X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flexs.ma;h=0ff659b1376689e45afb7fdca00469607a70a691;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=bd0a3af6b1f6e94066d47f93c415d332eb98aedb;hpb=7cf2232827c06c7e85a9bc3be005f9134d5b869d;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma index bd0a3af6b..0ff659b13 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -13,6 +13,7 @@ (**************************************************************************) include "ground_2/relocation/rtmap_sle.ma". +include "ground_2/relocation/rtmap_sdj.ma". include "basic_2/notation/relations/relationstar_5.ma". include "basic_2/syntax/lenv.ma". @@ -22,10 +23,10 @@ inductive lexs (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝ | lexs_atom: ∀f. lexs RN RP f (⋆) (⋆) | lexs_next: ∀f,I1,I2,L1,L2. lexs RN RP f L1 L2 → RN L1 I1 I2 → - lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) + lexs RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) | lexs_push: ∀f,I1,I2,L1,L2. lexs RN RP f L1 L2 → RP L1 I1 I2 → - lexs RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) + lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) . interpretation "generic entrywise extension (local environment)" @@ -35,16 +36,18 @@ definition R_pw_confluent2_lexs: relation3 lenv bind bind → relation3 lenv bin relation3 lenv bind bind → relation3 lenv bind bind → relation3 lenv bind bind → relation3 lenv bind bind → relation3 rtmap lenv bind ≝ - λR1,R2,RN1,RP1,RN2,RP2,f,L0,T0. - ∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → + λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0. + ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 → ∀L1. L0 ⪤*[RN1, RP1, f] L1 → ∀L2. L0 ⪤*[RN2, RP2, f] L2 → - ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + ∃∃I. R2 L1 I1 I & R1 L2 I2 I. -definition lexs_transitive: relation5 (relation3 lenv bind bind) - (relation3 lenv bind bind) … ≝ - λR1,R2,R3,RN,RP. - ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⪤*[RN, RP, f] L2 → - ∀T2. R2 L2 T T2 → R3 L1 T1 T2. +definition lexs_transitive: relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 rtmap lenv bind ≝ + λR1,R2,R3,RN,RP,f,L1,I1. + ∀I. R1 L1 I1 I → ∀L2. L1 ⪤*[RN, RP, f] L2 → + ∀I2. R2 L2 I I2 → R3 L1 I1 I2. (* Basic inversion lemmas ***************************************************) @@ -57,7 +60,7 @@ qed-. lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤*[RN, RP, f] Y → Y = ⋆. /2 width=6 by lexs_inv_atom1_aux/ qed-. -fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g → +fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g → ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct @@ -68,11 +71,11 @@ fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = qed-. (* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma lexs_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ⫯g] Y → +lemma lexs_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ↑g] Y → ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. /2 width=7 by lexs_inv_next1_aux/ qed-. -fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g → +fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g → ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct @@ -82,7 +85,7 @@ fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = ] qed-. -lemma lexs_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ↑g] Y → +lemma lexs_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ⫯g] Y → ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. /2 width=7 by lexs_inv_push1_aux/ qed-. @@ -95,7 +98,7 @@ qed-. lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆. /2 width=6 by lexs_inv_atom2_aux/ qed-. -fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g → +fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g → ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J2 #K2 #H destruct @@ -106,11 +109,11 @@ fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = qed-. (* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma lexs_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} → +lemma lexs_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ↑g] K2.ⓘ{J2} → ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. /2 width=7 by lexs_inv_next2_aux/ qed-. -fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g → +fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g → ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. #RN #RP #f #X #Y * -f -X -Y [ #f #J2 #K2 #g #H destruct @@ -120,20 +123,20 @@ fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = ] qed-. -lemma lexs_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ↑g] K2.ⓘ{J2} → +lemma lexs_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} → ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. /2 width=7 by lexs_inv_push2_aux/ qed-. (* Basic_2A1: includes lpx_sn_inv_pair *) lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} → + L1.ⓘ{I1} ⪤*[RN, RP, ↑f] L2.ⓘ{I2} → L1 ⪤*[RN, RP, f] L2 ∧ RN L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_next1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤*[RN, RP, ↑f] L2.ⓘ{I2} → + L1.ⓘ{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} → L1 ⪤*[RN, RP, f] L2 ∧ RP L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_push1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ @@ -170,10 +173,7 @@ lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤*[RN, RP, f] #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *) qed-. -(* Basic_2A1: uses: lpx_sn_refl *) -lemma lexs_refl: ∀RN,RP. - (∀L. reflexive … (RN L)) → - (∀L. reflexive … (RP L)) → +lemma lexs_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → ∀f.reflexive … (lexs RN RP f). #RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L // #L #I #IH #f elim (pn_split f) * @@ -194,16 +194,13 @@ lemma lexs_pair_repl: ∀RN,RP,f,I1,I2,L1,L2. L1.ⓘ{J1} ⪤*[RN, RP, f] L2.ⓘ{J2}. /3 width=3 by lexs_inv_tl, lexs_fwd_bind/ qed-. -lemma lexs_co: ∀RN1,RP1,RN2,RP2. - (∀L1,I1,I2. RN1 L1 I1 I2 → RN2 L1 I1 I2) → - (∀L1,I1,I2. RP1 L1 I1 I2 → RP2 L1 I1 I2) → +lemma lexs_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 → ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → L1 ⪤*[RN2, RP2, f] L2. #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by lexs_atom, lexs_next, lexs_push/ qed-. -lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. - (∀L1,I1,I2. RP1 L1 I1 I2 → RP2 L1 I1 I2) → +lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 → ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → 𝐈⦃f⦄ → L1 ⪤*[RN2, RP2, f] L2. #RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 // @@ -213,7 +210,19 @@ lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. ] qed-. -lemma sle_lexs_trans: ∀RN,RP. (∀L,I1,I2. RN L I1 I2 → RP L I1 I2) → +lemma lexs_sdj: ∀RN,RP. RP ⊆ RN → + ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 → + ∀f2. f1 ∥ f2 → L1 ⪤*[RP, RN, f2] L2. +#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // +#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 +[ elim (sdj_inv_nx … H12) -H12 [2,3: // ] + #g2 #H #H2 destruct /3 width=1 by lexs_push/ +| elim (sdj_inv_px … H12) -H12 [2,4: // ] * + #g2 #H #H2 destruct /3 width=1 by lexs_next, lexs_push/ +] +qed-. + +lemma sle_lexs_trans: ∀RN,RP. RN ⊆ RP → ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 → ∀f1. f1 ⊆ f2 → L1 ⪤*[RN, RP, f1] L2. #RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 // @@ -226,7 +235,7 @@ lemma sle_lexs_trans: ∀RN,RP. (∀L,I1,I2. RN L I1 I2 → RP L I1 I2) → ] qed-. -lemma sle_lexs_conf: ∀RN,RP. (∀L,I1,I2. RP L I1 I2 → RN L I1 I2) → +lemma sle_lexs_conf: ∀RN,RP. RP ⊆ RN → ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 → ∀f2. f1 ⊆ f2 → L1 ⪤*[RN, RP, f2] L2. #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // @@ -239,7 +248,7 @@ lemma sle_lexs_conf: ∀RN,RP. (∀L,I1,I2. RP L I1 I2 → RN L I1 I2) → ] qed-. -lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → +lemma lexs_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g → ∃∃L. L1 ⪤*[R1, RP, g] L & L ⪤*[R2, cfull, f] L2. #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2 @@ -252,6 +261,19 @@ lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. refle ] qed-. +lemma lexs_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → + ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ∥ g → + ∃∃L. L1 ⪤*[RP, R1, g] L & L ⪤*[R2, cfull, f] L2. +#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2 +[ /2 width=3 by lexs_atom, ex2_intro/ ] +#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H +[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct + elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/ +| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct + elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/ +] +qed-. + lemma lexs_dec: ∀RN,RP. (∀L,I1,I2. Decidable (RN L I1 I2)) → (∀L,I1,I2. Decidable (RP L I1 I2)) →