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/ex2/defs.ma".
19 include "LambdaDelta-1/nf2/defs.ma".
21 include "LambdaDelta-1/pr2/fwd.ma".
23 include "LambdaDelta-1/arity/fwd.ma".
28 \lambda (t2: T).(\lambda (H: (pr2 (CSort O) (THead (Flat Appl) (TSort O)
29 (TSort O)) t2)).(let H0 \def (pr2_gen_appl (CSort O) (TSort O) (TSort O) t2
30 H) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
31 (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
32 O) u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 (CSort O) (TSort O) t3))))
33 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
34 T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
35 (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
36 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
37 T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
38 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
39 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
40 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
41 b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
42 T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind b) y1
43 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
44 T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
45 Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
46 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort
47 O) u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
48 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b:
49 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
50 (y2: T).(pr2 (CHead (CSort O) (Bind b) y2) z1 z2)))))))) (eq T (THead (Flat
51 Appl) (TSort O) (TSort O)) t2) (\lambda (H1: (ex3_2 T T (\lambda (u2:
52 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
53 T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
54 (t3: T).(pr2 (CSort O) (TSort O) t3))))).(ex3_2_ind T T (\lambda (u2:
55 T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
56 T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))) (\lambda (_: T).(\lambda
57 (t3: T).(pr2 (CSort O) (TSort O) t3))) (eq T (THead (Flat Appl) (TSort O)
58 (TSort O)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
59 (THead (Flat Appl) x0 x1))).(\lambda (H3: (pr2 (CSort O) (TSort O)
60 x0)).(\lambda (H4: (pr2 (CSort O) (TSort O) x1)).(let H5 \def (eq_ind T x1
61 (\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H2 (TSort O)
62 (pr2_gen_sort (CSort O) x1 O H4)) in (let H6 \def (eq_ind T x0 (\lambda (t:
63 T).(eq T t2 (THead (Flat Appl) t (TSort O)))) H5 (TSort O) (pr2_gen_sort
64 (CSort O) x0 O H3)) in (eq_ind_r T (THead (Flat Appl) (TSort O) (TSort O))
65 (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (refl_equal
66 T (THead (Flat Appl) (TSort O) (TSort O))) t2 H6)))))))) H1)) (\lambda (H1:
67 (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
68 T).(eq T (TSort O) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
69 (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
70 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
71 T).(pr2 (CSort O) (TSort O) u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
72 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead (CSort O)
73 (Bind b) u) z1 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
74 T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O) (THead (Bind Abst) y1
75 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
76 T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
77 T).(\lambda (u2: T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))) (\lambda
78 (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
79 B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) z1 t3))))))) (eq T
80 (THead (Flat Appl) (TSort O) (TSort O)) t2) (\lambda (x0: T).(\lambda (x1:
81 T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H2: (eq T (TSort O) (THead
82 (Bind Abst) x0 x1))).(\lambda (H3: (eq T t2 (THead (Bind Abbr) x2
83 x3))).(\lambda (H4: (pr2 (CSort O) (TSort O) x2)).(\lambda (_: ((\forall (b:
84 B).(\forall (u: T).(pr2 (CHead (CSort O) (Bind b) u) x1 x3))))).(let H6 \def
85 (eq_ind T x2 (\lambda (t: T).(eq T t2 (THead (Bind Abbr) t x3))) H3 (TSort O)
86 (pr2_gen_sort (CSort O) x2 O H4)) in (eq_ind_r T (THead (Bind Abbr) (TSort O)
87 x3) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O)) t)) (let H7
88 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return (\lambda (_:
89 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
90 (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H2) in
91 (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead (Bind Abbr)
92 (TSort O) x3)) H7)) t2 H6)))))))))) H1)) (\lambda (H1: (ex6_6 B T T T T T
93 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
94 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
95 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
96 (TSort O) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
97 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
98 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
99 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
100 T).(\lambda (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_:
101 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
102 (y2: T).(pr2 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
103 (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort
104 O) (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
105 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
106 (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
107 (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TSort O)
108 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
109 T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
110 b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
111 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
112 (_: T).(pr2 (CSort O) (TSort O) u2))))))) (\lambda (_: B).(\lambda (y1:
113 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2
114 (CSort O) y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
115 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead (CSort O)
116 (Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) (TSort O) (TSort O)) t2)
117 (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
118 (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H3: (eq
119 T (TSort O) (THead (Bind x0) x1 x2))).(\lambda (H4: (eq T t2 (THead (Bind x0)
120 x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (H5: (pr2 (CSort O)
121 (TSort O) x4)).(\lambda (H6: (pr2 (CSort O) x1 x5)).(\lambda (_: (pr2 (CHead
122 (CSort O) (Bind x0) x5) x2 x3)).(let H_y \def (pr2_gen_csort x1 x5 O H6) in
123 (let H8 \def (eq_ind T x4 (\lambda (t: T).(eq T t2 (THead (Bind x0) x5 (THead
124 (Flat Appl) (lift (S O) O t) x3)))) H4 (TSort O) (pr2_gen_sort (CSort O) x4 O
125 H5)) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
126 (TSort O)) x3)) (\lambda (t: T).(eq T (THead (Flat Appl) (TSort O) (TSort O))
127 t)) (let H9 \def (eq_ind T (TSort O) (\lambda (ee: T).(match ee in T return
128 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
129 \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1
130 x2) H3) in (False_ind (eq T (THead (Flat Appl) (TSort O) (TSort O)) (THead
131 (Bind x0) x5 (THead (Flat Appl) (lift (S O) O (TSort O)) x3))) H9)) t2
132 H8))))))))))))))) H1)) H0))).
135 \forall (g: G).(\forall (a: A).((arity g ex2_c ex2_t a) \to (\forall (P:
138 \lambda (g: G).(\lambda (a: A).(\lambda (H: (arity g (CSort O) (THead (Flat
139 Appl) (TSort O) (TSort O)) a)).(\lambda (P: Prop).(let H0 \def
140 (arity_gen_appl g (CSort O) (TSort O) (TSort O) a H) in (ex2_ind A (\lambda
141 (a1: A).(arity g (CSort O) (TSort O) a1)) (\lambda (a1: A).(arity g (CSort O)
142 (TSort O) (AHead a1 a))) P (\lambda (x: A).(\lambda (_: (arity g (CSort O)
143 (TSort O) x)).(\lambda (H2: (arity g (CSort O) (TSort O) (AHead x a))).(let
144 H3 \def (match (arity_gen_sort g (CSort O) O (AHead x a) H2) in leq return
145 (\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0
146 (AHead x a)) \to ((eq A a1 (ASort O O)) \to P))))) with [(leq_sort h1 h2 n1
147 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1) (AHead x a))).(\lambda
148 (H5: (eq A (ASort h2 n2) (ASort O O))).((let H6 \def (eq_ind A (ASort h1 n1)
149 (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
150 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x a) H4) in
151 (False_ind ((eq A (ASort h2 n2) (ASort O O)) \to ((eq A (aplus g (ASort h1
152 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H6)) H5 H3))) | (leq_head a1 a2 H3
153 a3 a4 H4) \Rightarrow (\lambda (H5: (eq A (AHead a1 a3) (AHead x
154 a))).(\lambda (H6: (eq A (AHead a2 a4) (ASort O O))).((let H7 \def (f_equal A
155 A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
156 \Rightarrow a3 | (AHead _ a0) \Rightarrow a0])) (AHead a1 a3) (AHead x a) H5)
157 in ((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
158 (_: A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a0 _) \Rightarrow a0]))
159 (AHead a1 a3) (AHead x a) H5) in (eq_ind A x (\lambda (a0: A).((eq A a3 a)
160 \to ((eq A (AHead a2 a4) (ASort O O)) \to ((leq g a0 a2) \to ((leq g a3 a4)
161 \to P))))) (\lambda (H9: (eq A a3 a)).(eq_ind A a (\lambda (a0: A).((eq A
162 (AHead a2 a4) (ASort O O)) \to ((leq g x a2) \to ((leq g a0 a4) \to P))))
163 (\lambda (H10: (eq A (AHead a2 a4) (ASort O O))).(let H11 \def (eq_ind A
164 (AHead a2 a4) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
165 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O
166 O) H10) in (False_ind ((leq g x a2) \to ((leq g a a4) \to P)) H11))) a3
167 (sym_eq A a3 a H9))) a1 (sym_eq A a1 x H8))) H7)) H6 H3 H4)))]) in (H3
168 (refl_equal A (AHead x a)) (refl_equal A (ASort O O))))))) H0))))).