From 85ba2f09d81f44b8c75505cc470f1fc5c431b9f2 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 10 Feb 2016 12:03:01 +0000 Subject: [PATCH] - ground_2: update ... - basic_2: first commit for lexs ... --- .../notation/relations/relationstar_4.ma | 19 ++ .../lambdadelta/basic_2/relocation/lexs.ma | 189 +++++++++++++++++ .../basic_2/relocation/lexs_length.ma | 25 +++ .../lambdadelta/basic_2/relocation/lpx_sn.ma | 191 ------------------ .../contribs/lambdadelta/ground_2/lib/bool.ma | 23 ++- .../ground_2/relocation/nstream_after.ma | 16 ++ .../ground_2/relocation/nstream_id.ma | 6 +- .../ground_2/relocation/nstream_sle.ma | 75 +++++++ 8 files changed, 345 insertions(+), 199 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma create mode 100644 matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma create mode 100644 matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma create mode 100644 matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_sle.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma new file mode 100644 index 000000000..6015011d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) + +notation "hvbox( L1 ⦻ * break [ term 46 R , break term 46 f ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'RelationStar $R $f $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma new file mode 100644 index 000000000..38fd8e785 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -0,0 +1,189 @@ +(**************************************************************************) +(* ___ *) +(* ||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/nstream_sle.ma". +include "basic_2/notation/relations/relationstar_4.ma". +include "basic_2/grammar/lenv.ma". + +(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *) +inductive lexs (R:relation4 bool lenv term term): rtmap → relation lenv ≝ +| lexs_atom: ∀f. lexs R f (⋆) (⋆) +| lexs_next: ∀I,L1,L2,V1,V2,f. + lexs R f L1 L2 → R (Ⓣ) L1 V1 V2 → + lexs R (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) +| lexs_push: ∀I,L1,L2,V1,V2,f. + lexs R f L1 L2 → R (Ⓕ) L1 V1 V2 → + lexs R (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) +. + +interpretation "general entrywise extension (local environment)" + 'RelationStar R f L1 L2 = (lexs R f L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lexs_inv_atom1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → X = ⋆ → Y = ⋆. +#R #X #Y #f * -X -Y -f // +#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct +qed-. + +(* Basic_2A1: includes lpx_sn_inv_atom1 *) +lemma lexs_inv_atom1: ∀R,Y,f. ⋆ ⦻*[R, f] Y → Y = ⋆. +/2 width=6 by lexs_inv_atom1_aux/ qed-. + +fact lexs_inv_next1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g → + ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & Y = K2.ⓑ{J}W2. +#R #X #Y #f * -X -Y -f +[ #f #J #K1 #W1 #g #H destruct +| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K1 #W1 #g #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) +] +qed-. + +(* Basic_2A1: includes lpx_sn_inv_pair1 *) +lemma lexs_inv_next1: ∀R,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[R, ⫯g] Y → + ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & Y = K2.ⓑ{J}W2. +/2 width=7 by lexs_inv_next1_aux/ qed-. + + +fact lexs_inv_push1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g → + ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & Y = K2.ⓑ{J}W2. +#R #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 #HO #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lexs_inv_push1: ∀R,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[R, ↑g] Y → + ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & Y = K2.ⓑ{J}W2. +/2 width=7 by lexs_inv_push1_aux/ qed-. + +fact lexs_inv_atom2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → Y = ⋆ → X = ⋆. +#R #X #Y #f * -X -Y -f // +#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct +qed-. + +(* Basic_2A1: includes lpx_sn_inv_atom2 *) +lemma lexs_inv_atom2: ∀R,X,f. X ⦻*[R, f] ⋆ → X = ⋆. +/2 width=6 by lexs_inv_atom2_aux/ qed-. + +fact lexs_inv_next2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g → + ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & X = K1.ⓑ{J}W1. +#R #X #Y #f * -X -Y -f +[ #f #J #K2 #W2 #g #H destruct +| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K2 #W2 #g #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) +] +qed-. + +(* Basic_2A1: includes lpx_sn_inv_pair2 *) +lemma lexs_inv_next2: ∀R,J,X,K2,W2,g. X ⦻*[R, ⫯g] K2.ⓑ{J}W2 → + ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & X = K1.ⓑ{J}W1. +/2 width=7 by lexs_inv_next2_aux/ qed-. + +fact lexs_inv_push2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g → + ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & X = K1.ⓑ{J}W1. +#R #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 #HO #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lexs_inv_push2: ∀R,J,X,K2,W2,g. X ⦻*[R, ↑g] K2.ⓑ{J}W2 → + ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓕ) 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: ∀R,I1,I2,L1,L2,V1,V2,f. + L1.ⓑ{I1}V1 ⦻*[R, ⫯f] (L2.ⓑ{I2}V2) → + ∧∧ L1 ⦻*[R, f] L2 & R (Ⓣ) L1 V1 V2 & I1 = I2. +#R #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/ +qed-. + +lemma lexs_inv_push: ∀R,I1,I2,L1,L2,V1,V2,f. + L1.ⓑ{I1}V1 ⦻*[R, ↑f] (L2.ⓑ{I2}V2) → + ∧∧ L1 ⦻*[R, f] L2 & R (Ⓕ) L1 V1 V2 & I1 = I2. +#R #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/ +qed-. + +(* Basic properties *********************************************************) + +lemma lexs_eq_repl_back: ∀R,L1,L2. eq_stream_repl_back … (λf. L1 ⦻*[R, f] L2). +#R #L1 #L2 #f1 #H elim H -L1 -L2 -f1 // +[ #I #L1 #L2 #V1 #v2 #f1 #_ #HS #IH #f2 #H elim (next_inv_sn … H) -H /3 width=1 by lexs_next/ +| #I #L1 #L2 #V1 #v2 #f1 #_ #HO #IH #f2 #H elim (push_inv_sn … H) -H /3 width=1 by lexs_push/ +] +qed-. + +lemma lexs_eq_repl_fwd: ∀R,L1,L2. eq_stream_repl_fwd … (λf. L1 ⦻*[R, f] L2). +#R #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *) +qed-. + +(* Basic_2A1: includes: lpx_sn_refl *) +lemma lexs_refl: ∀R,f. + (∀b,L. reflexive … (R b L)) → + reflexive … (lexs R f). +#R #f #HR #L generalize in match f; -f elim L -L // +#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/ +qed. + +lemma sle_lexs_trans: ∀R. (∀L,T1,T2. R (Ⓣ) L T1 T2 → R (Ⓕ) L T1 T2) → + ∀L1,L2,f2. L1 ⦻*[R, f2] L2 → + ∀f1. f1 ⊆ f2 → L1 ⦻*[R, f1] L2. +#R #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_SS_aux/ +| /4 width=5 by lexs_push, sle_inv_OS_aux/ +| elim (sle_inv_xO_aux … H) -H [3: // |2: skip ] + #g1 #H #H1 destruct /3 width=5 by lexs_push/ +] +qed-. + +lemma sle_lexs_conf: ∀R. (∀L,T1,T2. R (Ⓕ) L T1 T2 → R (Ⓣ) L T1 T2) → + ∀L1,L2,f1. L1 ⦻*[R, f1] L2 → + ∀f2. f1 ⊆ f2 → L1 ⦻*[R, f2] L2. +#R #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_OS_aux/ +| /4 width=5 by lexs_push, sle_inv_OO_aux/ +| elim (sle_inv_Sx_aux … H) -H [3: // |2: skip ] + #g2 #H #H2 destruct /3 width=5 by lexs_next/ +] +qed-. + +lemma lexs_co: ∀R1,R2. + (∀b,L1,T1,T2. R1 b L1 T1 T2 → R2 b L1 T1 T2) → + ∀L1,L2,f. L1 ⦻*[R1, f] L2 → L1 ⦻*[R2, f] L2. +#R1 #R2 #HR #L1 #L2 #f #H elim H -L1 -L2 -f +/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 +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma new file mode 100644 index 000000000..c3626ee27 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "basic_2/grammar/lenv_length.ma". +include "basic_2/relocation/lexs.ma". + +(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) + +(* Forward lemmas on length for local environments **************************) + +(* Basic_2A1: includes: lpx_sn_fwd_length *) +lemma lexs_fwd_length: ∀R,L1,L2,f. L1 ⦻*[R, f] L2 → |L1| = |L2|. +#R #L1 #L2 #f #H elim H -L1 -L2 -f // +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma deleted file mode 100644 index 513d62019..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn.ma +++ /dev/null @@ -1,191 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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/nstream_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 ****) - -(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *) -inductive lexs (RS,RO:relation3 lenv term term): rtmap → relation lenv ≝ -| lexs_atom: ∀f. lexs RS RO f (⋆) (⋆) -| lexs_next: ∀I,L1,L2,V1,V2,f. - lexs RS RO f L1 L2 → RS L1 V1 V2 → - lexs RS RO (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) -| lexs_push: ∀I,L1,L2,V1,V2,f. - lexs RS RO f L1 L2 → RO L1 V1 V2 → - lexs RS RO (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) -. - -interpretation "general entrywise extension (local environment)" - 'RelationStar RS RO f L1 L2 = (lexs RS RO f L1 L2). - -(* Basic inversion lemmas ***************************************************) - -fact lexs_inv_atom1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → X = ⋆ → Y = ⋆. -#RS #RO #X #Y #f * -X -Y -f // -#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct -qed-. - -(* Basic_2A1: includes lpx_sn_inv_atom1 *) -lemma lexs_inv_atom1: ∀RS,RO,Y,f. ⋆ ⦻*[RS, RO, f] Y → Y = ⋆. -/2 width=6 by lexs_inv_atom1_aux/ qed-. - -fact lexs_inv_next1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g → - ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & Y = K2.ⓑ{J}W2. -#RS #RO #X #Y #f * -X -Y -f -[ #f #J #K1 #W1 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K1 #W1 #g #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) -] -qed-. - -(* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma lexs_inv_next1: ∀RS,RO,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RS, RO, ⫯g] Y → - ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & Y = K2.ⓑ{J}W2. -/2 width=7 by lexs_inv_next1_aux/ qed-. - - -fact lexs_inv_push1_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g → - ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & Y = K2.ⓑ{J}W2. -#RS #RO #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 #HO #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct - /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma lexs_inv_push1: ∀RS,RO,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RS, RO, ↑g] Y → - ∃∃K2,W2. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & Y = K2.ⓑ{J}W2. -/2 width=7 by lexs_inv_push1_aux/ qed-. - -fact lexs_inv_atom2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → Y = ⋆ → X = ⋆. -#RS #RO #X #Y #f * -X -Y -f // -#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct -qed-. - -(* Basic_2A1: includes lpx_sn_inv_atom2 *) -lemma lexs_inv_atom2: ∀RS,RO,X,f. X ⦻*[RS, RO, f] ⋆ → X = ⋆. -/2 width=6 by lexs_inv_atom2_aux/ qed-. - -fact lexs_inv_next2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g → - ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & X = K1.ⓑ{J}W1. -#RS #RO #X #Y #f * -X -Y -f -[ #f #J #K2 #W2 #g #H destruct -| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K2 #W2 #g #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) -] -qed-. - -(* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma lexs_inv_next2: ∀RS,RO,J,X,K2,W2,g. X ⦻*[RS, RO, ⫯g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RS K1 W1 W2 & X = K1.ⓑ{J}W1. -/2 width=7 by lexs_inv_next2_aux/ qed-. - -fact lexs_inv_push2_aux: ∀RS,RO,X,Y,f. X ⦻*[RS, RO, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g → - ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RO K1 W1 W2 & X = K1.ⓑ{J}W1. -#RS #RO #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 #HO #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct - /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma lexs_inv_push2: ∀RS,RO,J,X,K2,W2,g. X ⦻*[RS, RO, ↑g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RS, RO, g] K2 & RO 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: ∀RS,RO,I1,I2,L1,L2,V1,V2,f. - L1.ⓑ{I1}V1 ⦻*[RS, RO, ⫯f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RS, RO, f] L2 & RS L1 V1 V2 & I1 = I2. -#RS #RO #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/ -qed-. - -lemma lexs_inv_push: ∀RS,RO,I1,I2,L1,L2,V1,V2,f. - L1.ⓑ{I1}V1 ⦻*[RS, RO, ↑f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RS, RO, f] L2 & RO L1 V1 V2 & I1 = I2. -#RS #RO #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/ -qed-. - -(* Basic properties *********************************************************) - -lemma lexs_eq_repl_back: ∀RS,RO,L1,L2. eq_stream_repl_back … (λf. L1 ⦻*[RS, RO, f] L2). -#RS #RO #L1 #L2 #f1 #H elim H -L1 -L2 -f1 // -[ #I #L1 #L2 #V1 #v2 #f1 #_ #HS #IH #f2 #H elim (next_inv_sn … H) -H /3 width=1 by lexs_next/ -| #I #L1 #L2 #V1 #v2 #f1 #_ #HO #IH #f2 #H elim (push_inv_sn … H) -H /3 width=1 by lexs_push/ -] -qed-. - -lemma lexs_eq_repl_fwd: ∀RS,RO,L1,L2. eq_stream_repl_fwd … (λf. L1 ⦻*[RS, RO, f] L2). -#RS #RO #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *) -qed-. - -(* Basic_2A1: includes: lpx_sn_refl *) -lemma lexs_refl: ∀RS,RO,f. - (∀L. reflexive … (RS L)) → - (∀L. reflexive … (RO L)) → - reflexive … (lexs RS RO f). -#RS #RO #f #HS #HO #L generalize in match f; -f elim L -L // -#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/ -qed. - -lemma sle_lexs_trans: ∀RS,RO. (∀L,T1,T2. RO L T1 T2 → RS L T1 T2) → - ∀L1,L2,f2. L1 ⦻*[RS, RO, f2] L2 → - ∀f1. f1 ⊆ f2 → L1 ⦻*[RS, RO, f1] L2. -#RS #RO #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 // -#I #L1 #L2 #V1 #V2 #f2 #_ #HV12 #IH -[2: * * [2: #n1 ] ] #f1 #H -[