X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flexs.ma;h=7fd35071acaea51d259b3f7f828de5ab4ee4970a;hb=5b93ea047903b606979705ed25a6df6504fd027c;hp=0bed76cfd05d77e09c8ce748845b841719212e04;hpb=bb6e68b2cf746bb3108543807207a1ca628ab442;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 0bed76cfd..7fd35071a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -12,132 +12,163 @@ (* *) (**************************************************************************) -include "ground_2/relocation/nstream_sle.ma". +include "ground_2/relocation/rtmap_sle.ma". include "basic_2/notation/relations/relationstar_5.ma". include "basic_2/grammar/lenv.ma". -(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) +(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****) (* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *) inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝ | lexs_atom: ∀f. lexs RN RP f (⋆) (⋆) -| lexs_next: ∀I,L1,L2,V1,V2,f. +| lexs_next: ∀f,I,L1,L2,V1,V2. 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_push: ∀f,I,L1,L2,V1,V2. lexs RN RP f L1 L2 → RP L1 V1 V2 → lexs RN RP (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) . -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 lexs_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: relation5 (relation3 lenv term term) + (relation3 lenv term term) … ≝ + 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. + (* 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 #I #L1 #L2 #V1 #V2 #_ #_ #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 → +fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. 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 +#RN #RP #f #X #Y * -f -X -Y +[ #f #g #J #K1 #W1 #H destruct +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #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 #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #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 → +lemma lexs_inv_next1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y → ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. /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 → +fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. 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 +#RN #RP #f #X #Y * -f -X -Y +[ #f #g #J #K1 #W1 #H destruct +| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_next_push … H) +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #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 → +lemma lexs_inv_push1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y → ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. /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 #I #L1 #L2 #V1 #V2 #_ #_ #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 → +fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. 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 +#RN #RP #f #X #Y * -f -X -Y +[ #f #g #J #K2 #W2 #H destruct +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #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 #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #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 → +lemma lexs_inv_next2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 → ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. /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 → +fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. 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 +#RN #RP #f #X #Y * -f -X -Y [ #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 +| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_next_push … H) +| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #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 → +lemma lexs_inv_push2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 → ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. /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. +lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. 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 +#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_next1 … H) -H #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ qed-. -lemma lexs_inv_push: ∀RN,RP,I1,I2,L1,L2,V1,V2,f. +lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. 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 +#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push1 … H) -H #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ qed-. +lemma lexs_inv_tl: ∀RN,RP,f,I,L1,L2,V1,V2. L1 ⦻*[RN, RP, ⫱f] L2 → + RN L1 V1 V2 → RP L1 V1 V2 → + L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2. +#RN #RP #f #I #L2 #L2 #V1 #V2 elim (pn_split f) * +/2 width=1 by lexs_next, lexs_push/ +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lexs_fwd_pair: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. + L1.ⓑ{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 → + L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2. +#RN #RP #f #I1 #I2 #L2 #L2 #V1 #V2 #Hf +elim (pn_split f) * #g #H destruct +[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf +/2 width=1 by conj/ +qed-. + (* Basic properties *********************************************************) -lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_stream_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 -[ elim (next_inv_sn … H) -H /3 width=1 by lexs_next/ -| elim (push_inv_sn … H) -H /3 width=1 by lexs_push/ +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 #I #L1 #L2 #V1 #v2 #_ #HV #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_stream_repl_fwd … (λf. L1 ⦻*[RN, RP, f] L2). -#RN #RP #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *) +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 *) @@ -151,27 +182,27 @@ lemma lexs_refl: ∀RN,RP,f. 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 → + ∀f2,L1,L2. 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 +#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 // +#f2 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH [ * * [2: #n1 ] ] #f1 #H -[ /4 width=5 by lexs_next, sle_inv_SS_aux/ -| /4 width=5 by lexs_push, sle_inv_OS_aux/ -| elim (sle_inv_xO_aux … H) -H [3: // |2: skip ] +[ /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: // ] #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 → + ∀f1,L1,L2. 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 +#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // +#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH [2: * * [2: #n2 ] ] #f2 #H -[ /4 width=5 by lexs_next, sle_inv_OS_aux/ -| /4 width=5 by lexs_push, sle_inv_OO_aux/ -| elim (sle_inv_Sx_aux … H) -H [3: // |2: skip ] +[ /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: // ] #g2 #H #H2 destruct /3 width=5 by lexs_next/ ] qed-. @@ -179,15 +210,7 @@ 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 + ∀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-. - -(* 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 -*)