b1)))) (eq_ind A a3 (\lambda (a: A).(ex2 A (\lambda
83 (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))
84 (ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead
85 a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H12) a3 (sym_eq A a3 b2 H12))) a7
86 (sym_eq A a7 a5 H11))) a6 (sym_eq A a6 a3 H10))) H9)) H8)))) | (aprem_succ a6
87 a i0 H6 a7) \Rightarrow (\lambda (H7: (eq nat (S i0) O)).(\lambda (H8: (eq A
88 (AHead a7 a6) (AHead a3 a5))).(\lambda (H9: (eq A a b2)).((let H10 \def
89 (eq_ind nat (S i0) (\lambda (e: nat).(match e in nat return (\lambda (_:
90 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
91 (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem
92 i0 a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O
93 (AHead a0 a4) b1)))))) H10)) H8 H9 H6))))]) in (H6 (refl_equal nat O)
94 (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) (\lambda (i0: nat).(\lambda
95 (_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
96 (\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0)
97 (AHead a3 a5) b2)).(let H6 \def (match H5 in aprem return (\lambda (n:
98 nat).(\lambda (a: A).(\lambda (a6: A).(\lambda (_: (aprem n a a6)).((eq nat n
99 (S i0)) \to ((eq A a (AHead a3 a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda
100 (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4)
101 b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (H6: (eq nat O (S
102 i0))).(\lambda (H7: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H8: (eq A
103 a6 b2)).((let H9 \def (eq_ind nat O (\lambda (e: nat).(match e in nat return
104 (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
105 I (S i0) H6) in (False_ind ((eq A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6
106 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0)
107 (AHead a0 a4) b1))))) H9)) H7 H8)))) | (aprem_succ a6 a i1 H6 a7) \Rightarrow
108 (\lambda (H7: (eq nat (S i1) (S i0))).(\lambda (H8: (eq A (AHead a7 a6)
109 (AHead a3 a5))).(\lambda (H9: (eq A a b2)).((let H10 \def (f_equal nat nat
110 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).nat) with [O
111 \Rightarrow i1 | (S n) \Rightarrow n])) (S i1) (S i0) H7) in (eq_ind nat i0
112 (\lambda (n: nat).((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to
113 ((aprem n a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
114 A).(aprem (S i0) (AHead a0 a4) b1))))))) (\lambda (H11: (eq A (AHead a7 a6)
115 (AHead a3 a5))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A
116 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a8)
117 \Rightarrow a8])) (AHead a7 a6) (AHead a3 a5) H11) in ((let H13 \def (f_equal
118 A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
119 \Rightarrow a7 | (AHead a8 _) \Rightarrow a8])) (AHead a7 a6) (AHead a3 a5)
120 H11) in (eq_ind A a3 (\lambda (_: A).((eq A a6 a5) \to ((eq A a b2) \to
121 ((aprem i0 a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
122 A).(aprem (S i0) (AHead a0 a4) b1))))))) (\lambda (H14: (eq A a6 a5)).(eq_ind
123 A a5 (\lambda (a8: A).((eq A a b2) \to ((aprem i0 a8 a) \to (ex2 A (\lambda
124 (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))))))
125 (\lambda (H15: (eq A a b2)).(eq_ind A b2 (\lambda (a8: A).((aprem i0 a5 a8)
126 \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0)
127 (AHead a0 a4) b1))))) (\lambda (H16: (aprem i0 a5 b2)).(let H_x \def (H3 i0
128 b2 H16) in (let H17 \def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b2))
129 (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: A).(leq g b1 b2))
130 (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda (x: A).(\lambda
131 (H18: (leq g x b2)).(\lambda (H19: (aprem i0 a4 x)).(ex_intro2 A (\lambda
132 (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1)) x
133 H18 (aprem_succ a4 x i0 H19 a0))))) H17)))) a (sym_eq A a b2 H15))) a6
134 (sym_eq A a6 a5 H14))) a7 (sym_eq A a7 a3 H13))) H12))) i1 (sym_eq nat i1 i0
135 H10))) H8 H9 H6))))]) in (H6 (refl_equal nat (S i0)) (refl_equal A (AHead a3
136 a5)) (refl_equal A b2)))))) i H4)))))))))))) a1 a2 H)))).
139 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i
140 a1 a2) \to (aprem i (asucc g a1) a2)))))
142 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
143 (H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
144 (a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
145 A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
146 (i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
147 a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2