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=a79a3e4d9a3559d2077584c65778844da9aa1d77;hpb=ebc170efe71cf4ee842acfbe58bb6864e76ba98c;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 a79a3e4d9..0ff659b13 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -13,207 +13,284 @@ (**************************************************************************) 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/grammar/lenv.ma". +include "basic_2/syntax/lenv.ma". -(* GENERAL ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) +(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) -definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] -≝ λA,B,C,D,E.A→B→C→D→E→Prop. - -definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] -≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop. - -(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *) -inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝ +inductive lexs (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝ | lexs_atom: ∀f. lexs RN RP f (⋆) (⋆) -| lexs_next: ∀I,L1,L2,V1,V2,f. - lexs RN RP f L1 L2 → RN L1 V1 V2 → - lexs RN RP (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) -| lexs_push: ∀I,L1,L2,V1,V2,f. - lexs RN RP f L1 L2 → RP L1 V1 V2 → - lexs RN RP (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) +| 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_push: ∀f,I1,I2,L1,L2. + lexs RN RP f L1 L2 → RP L1 I1 I2 → + lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) . -interpretation "general entrywise extension (local environment)" +interpretation "generic entrywise extension (local environment)" 'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2). -definition lpx_sn_confluent: relation6 (relation3 lenv term term) - (relation3 lenv term term) … ≝ - λR1,R2,RN1,RP1,RN2,RP2. - ∀f,L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 → - ∃∃T. R2 L1 T1 T & R1 L2 T2 T. - -definition lexs_transitive: relation4 (relation3 lenv term term) - (relation3 lenv term term) … ≝ - λR1,R2,RN,RP. - ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⦻*[RN, RP, f] L2 → - ∀T2. R2 L2 T T2 → R1 L1 T1 T2. +definition R_pw_confluent2_lexs: 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 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 ***************************************************) -fact lexs_inv_atom1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆. -#RN #RP #X #Y #f * -X -Y -f // -#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct +fact lexs_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 lexs_inv_atom1: ∀RN,RP,Y,f. ⋆ ⦻*[RN, RP, f] Y → Y = ⋆. +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,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. -#RN #RP #X #Y #f * -X -Y -f -[ #f #J #K1 #W1 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_next … H2) -g destruct +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 +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct /2 width=5 by ex3_2_intro/ -| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_push_next … H) +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H) ] qed-. (* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma lexs_inv_next1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. +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,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. -#RN #RP #X #Y #f * -X -Y -f -[ #f #J #K1 #W1 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_next_push … H) -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct +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 +| #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 lexs_inv_push1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. +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-. -fact lexs_inv_atom2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆. -#RN #RP #X #Y #f * -X -Y -f // -#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct +fact lexs_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 lexs_inv_atom2: ∀RN,RP,X,f. X ⦻*[RN, RP, f] ⋆ → X = ⋆. +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,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. -#RN #RP #X #Y #f * -X -Y -f -[ #f #J #K2 #W2 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_next … H2) -g destruct +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 +| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct /2 width=5 by ex3_2_intro/ -| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_push_next … H) +| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H) ] qed-. (* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma lexs_inv_next2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. +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,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. -#RN #RP #X #Y #f * -X -Y -f -[ #f #J #K2 #W2 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_next_push … H) -| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct +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 +| #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 lexs_inv_push2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. +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,I1,I2,L1,L2,V1,V2,f. - L1.ⓑ{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2. -#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_next1 … H) -H -#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ +lemma lexs_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 (lexs_inv_next1 … H) -H +#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. -lemma lexs_inv_push: ∀RN,RP,I1,I2,L1,L2,V1,V2,f. - L1.ⓑ{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2. -#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push1 … H) -H -#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ +lemma lexs_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 (lexs_inv_push1 … H) -H +#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. -lemma lexs_inv_tl: ∀RN,RP,I,L1,L2,V1,V2,f. L1 ⦻*[RN, RP, ⫱f] L2 → - RN L1 V1 V2 → RP L1 V1 V2 → - L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2. -#RN #RP #I #L2 #L2 #V1 #V2 #f elim (pn_split f) * +lemma lexs_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 lexs_next, lexs_push/ qed-. +(* Basic forward lemmas *****************************************************) + +lemma lexs_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 (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf // +qed-. + (* Basic properties *********************************************************) -lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP, f] L2). -#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 // -#I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H +lemma lexs_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 lexs_next/ | elim (eq_inv_px … H) -H /3 width=3 by lexs_push/ ] qed-. -lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⦻*[RN, RP, f] L2). +lemma lexs_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 lexs_eq_repl_back/ (**) (* full auto fails *) qed-. -(* Note: fexs_sym and fexs_trans hold, but lexs_sym and lexs_trans do not *) -(* Basic_2A1: includes: lpx_sn_refl *) -lemma lexs_refl: ∀RN,RP,f. - (∀L. reflexive … (RN L)) → - (∀L. reflexive … (RP L)) → - reflexive … (lexs RN RP f). -#RN #RP #f #HRN #HRP #L generalize in match f; -f elim L -L // -#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/ +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) * +#g #H destruct /2 width=1 by lexs_next, lexs_push/ qed. -lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) → - ∀L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 → - ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2. -#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 // -#I #L1 #L2 #V1 #V2 #f2 #_ #HV12 #IH -[ * * [2: #n1 ] ] #f1 #H -[ /4 width=5 by lexs_next, sle_inv_nn/ -| /4 width=5 by lexs_push, sle_inv_pn/ -| elim (sle_inv_xp … H) -H [2,3: // ] +lemma lexs_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 … (lexs RN RP f). +#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f +/3 width=2 by lexs_next, lexs_push/ +qed-. + +lemma lexs_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 lexs_inv_tl, lexs_fwd_bind/ qed-. + +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. 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 lexs_push, isid_inv_push/ +] +qed-. + +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 // +#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12 +[ elim (pn_split f1) * ] +[ /4 width=5 by lexs_push, sle_inv_pn/ +| /4 width=5 by lexs_next, sle_inv_nn/ +| elim (sle_inv_xp … H12) -H12 [2,3: // ] #g1 #H #H1 destruct /3 width=5 by lexs_push/ ] qed-. -lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) → - ∀L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 → - ∀f2. f1 ⊆ f2 → L1 ⦻*[RN, RP, f2] L2. -#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 // -#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH -[2: * * [2: #n2 ] ] #f2 #H -[ /4 width=5 by lexs_next, sle_inv_pn/ -| /4 width=5 by lexs_push, sle_inv_pp/ -| elim (sle_inv_nx … H) -H [2,3: // ] +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 // +#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 +[2: elim (pn_split f2) * ] +[ /4 width=5 by lexs_push, sle_inv_pp/ +| /4 width=5 by lexs_next, sle_inv_pn/ +| elim (sle_inv_nx … H12) -H12 [2,3: // ] #g2 #H #H2 destruct /3 width=5 by lexs_next/ ] qed-. -lemma lexs_co: ∀RN1,RP1,RN2,RP2. - (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) → - (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) → - ∀L1,L2,f. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2. -#RN1 #RP1 #RN2 #RP2 #HRN #HRP #L1 #L2 #f #H elim H -L1 -L2 -f -/3 width=1 by lexs_atom, lexs_next, lexs_push/ +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 +[ /2 width=3 by lexs_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 lexs_next, ex2_intro/ +| elim (sle_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_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-. -(* Basic_2A1: removed theorems 17: - llpx_sn_inv_bind llpx_sn_inv_flat - llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length - llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx - llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co - llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx -*) +lemma lexs_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 lexs_atom, or_introl/ +| #L2 #I2 #f @or_intror #H + lapply (lexs_inv_atom1 … H) -H #H destruct +| #f @or_intror #H + lapply (lexs_inv_atom2 … H) -H #H destruct +| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12 + [2: /4 width=3 by lexs_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 lexs_push, lexs_next, or_introl/ ] + @or_intror #H + [ elim (lexs_inv_push … H) | elim (lexs_inv_next … H) ] -H + /2 width=1 by/ +] +qed-.