X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fsex.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fsex.ma;h=96f62f93e7d89b840b9af0ff2b3d57991fa0bab8;hb=222044da28742b24584549ba86b1805a87def070;hp=0000000000000000000000000000000000000000;hpb=5c186c72f508da0849058afeecc6877cd9ed6303;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/sex.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/sex.ma new file mode 100644 index 000000000..96f62f93e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/sex.ma @@ -0,0 +1,296 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/relocation/rtmap_sle.ma". +include "ground_2/relocation/rtmap_sdj.ma". +include "basic_2/notation/relations/relation_5.ma". +include "basic_2/syntax/lenv.ma". + +(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) + +inductive sex (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝ +| sex_atom: ∀f. sex RN RP f (⋆) (⋆) +| sex_next: ∀f,I1,I2,L1,L2. + sex RN RP f L1 L2 → RN L1 I1 I2 → + sex RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) +| sex_push: ∀f,I1,I2,L1,L2. + sex RN RP f L1 L2 → RP L1 I1 I2 → + sex RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) +. + +interpretation "generic entrywise extension (local environment)" + 'Relation RN RP f L1 L2 = (sex RN RP f L1 L2). + +definition R_pw_confluent2_sex: relation3 lenv bind bind → 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,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 → + ∃∃I. R2 L1 I1 I & R1 L2 I2 I. + +definition sex_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 ***************************************************) + +fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆. +#RN #RP #f #X #Y * -f -X -Y // +#f #I1 #I2 #L1 #L2 #_ #_ #H destruct +qed-. + +(* Basic_2A1: includes lpx_sn_inv_atom1 *) +lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆. +/2 width=6 by sex_inv_atom1_aux/ qed-. + +fact sex_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 +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct + /2 width=5 by ex3_2_intro/ +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H) +] +qed-. + +(* Basic_2A1: includes lpx_sn_inv_pair1 *) +lemma sex_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 sex_inv_next1_aux/ qed-. + +fact sex_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 +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H) +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma sex_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 sex_inv_push1_aux/ qed-. + +fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆. +#RN #RP #f #X #Y * -f -X -Y // +#f #I1 #I2 #L1 #L2 #_ #_ #H destruct +qed-. + +(* Basic_2A1: includes lpx_sn_inv_atom2 *) +lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆. +/2 width=6 by sex_inv_atom2_aux/ qed-. + +fact sex_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 +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct + /2 width=5 by ex3_2_intro/ +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H) +] +qed-. + +(* Basic_2A1: includes lpx_sn_inv_pair2 *) +lemma sex_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 sex_inv_next2_aux/ qed-. + +fact sex_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 +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H) +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma sex_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 sex_inv_push2_aux/ qed-. + +(* Basic_2A1: includes lpx_sn_inv_pair *) +lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2. + 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 (sex_inv_next1 … H) -H +#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ +qed-. + +lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2. + 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 (sex_inv_push1 … H) -H +#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ +qed-. + +lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 → + RN L1 I1 I2 → RP L1 I1 I2 → + L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}. +#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) * +/2 width=1 by sex_next, sex_push/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2. + L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → + L1 ⪤[RN, RP, ⫱f] L2. +#RN #RP #f #I1 #I2 #L1 #L2 #Hf +elim (pn_split f) * #g #H destruct +[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf // +qed-. + +(* Basic properties *********************************************************) + +lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2). +#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // +#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H +[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/ +| elim (eq_inv_px … H) -H /3 width=3 by sex_push/ +] +qed-. + +lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2). +#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *) +qed-. + +lemma sex_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → + ∀f.reflexive … (sex 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) * +#g #H destruct /2 width=1 by sex_next, sex_push/ +qed. + +lemma sex_sym: ∀RN,RP. + (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) → + (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) → + ∀f. symmetric … (sex RN RP f). +#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f +/3 width=2 by sex_next, sex_push/ +qed-. + +lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2. + L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → + ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 → + L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}. +/3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-. + +lemma sex_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 sex_atom, sex_next, sex_push/ +qed-. + +lemma sex_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 // +#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H +[ elim (isid_inv_next … H) -H // +| /4 width=3 by sex_push, isid_inv_push/ +] +qed-. + +lemma sex_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 sex_push/ +| elim (sdj_inv_px … H12) -H12 [2,4: // ] * + #g2 #H #H2 destruct /3 width=1 by sex_next, sex_push/ +] +qed-. + +lemma sle_sex_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 // +#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12 +[ elim (pn_split f1) * ] +[ /4 width=5 by sex_push, sle_inv_pn/ +| /4 width=5 by sex_next, sle_inv_nn/ +| elim (sle_inv_xp … H12) -H12 [2,3: // ] + #g1 #H #H1 destruct /3 width=5 by sex_push/ +] +qed-. + +lemma sle_sex_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 // +#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 +[2: elim (pn_split f2) * ] +[ /4 width=5 by sex_push, sle_inv_pp/ +| /4 width=5 by sex_next, sle_inv_pn/ +| elim (sle_inv_nx … H12) -H12 [2,3: // ] + #g2 #H #H2 destruct /3 width=5 by sex_next/ +] +qed-. + +lemma sex_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 +[ /2 width=3 by sex_atom, ex2_intro/ ] +#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H +[ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct + elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, ex2_intro/ +| elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct + elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/ +] +qed-. + +lemma sex_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 sex_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 sex_next, sex_push, ex2_intro/ +| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct + elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/ +] +qed-. + +lemma sex_dec: ∀RN,RP. + (∀L,I1,I2. Decidable (RN L I1 I2)) → + (∀L,I1,I2. Decidable (RP L I1 I2)) → + ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2). +#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ] +[ /2 width=1 by sex_atom, or_introl/ +| #L2 #I2 #f @or_intror #H + lapply (sex_inv_atom1 … H) -H #H destruct +| #f @or_intror #H + lapply (sex_inv_atom2 … H) -H #H destruct +| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12 + [2: /4 width=3 by sex_fwd_bind, or_intror/ ] + elim (pn_split f) * #g #H destruct + [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12 + [1,3: /3 width=1 by sex_push, sex_next, or_introl/ ] + @or_intror #H + [ elim (sex_inv_push … H) | elim (sex_inv_next … H) ] -H + /2 width=1 by/ +] +qed-.