(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props".
+include "LambdaDelta-1/leq/defs.ma".
-include "leq/defs.ma".
-
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
theorem ahead_inj_snd:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1)
(ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus
(minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g
-(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0
-H10)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0)
-(aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 H9))))) (\lambda
-(H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0
-k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus (minus k k0) k0)
-(\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n)))
-H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A
-(aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k)
-H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 (sym_eq nat h0 h2 H6)))
-H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A
-(AHead a0 a5) (ASort h2 n2))).(\lambda (H5: (eq A (AHead a4 a6) a3)).((let H6
-\def (eq_ind A (AHead a0 a5) (\lambda (e: A).(match e in A return (\lambda
-(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
-True])) I (ASort h2 n2) H4) in (False_ind ((eq A (AHead a4 a6) a3) \to ((leq
-g a0 a4) \to ((leq g a5 a6) \to (leq g (ASort h1 n1) a3)))) H6)) H5 H2
-H3)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3)))))))))))
-(\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1:
-((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5:
-A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7:
-A).((leq g a6 a7) \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g
-(AHead a4 a6) a0)).(let H5 \def (match H4 in leq return (\lambda (a:
-A).(\lambda (a7: A).(\lambda (_: (leq ? a a7)).((eq A a (AHead a4 a6)) \to
-((eq A a7 a0) \to (leq g (AHead a3 a5) a0)))))) with [(leq_sort h1 h2 n1 n2 k
-H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda
-(H7: (eq A (ASort h2 n2) a0)).((let H8 \def (eq_ind A (ASort h1 n1) (\lambda
-(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a4 a6) H6) in
-(False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k)
-(aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H8)) H7 H5))) |
-(leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda (H7: (eq A (AHead a7 a9)
-(AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) a0)).((let H9 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) (AHead a7 a9)
-(AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: A).(match e in
-A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _)
-\Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 (\lambda (a:
-A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) \to ((leq g
-a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9
+(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_trans k (S k) k0 (le_S
+k k (le_n k)) H10))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g
+(ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11
+H9))))) (\lambda (H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2)
+(ASort h3 n3) k0 k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus
+(minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g
+(ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3
+n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)
+(aplus g (ASort h3 n3) k) H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0
+(sym_eq nat h0 h2 H6))) H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3)
+\Rightarrow (\lambda (H4: (eq A (AHead a0 a5) (ASort h2 n2))).(\lambda (H5:
+(eq A (AHead a4 a6) a3)).((let H6 \def (eq_ind A (AHead a0 a5) (\lambda (e:
+A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind
+((eq A (AHead a4 a6) a3) \to ((leq g a0 a4) \to ((leq g a5 a6) \to (leq g
+(ASort h1 n1) a3)))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort h2 n2))
+(refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
+(leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3
+a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5
+a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5
+a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def
+(match H4 in leq return (\lambda (a: A).(\lambda (a7: A).(\lambda (_: (leq ?
+a a7)).((eq A a (AHead a4 a6)) \to ((eq A a7 a0) \to (leq g (AHead a3 a5)
+a0)))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A
+(ASort h1 n1) (AHead a4 a6))).(\lambda (H7: (eq A (ASort h2 n2) a0)).((let H8
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a4 a6) H6) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq
+A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3
+a5) a0))) H8)) H7 H5))) | (leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda
+(H7: (eq A (AHead a7 a9) (AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10)
+a0)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a]))
+(AHead a7 a9) (AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 |
+(AHead a _) \Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4
+(\lambda (a: A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8)
+\to ((leq g a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9
a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4
a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A
(AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8)
H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A
a0))))))))))))) a1 a2 H)))).
-theorem leq_ahead_false:
+theorem leq_ahead_false_1:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
\to (\forall (P: Prop).P))))
\def
H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a
a0))))))))))) a1)).
+theorem leq_ahead_false_2:
+ \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2)
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1:
+A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort
+n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda
+(a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort
+O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
+H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O
+n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O n0))).((let H4 \def (eq_ind
+A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
+(AHead a1 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O n0))
+\to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H4))
+H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) \Rightarrow (\lambda (H3: (eq A
+(AHead a0 a4) (AHead a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5)
+(ASort O n0))).((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort O n0)) H3) in ((let H6 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4)
+(AHead a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort
+O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g
+a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O
+n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to
+((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O n0))).(let
+H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda
+(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
+True])) I (ASort O n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O
+n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1
+H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0)))
+(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
+(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
+a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq
+return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
+(AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P)))))
+with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1
+n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort
+(S n1) n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
+_ _) \Rightarrow False])) I (AHead a1 (ASort (S n1) n0)) H2) in (False_ind
+((eq A (ASort h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n2) k)
+(aplus g (ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5
+H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead a1 (ASort (S n1)
+n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S n1) n0))).((let H5 \def
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4)
+(AHead a1 (ASort (S n1) n0)) H3) in ((let H6 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
+(AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 (ASort (S n1) n0)) H3)
+in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A
+(AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to
+P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0)
+(\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3)
+\to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S n1)
+n0))).(let H9 \def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort (S n1) n0) H8) in (False_ind ((leq g a1 a3)
+\to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1)
+n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A
+(AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H))))))
+(\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to
+(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1:
+A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1:
+A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P:
+Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4:
+A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a1 (AHead a a0))) \to ((eq A
+a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
+(\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq
+A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda
+(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 (AHead a a0))
+H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head
+a3 a4 H2 a5 a6 H3) \Rightarrow (\lambda (H4: (eq A (AHead a3 a5) (AHead a1
+(AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a a0))).((let H6
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3
+a5) (AHead a1 (AHead a a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 |
+(AHead a7 _) \Rightarrow a7])) (AHead a3 a5) (AHead a1 (AHead a a0)) H4) in
+(eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4
+a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda
+(H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A
+(AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P))))
+(\lambda (H9: (eq A (AHead a4 a6) (AHead a a0))).(let H10 \def (f_equal A A
+(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a a0)
+H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _)
+\Rightarrow a7])) (AHead a4 a6) (AHead a a0) H9) in (eq_ind A a (\lambda (a7:
+A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P))))
+(\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to
+((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14:
+(leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq
+A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1
+H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0)))
+(refl_equal A (AHead a a0))))))))))) a2)).
+
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