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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/fwd".
19 include "leq/defs.ma".
22 \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
23 g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
24 nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda
25 (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
28 \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
29 A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return
30 (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort
31 h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
32 (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2:
33 nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k)
34 (aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0)
35 \Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2:
36 (eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match
37 e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _
38 _) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def
39 (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
40 [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
41 (ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to
42 ((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort
43 h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
44 nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
45 (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
46 h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n:
47 nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g
48 (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
49 nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
50 (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
51 h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2
52 n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
53 k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
54 nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
55 (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))
56 (\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
57 k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
58 (_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
59 (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
60 h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq
61 nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0
62 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1
63 n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead
64 a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
65 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1
66 n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g
67 a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
68 (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2:
69 nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
70 k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal
74 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
75 (AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a
76 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda
77 (_: A).(\lambda (a4: A).(leq g a2 a4))))))))
79 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
80 (H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda
81 (a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2))
82 \to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a
83 (AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
84 (_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k
85 H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda
86 (H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda
87 (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
88 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in
89 (False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k)
90 (aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
91 A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3)))
92 (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head
93 a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1
94 a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A
95 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
96 \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2)
97 H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return
98 (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _)
99 \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda
100 (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to
101 ((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
102 (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
103 (_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4
104 a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3)
105 \to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
106 (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
107 (_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3
108 a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g
109 a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7
110 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
111 A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda
112 (H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A
113 (AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1
114 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A
115 (AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1
116 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A