X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fprops.ma;h=d879be152b236b2d3ca6620e84c1f2844a5ebfc7;hb=783fbc6eb42d5db260d93d8d213a7d5c035e99e3;hp=2fda46a6e65e001a88d2be06c4e234bd0d4da5c0;hpb=831af787465e1bff886e22ee14b68c8f1bb0177c;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma index 2fda46a6e..d879be152 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props". + include "leq/defs.ma". @@ -118,44 +118,44 @@ n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) 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) @@ -165,7 +165,7 @@ 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: +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 @@ -268,3 +268,106 @@ a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 (AHead a a0) H7))) H6)) H5 H2 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)). +