include "leq/defs.ma".
-theorem leq_gen_sort:
- \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
-g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
-h2 n2) k))))))))))
+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))))))
\def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return
-(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort
-h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
-(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2:
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k)
-(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0)
-\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2:
-(eq A (ASort h2 n2) a2)).((let H3 \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 h1 n1) H1) in ((let H4 \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 h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to
-((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort
-h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n:
-nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g
-(ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3:
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2
-n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
-nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))
-(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
-(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
-h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq
-nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0
-a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1
-n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead
-a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1
-n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g
-a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda
-(_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
-k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal
-A a2))))))).
-
-theorem leq_gen_head:
- \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
-(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda
-(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
-\def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
-(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda
-(a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2))
-\to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a
-(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda
-(_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k
-H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda
-(H2: (eq A (ASort h2 n2) a)).((let H3 \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 a2) H1) in
-(False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k)
-(aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
-A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3)))
-(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head
-a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1
-a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2)
-H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _)
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda
-(a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to
-((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
-(_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4
-a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3)
-\to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda
-(_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3
-a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g
-a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7
-a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_:
-A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda
-(H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A
-(AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1
-a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A
-(AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1
-H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A
-a))))))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
+H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
+a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2
+a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
+(ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3
+a4))).((let H3 \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 a2) H1) in (False_ind ((eq A (ASort h2 n2)
+(AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3:
+(eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e:
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 |
+(AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \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 a6)
+(AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A
+(AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g
+a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A
+(AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g
+a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \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 a5 a7)
+(AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
+\Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a:
+A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4)))))
+(\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to
+((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12:
+(leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8)))
+a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0
+(refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
theorem leq_refl:
\forall (g: G).(\forall (a: A).(leq g a a))
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