include "leq/defs.ma".
+include "aplus/props.ma".
+
theorem ahead_inj_snd:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
H1 a6 a5 H3))))))))) a1 a2 H)))).
-axiom leq_trans:
+theorem leq_trans:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
-.
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
+(ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a:
+A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to
+((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3
+k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda
+(H4: (eq A (ASort h3 n3) a3)).((let H5 \def (f_equal A nat (\lambda (e:
+A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n
+| (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H3) in ((let H6
+\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
+with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0)
+(ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to
+((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort
+h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0
+n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A
+(aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1
+n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3)
+(\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0))
+\to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0)
+(aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3
+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
+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)
+\to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4
+a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3
+a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9))
+H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A
+a0))))))))))))) a1 a2 H)))).
theorem leq_ahead_false:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
\lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
-n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n in nat return (\lambda
-(n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O
-\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (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 (ASort O n0) a2)) \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 (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2)
-(ASort O n0))).((let H4 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
+n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead
+(ASort O n0) a2) (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 (ASort O
+n0) a2)) \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 (ASort O n0)
+a2))).(\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 (ASort O n0) a2) 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 (ASort O n0) a2))).(\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 (ASort O n0) a2) 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 (ASort O n0) a2) H3) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A
+a4 a2) \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 a2)).(eq_ind A a2 (\lambda (a:
+A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g (ASort O n0) 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 (ASort O n0) a3) \to ((leq g
+a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0)
+H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2))
+(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g
+(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead
+(ASort (S n1) n0) a2) (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 (ASort (S n1) n0) a2)) \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 (ASort (S n1) n0) a2))).(\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 (ASort O n0) a2) 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 (ASort O n0)
-a2))).(\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 (ASort O
-n0) a2) 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 (ASort O n0) a2) H3) in (eq_ind A
-(ASort O n0) (\lambda (a: A).((eq A a4 a2) \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
-a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to
-((leq g (ASort O n0) 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 (ASort O n0) a3) \to ((leq g a2 a5) \to P)) H9))) a4
-(sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort O n0) H6))) H5)) H4 H1 H2)))])
-in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0)))))
-| (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (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 (ASort (S n1) n0) a2)) \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 (ASort (S n1) n0)
-a2))).(\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 (ASort (S n1) n0) a2) 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 (ASort (S n1) n0) a2))).(\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 (ASort (S
-n1) n0) a2) 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 (ASort (S n1) n0) a2) H3) in (eq_ind A
-(ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \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 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) (ASort
-(S n1) n0)) \to ((leq g (ASort (S n1) n0) 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 (ASort (S n1) n0) a3) \to ((leq g a2 a5)
-\to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0)
-H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
-(refl_equal A (ASort (S n1) n0)))))]) H)))))) (\lambda (a: A).(\lambda (H:
-((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
+_ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) 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 (ASort (S n1) n0)
+a2))).(\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 (ASort (S n1) n0) a2) 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 (ASort (S n1) n0) a2) H3)
+in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \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 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead
+a3 a5) (ASort (S n1) n0)) \to ((leq g (ASort (S n1) n0) 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 (ASort (S n1) n0) a3)
+\to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0
+(ASort (S n1) n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort
+(S n1) n0) a2)) (refl_equal A (ASort (S n1) n0))))))) n H)))))) (\lambda (a:
+A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P:
Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq
g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def
projects / helm.git / blobdiff