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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/props".
19 include "leq/defs.ma".
21 theorem ahead_inj_snd:
22 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
23 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
25 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
26 (a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
27 H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
28 a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2
29 a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
30 (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3
31 a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A
32 return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
33 \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2)
34 (AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
35 k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1)
36 \Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3:
37 (eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e:
38 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
39 (AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def
40 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
41 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6)
42 (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A
43 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g
44 a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A
45 (AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g
46 a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def
47 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
48 [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7)
49 (AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A
50 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
51 \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a:
52 A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4)))))
53 (\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to
54 ((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12:
55 (leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8)))
56 a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0
57 (refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
60 \forall (g: G).(\forall (a: A).(leq g a a))
62 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
63 (\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
64 (aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
65 a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
69 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
72 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
73 a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
76 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
79 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
80 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
81 nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
82 nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
83 k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
84 (ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
85 (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
86 A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
87 H1 a6 a5 H3))))))))) a1 a2 H)))).
90 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
91 (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
94 theorem leq_ahead_false:
95 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
96 \to (\forall (P: Prop).P))))
98 \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
99 A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
100 nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
101 n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n in nat return (\lambda
102 (n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O
103 \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O n0))).(let
104 H1 \def (match H0 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_:
105 (leq ? a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O n0))
106 \to P))))) with [(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A
107 (ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2)
108 (ASort O n0))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
109 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
110 _ _) \Rightarrow False])) I (AHead (ASort O n0) a2) H2) in (False_ind ((eq A
111 (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
112 (ASort h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2)
113 \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0)
114 a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort O n0))).((let H5 \def (f_equal
115 A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
116 \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O
117 n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e: A).(match e in A
118 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _)
119 \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in (eq_ind A
120 (ASort O n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (ASort O
121 n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P))))) (\lambda (H7: (eq A a4
122 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to
123 ((leq g (ASort O n0) a3) \to ((leq g a a5) \to P)))) (\lambda (H8: (eq A
124 (AHead a3 a5) (ASort O n0))).(let H9 \def (eq_ind A (AHead a3 a5) (\lambda
125 (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
126 \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n0) H8) in
127 (False_ind ((leq g (ASort O n0) a3) \to ((leq g a2 a5) \to P)) H9))) a4
128 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0) H6))) H5)) H4 H1 H2)))])
129 in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0)))))
130 | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort
131 (S n1) n0))).(let H1 \def (match H0 in leq return (\lambda (a: A).(\lambda
132 (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort (S n1) n0) a2)) \to
133 ((eq A a0 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n2 n3 k H1)
134 \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (AHead (ASort (S n1) n0)
135 a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def
136 (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A return (\lambda (_:
137 A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
138 False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A (ASort h2
139 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort
140 h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow
141 (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0) a2))).(\lambda
142 (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def (f_equal A A
143 (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
144 \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S
145 n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e: A).(match e in A
146 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _)
147 \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3) in (eq_ind A
148 (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5)
149 (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P))))) (\lambda
150 (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) (ASort
151 (S n1) n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5) \to P))))
152 (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1) n0))).(let H9 \def (eq_ind A
153 (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
154 [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort (S
155 n1) n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3) \to ((leq g a2 a5)
156 \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0)
157 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
158 (refl_equal A (ASort (S n1) n0)))))]) H)))))) (\lambda (a: A).(\lambda (H:
159 ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
160 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
161 a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
162 g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def
163 (match H1 in leq return (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq ?
164 a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A a4 (AHead a a0)) \to
165 P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
166 (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2)
167 (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
168 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
169 _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A
170 (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
171 (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3)
172 \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0)
173 a2))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6 \def (f_equal
174 A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
175 \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a
176 a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
177 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _)
178 \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A
179 (AHead a a0) (\lambda (a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a
180 a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5
181 a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a a0)) \to
182 ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: (eq A
183 (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A (\lambda (e:
184 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
185 (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in ((let H11
186 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
187 with [(ASort _ _) \Rightarrow a4 | (AHead a7 _) \Rightarrow a7])) (AHead a4
188 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7: A).((eq A a6 a0) \to ((leq
189 g (AHead a a0) a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6
190 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7)
191 \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2
192 a0)).(H a0 H13 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq A a4 a H11))) H10)))
193 a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 (AHead a a0) H7))) H6)) H5 H2
194 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a