From: Ferruccio Guidi Date: Wed, 1 Feb 2017 19:52:07 +0000 (+0000) Subject: - advances in rt_transition X-Git-Tag: make_still_working~507 X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=commitdiff_plain;h=a9e994f4ca8e04743c53ed8cb057ee5dae80c8f9 - advances in rt_transition - bugs fixed and refactoring of some parked files --- diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc deleted file mode 100644 index 57ecbaf68..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq.etc +++ /dev/null @@ -1,42 +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 "basic_2/syntax/lenv.ma". - -(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) - -definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2. - -definition cfull: relation3 lenv term term ≝ λL,T1,T2. ⊤. - -(* Basic properties *********************************************************) - -lemma ceq_refl (L): reflexive … (ceq L). -// qed. - -lemma cfull_refl (L): reflexive … (cfull L). -// qed. - -lemma ceq_sym (L): symmetric … (ceq L). -// qed-. - -lemma cfull_sym (L): symmetric … (cfull L). -// qed-. - -lemma cfull_top (R:relation3 lenv term term) (L) (T1) (T2): - R L T1 T2 → cfull L T1 T2. -// qed-. - -lemma ceq_cfull (L) (T1) (T2): ceq L T1 T2 → cfull L T1 T2. -// qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc deleted file mode 100644 index b17b186e2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/ceq_ceq.etc +++ /dev/null @@ -1,37 +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 "basic_2/syntax/ceq.ma". - -(* CONTEXT-SENSITIVE EQUIVALENCES FOR TERMS *********************************) - -(* Main properties **********************************************************) - -theorem ceq_trans (L): Transitive … (ceq L). -// qed-. - -lemma ceq_canc_sn (L): left_cancellable … (ceq L). -// qed-. - -lemma ceq_canc_dx (L): right_cancellable … (ceq L). -// qed-. - -theorem cfull_trans (L): Transitive … (cfull L). -// qed-. - -lemma cfull_canc_sn (L): left_cancellable … (cfull L). -// qed-. - -lemma cfull_canc_dx (L): right_cancellable … (cfull L). -// qed-. \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpx_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx_lreq.ma new file mode 100644 index 000000000..9c41e252a --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpx_lreq.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||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/substitution/drop_lreq.ma". +include "basic_2/reduction/cpx.ma". + +(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) + +(* Properties on equivalence for local environments *************************) + +lemma lreq_cpx_trans: ∀h,o,G. lsub_trans … (cpx h o G) (lreq 0 (∞)). +#h #o #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 +[ // +| /2 width=2 by cpx_st/ +| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (lreq_drop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/ +|4,9: /4 width=1 by cpx_bind, cpx_beta, lreq_pair_O_Y/ +|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/ +|6,10: /4 width=3 by cpx_zeta, cpx_theta, lreq_pair_O_Y/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/psubst_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/psubst_6.etc new file mode 100644 index 000000000..56cb72ec4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpy/psubst_6.etc @@ -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( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubst $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststar_6.etc new file mode 100644 index 000000000..90362edb6 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststar_6.etc @@ -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( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststaralt_6.etc new file mode 100644 index 000000000..80a29a841 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpys/psubststaralt_6.etc @@ -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( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" + non associative with precedence 45 + for @{ 'PSubstStarAlt $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc deleted file mode 100644 index b21fe5117..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/droppreds_3.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⫱ * [ term 46 L , break term 46 K ] term 46 f )" - non associative with precedence 46 - for @{ 'DropPreds $L $K $f }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/fpb_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/fpb_lift.ma new file mode 100644 index 000000000..6fae399d9 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/fpb_lift.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||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/unfold/lstas_da.ma". +include "basic_2/reduction/cpx_lift.ma". +include "basic_2/reduction/fpb.ma". + +(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************) + +(* Advanced properties ******************************************************) + +lemma sta_fpb: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → + ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄. +#h #o #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2) +/3 width=2 by fpb_cpx, sta_cpx/ #H destruct +elim (lstas_inv_refl_pos h G L T2 0) // +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/fpbq_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/etc/fpbq_lift.ma new file mode 100644 index 000000000..4029d9417 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/fpbq_lift.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/reduction/cpx_lift.ma". +include "basic_2/reduction/fpbq.ma". + +(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************) + +(* Advanced properties ******************************************************) + +lemma sta_fpbq: ∀h,o,G,L,T1,T2,d. + ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → + ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄. +/3 width=4 by fpbq_cpx, sta_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc deleted file mode 100644 index 70d3a1c66..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lazyor_5.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( L1 ⋓ break [ term 46 T , break term 46 f ] break term 46 L2 ≡ break term 46 L )" - non associative with precedence 45 - for @{ 'LazyOr $L1 $T $f $L2 $L }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc deleted file mode 100644 index bfe23e721..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfpr_main.etc +++ /dev/null @@ -1,80 +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 "basic_2/rt_transition/lfpr_lfpr.ma". - -(* PARALLEL R-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES ****************) - -definition lfxs_confluent_R: relation2 … ≝ - λRP1,RP2. - ∀L0,T0,T1. RP1 L0 T0 T1 → ∀T2. RP2 L0 T0 T2 → - ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → - ∃∃L. L1 ⦻*[RP2, T1] L & L2 ⦻*[RP1, T2] L. - -(* Main properties **********************************************************) - -fact lfpr_conf_cpr_atom_atom: - ∀h,I,G,L0. ( - ∀L,T. ⦃G, L0, ⓪{I}⦄ ⊐+ ⦃G, L, T⦄ → - ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → - ∀L1. ⦃G, L⦄ ⊢ ➡[h, T] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h, T] L2 → - ∃∃K0. ⦃G, L1⦄ ⊢ ➡[h, T1] K0 & ⦃G, L2⦄ ⊢ ➡[h, T2] K0 - ) → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h, ⓪{I}] L2 → - ∃∃L. ⦃G, L1⦄ ⊢ ➡[h, ⓪{I}] L & ⦃G, L2⦄ ⊢ ➡[h, ⓪{I}] L. -#h #I #G * [ | #K0 #J #V0 cases I -I [ | * | ] ] -[ #_ #L1 #HL01 #L2 #HL02 - lapply (lfpr_inv_atom_sn … HL01) -HL01 #H destruct - lapply (lfpr_inv_atom_sn … HL02) -HL02 #H destruct - /2 width=3 by ex2_intro/ -| #s #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_sort_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_sort_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_sort, fqu_fqup, fqu_drop, ex2_intro/ -| #IH #L1 #HL01 #L2 #HL02 - elim (lfpr_inv_zero_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #HV01 #H destruct - elim (lfpr_inv_zero_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #HV02 #H destruct - elim (cpr_conf_lfpr … HV01 … HV02 … HK01 … HK02) #V #HV1 #HV2 - elim (IH … HV01 … HV02 … HK01 … HK02) -IH -HV01 -HV02 -HK01 -HK02 - /3 width=5 by lfpr_zero, fqu_fqup, fqu_drop, ex2_intro/ -| #i #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_lref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_lref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_lref, fqu_fqup, fqu_drop, ex2_intro/ -| #l #IH #L1 #HL01 #L2 #HL02 - elim (lfxs_inv_gref_pair_sn … HL01) -HL01 #K1 #V1 #HK01 #H destruct - elim (lfxs_inv_gref_pair_sn … HL02) -HL02 #K2 #V2 #HK02 #H destruct - elim (IH … HK01 … HK02) -IH -HK01 -HK02 - /3 width=5 by lfpr_gref, fqu_fqup, fqu_drop, ex2_intro/ -] -qed-. - -theorem lfpr_conf_cpr: ∀h,G. lfxs_confluent_R (cpm 0 h G) (cpm 0 h G). -#h #G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ] -[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_atom1_drops … H1) -H1 - elim (cpr_inv_atom1_drops … H2) -H2 - [ #H2 #H1 destruct - /3 width=7 by lfpr_conf_cpr_atom_atom/ - | * #K0 #V0 #V2 * [2: #i2 ] #HLK0 #HV02 #HVT2 #H2 #H1 destruct - -(* - -theorem lpr_conf: ∀G. confluent … (lpr G). -/3 width=6 by lpx_sn_conf, cpr_conf_lpr/ -qed-. - -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc deleted file mode 100644 index f09a2cfc2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs.etc +++ /dev/null @@ -1,264 +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/rtmap_id.ma". -include "basic_2/notation/relations/relationstar_4.ma". -include "basic_2/relocation/lexs.ma". -include "basic_2/static/frees.ma". - -(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) - -definition lfxs (R) (T): relation lenv ≝ - λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2. - -interpretation "generic extension on referred entries (local environment)" - 'RelationStar R T L1 L2 = (lfxs R T L1 L2). - -definition R_frees_confluent: predicate (relation3 lenv term term) ≝ - λRN. - ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 → - ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1. - -definition lexs_frees_confluent: relation (relation3 lenv term term) ≝ - λRN,RP. - ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → - ∀L2. L1 ⦻*[RN, RP, f1] L2 → - ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. - -definition R_confluent2_lfxs: relation4 (relation3 lenv term term) - (relation3 lenv term term) … ≝ - λR1,R2,RP1,RP2. - ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → - ∃∃T. R2 L1 T1 T & R1 L2 T2 T. - -(* Basic properties ***********************************************************) - -lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. -/3 width=3 by lexs_atom, frees_atom, ex2_intro/ -qed. - -lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s. - L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2. -#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/ -qed. - -lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2. -#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/ -qed. - -lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i. - L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2. -#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/ -qed. - -lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l. - L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2. -#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/ -qed. - -lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. - L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 → - ∀V2. R L1 V V2 → - L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2. -#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR -/3 width=5 by lexs_pair_repl, ex2_intro/ -qed-. - -lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull → - (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → - ∀T. symmetric … (lfxs R T). -#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1 -/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/ -qed-. - -lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → - ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. -#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆. -#R #I #Y2 * /2 width=4 by lexs_inv_atom1/ -qed-. - -lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. -#R #I #Y1 * /2 width=4 by lexs_inv_atom2/ -qed-. - -lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 -[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ -| lapply (frees_inv_sort … H1) -H1 #Hf - elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct - elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct - /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/ -] -qed-. - -lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * -[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ -| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg - /4 width=9 by ex4_5_intro, ex2_intro, or_intror/ -] -qed-. - -lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * -[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ -| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg - /4 width=8 by ex3_5_intro, ex2_intro, or_intror/ -] -qed-. - -lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2 -[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ -| lapply (frees_inv_gref … H1) -H1 #Hf - elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct - elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct - /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/ -] -qed-. - -lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → - L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. -#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf -/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ -qed-. - -lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → - L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2. -#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf -/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 → - ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. -#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * -[ #H destruct -| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. -#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * -[ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → - ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & - Y2 = L2.ⓑ{I}V2. -#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H * -[ #H destruct -| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct - /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & - Y1 = L1.ⓑ{I}V1. -#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H * -[ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct - /2 width=5 by ex3_2_intro/ -] -qed-. - -lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 → - ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2. -#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H * -[ #H destruct -| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1. -#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H * -[ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 → - ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. -#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H * -[ #H destruct -| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. -#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H * -[ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ -] -qed-. - -(* Basic forward lemmas *****************************************************) - -lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2. -#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf -/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/ -qed-. - -lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. -#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV // -qed-. - -lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2. -#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // -qed-. - -lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2. -#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // -qed-. - -lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2. -#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/ -qed-. - -(* Basic_2A1: removed theorems 24: - llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref - llpx_sn_bind llpx_sn_flat - 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/etc/lfxs/lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs.etc new file mode 100644 index 000000000..f09a2cfc2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs.etc @@ -0,0 +1,264 @@ +(**************************************************************************) +(* ___ *) +(* ||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_id.ma". +include "basic_2/notation/relations/relationstar_4.ma". +include "basic_2/relocation/lexs.ma". +include "basic_2/static/frees.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +definition lfxs (R) (T): relation lenv ≝ + λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2. + +interpretation "generic extension on referred entries (local environment)" + 'RelationStar R T L1 L2 = (lfxs R T L1 L2). + +definition R_frees_confluent: predicate (relation3 lenv term term) ≝ + λRN. + ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 → + ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1. + +definition lexs_frees_confluent: relation (relation3 lenv term term) ≝ + λRN,RP. + ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → + ∀L2. L1 ⦻*[RN, RP, f1] L2 → + ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. + +definition R_confluent2_lfxs: relation4 (relation3 lenv term term) + (relation3 lenv term term) … ≝ + λR1,R2,RP1,RP2. + ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → + ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + +(* Basic properties ***********************************************************) + +lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. +/3 width=3 by lexs_atom, frees_atom, ex2_intro/ +qed. + +lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s. + L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/ +qed. + +lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/ +qed. + +lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i. + L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/ +qed. + +lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l. + L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2. +#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/ +qed. + +lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 → + ∀V2. R L1 V V2 → + L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2. +#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR +/3 width=5 by lexs_pair_repl, ex2_intro/ +qed-. + +lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull → + (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → + ∀T. symmetric … (lfxs R T). +#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1 +/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/ +qed-. + +lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → + ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2. +#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆. +#R #I #Y2 * /2 width=4 by lexs_inv_atom1/ +qed-. + +lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. +#R #I #Y1 * /2 width=4 by lexs_inv_atom2/ +qed-. + +lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_sort … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * +[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ +| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg + /4 width=9 by ex4_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * +[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ +| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg + /4 width=8 by ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → + (Y1 = ⋆ ∧ Y2 = ⋆) ∨ + ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_gref … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct + /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/ +] +qed-. + +lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → + L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. +#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf +/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ +qed-. + +lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → + L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2. +#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf +/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ +qed-. + +(* Advanced inversion lemmas ************************************************) + +lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 → + ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → + ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & + Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct + /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 → + ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 → + ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. +#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H * +[ #H destruct +| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. +#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H * +[ #_ #H destruct +| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2. +#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf +/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/ +qed-. + +lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2. +#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV // +qed-. + +lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2. +#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // +qed-. + +lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2. +#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // +qed-. + +lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2. +#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/ +qed-. + +(* Basic_2A1: removed theorems 24: + llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref + llpx_sn_bind llpx_sn_flat + 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/etc/lfxs/lfxs_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_drops.etc new file mode 100644 index 000000000..0a3f287bb --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_drops.etc @@ -0,0 +1,93 @@ +(**************************************************************************) +(* ___ *) +(* ||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/relocation/drops_ceq.ma". +include "basic_2/relocation/drops_lexs.ma". +include "basic_2/static/frees_drops.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +definition dedropable_sn: predicate (relation3 lenv term term) ≝ + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → + ∀K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U → + ∃∃L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. + +definition dropable_sn: predicate (relation3 lenv term term) ≝ + λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ → + ∀L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U → + ∃∃K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2. + +definition dropable_dx: predicate (relation3 lenv term term) ≝ + λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 → + ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U → + ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[R, T] K2. + +(* Properties with generic slicing for local environments *******************) + +(* Basic_2A1: includes: llpx_sn_lift_le llpx_sn_lift_ge *) +lemma lfxs_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) → + d_liftable2 R → dedropable_sn R. +#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU +elim (frees_total L1 U) #f2 #Hf2 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf +elim (lexs_liftable_co_dedropable … H1R … H2R … HLK1 … HK12 … Hf) -f1 -K1 +/3 width=6 by cfull_lift, ex3_intro, ex2_intro/ +qed-. + +(* Inversion lemmas with generic slicing for local environments *************) + +(* Basic_2A1: restricts: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *) +(* Basic_2A1: was: llpx_sn_drop_conf_O *) +lemma lfxs_dropable_sn: ∀R. dropable_sn R. +#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU +elim (frees_total K1 T) #f1 #Hf1 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f +elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1 +/3 width=3 by ex2_intro/ +qed-. + +(* Basic_2A1: was: llpx_sn_drop_trans_O *) +(* Note: the proof might be simplified *) +lemma lfxs_dropable_dx: ∀R. dropable_dx R. +#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU +elim (drops_isuni_ex … H1f L1) #K1 #HLK1 +elim (frees_total K1 T) #f1 #Hf1 +lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f +elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2 +/4 width=9 by frees_inv_lifts, ex2_intro/ +qed-. + +(* Basic_2A1: was: llpx_sn_inv_lift_O *) +lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 → + ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 → + ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2. +#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU +elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY +lapply (drops_mono … HY … HLK2) -L2 -i #H destruct // +qed-. + +lemma lfxs_inv_lref_sn: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // +#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY +#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. + +lemma lfxs_inv_lref_dx: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. +#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // +#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY +#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_fqup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_fqup.etc new file mode 100644 index 000000000..62c5b0564 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_fqup.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||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/static/frees_fqup.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Advanced properties ******************************************************) + +lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L. +#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/ +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_length.etc new file mode 100644 index 000000000..01dd82cf4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_length.etc @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||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/relocation/lexs_length.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Forward lemmas with length for local environments ************************) + +lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|. +#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_lfxs.etc new file mode 100644 index 000000000..307740bea --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs/lfxs_lfxs.etc @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||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/relocation/lexs_lexs.ma". +include "basic_2/static/frees_fqup.ma". +include "basic_2/static/frees_frees.ma". +include "basic_2/static/lfxs.ma". + +(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) + +(* Main properties **********************************************************) + +theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T. + L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 → + L1 ⦻*[R, ⓑ{p,I}V1.T] L2. +#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 +elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2)) +/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/ +qed. + +theorem lfxs_flat: ∀R,I,L1,L2,V,T. + L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 → + L1 ⦻*[R, ⓕ{I}V.T] L2. +#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) +/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/ +qed. + +theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull → + ∀T. Transitive … (lfxs R T). +#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 +elim (H1R … Hf1 … HL1) #f #H0 #H1 +lapply (frees_mono … Hf2 … H0) -Hf2 -H0 #Hf2 +lapply (lexs_eq_repl_back … HL2 … Hf2) -f2 #HL2 +lapply (sle_lexs_trans … HL1 … H1) -HL1 // #Hl1 +@(ex2_intro … f) + +/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/ +qed-. + +theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull → + R_confluent2_lfxs R R R R → + ∀T. confluent … (lfxs R T). +#R #H1R #H2R #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02 +lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12 +lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01 +elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ] +[ #L #HL1 #HL2 + elim (H1R … Hf … HL01) -HL01 #f1 #Hf1 #H1 + elim (H1R … Hf … HL02) -HL02 #f2 #Hf2 #H2 + lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1 + lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2 + /3 width=5 by ex2_intro/ +| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02 + elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0 + lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01 + lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02 + elim (H2R … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc deleted file mode 100644 index 0a3f287bb..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_drops.etc +++ /dev/null @@ -1,93 +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 "basic_2/relocation/drops_ceq.ma". -include "basic_2/relocation/drops_lexs.ma". -include "basic_2/static/frees_drops.ma". -include "basic_2/static/lfxs.ma". - -(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) - -definition dedropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → - ∀K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U → - ∃∃L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. - -definition dropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ → - ∀L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U → - ∃∃K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2. - -definition dropable_dx: predicate (relation3 lenv term term) ≝ - λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 → - ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U → - ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[R, T] K2. - -(* Properties with generic slicing for local environments *******************) - -(* Basic_2A1: includes: llpx_sn_lift_le llpx_sn_lift_ge *) -lemma lfxs_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) → - d_liftable2 R → dedropable_sn R. -#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU -elim (frees_total L1 U) #f2 #Hf2 -lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf -elim (lexs_liftable_co_dedropable … H1R … H2R … HLK1 … HK12 … Hf) -f1 -K1 -/3 width=6 by cfull_lift, ex3_intro, ex2_intro/ -qed-. - -(* Inversion lemmas with generic slicing for local environments *************) - -(* Basic_2A1: restricts: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *) -(* Basic_2A1: was: llpx_sn_drop_conf_O *) -lemma lfxs_dropable_sn: ∀R. dropable_sn R. -#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU -elim (frees_total K1 T) #f1 #Hf1 -lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f -elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1 -/3 width=3 by ex2_intro/ -qed-. - -(* Basic_2A1: was: llpx_sn_drop_trans_O *) -(* Note: the proof might be simplified *) -lemma lfxs_dropable_dx: ∀R. dropable_dx R. -#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU -elim (drops_isuni_ex … H1f L1) #K1 #HLK1 -elim (frees_total K1 T) #f1 #Hf1 -lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f -elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2 -/4 width=9 by frees_inv_lifts, ex2_intro/ -qed-. - -(* Basic_2A1: was: llpx_sn_inv_lift_O *) -lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 → - ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 → - ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2. -#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU -elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY -lapply (drops_mono … HY … HLK2) -L2 -i #H destruct // -qed-. - -lemma lfxs_inv_lref_sn: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 → - ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. -#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // -#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY -#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ -qed-. - -lemma lfxs_inv_lref_dx: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 → - ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2. -#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // -#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY -#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc deleted file mode 100644 index 62c5b0564..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_fqup.etc +++ /dev/null @@ -1,24 +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 "basic_2/static/frees_fqup.ma". -include "basic_2/static/lfxs.ma". - -(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) - -(* Advanced properties ******************************************************) - -lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L. -#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/ -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc deleted file mode 100644 index 01dd82cf4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_length.etc +++ /dev/null @@ -1,24 +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 "basic_2/relocation/lexs_length.ma". -include "basic_2/static/lfxs.ma". - -(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) - -(* Forward lemmas with length for local environments ************************) - -lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|. -#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc deleted file mode 100644 index 307740bea..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lfxs_lfxs.etc +++ /dev/null @@ -1,70 +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 "basic_2/relocation/lexs_lexs.ma". -include "basic_2/static/frees_fqup.ma". -include "basic_2/static/frees_frees.ma". -include "basic_2/static/lfxs.ma". - -(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) - -(* Main properties **********************************************************) - -theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T. - L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 → - L1 ⦻*[R, ⓑ{p,I}V1.T] L2. -#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 -elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2)) -/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/ -qed. - -theorem lfxs_flat: ∀R,I,L1,L2,V,T. - L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 → - L1 ⦻*[R, ⓕ{I}V.T] L2. -#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) -/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/ -qed. - -theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull → - ∀T. Transitive … (lfxs R T). -#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 -elim (H1R … Hf1 … HL1) #f #H0 #H1 -lapply (frees_mono … Hf2 … H0) -Hf2 -H0 #Hf2 -lapply (lexs_eq_repl_back … HL2 … Hf2) -f2 #HL2 -lapply (sle_lexs_trans … HL1 … H1) -HL1 // #Hl1 -@(ex2_intro … f) - -/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/ -qed-. - -theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull → - R_confluent2_lfxs R R R R → - ∀T. confluent … (lfxs R T). -#R #H1R #H2R #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02 -lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12 -lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01 -elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ] -[ #L #HL1 #HL2 - elim (H1R … Hf … HL01) -HL01 #f1 #Hf1 #H1 - elim (H1R … Hf … HL02) -HL02 #f2 #Hf2 #H2 - lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1 - lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2 - /3 width=5 by ex2_intro/ -| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02 - elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0 - lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01 - lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02 - elim (H2R … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc deleted file mode 100644 index 6135be4d4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lrsubeq_4.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( L1 break ⊆ [ term 46 l , break term 46 m ] break term 46 L2 )" - non associative with precedence 45 - for @{ 'LRSubEq $L1 $l $k $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lrsubeq_4.etc new file mode 100644 index 000000000..6135be4d4 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lrsubeq_4.etc @@ -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 l , break term 46 m ] break term 46 L2 )" + non associative with precedence 45 + for @{ 'LRSubEq $L1 $l $k $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ltls.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ltls.etc deleted file mode 100644 index c68bf4bf2..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/ltls.etc +++ /dev/null @@ -1,47 +0,0 @@ -include "basic_2/notation/functions/droppreds_3.ma". -include "basic_2/grammar/lenv_length.ma". - -axiom pred_minus: ∀x,y. y < x → ⫰(x - y) = x - ⫯y. - -(* -axiom drops_T_isuni_inv_refl: ∀n,L. ⬇*[n] L ≡ L → n = 0. - -lemma le_succ_trans: ∀m,n. ⫯m ≤ n → m ≤ n. -/2 width=1 by lt_to_le/ qed-. -*) - -lemma tls_pred: ∀f,n. 0 < n → ⫱*[n] f = ⫱ ⫱*[⫰n] f. -#f #n #Hn >tls_S >S_pred // -qed-. - -definition ltls (f): lenv → lenv → rtmap ≝ λL,K. ⫱*[|L|-|K|] f. - -interpretation "ltls (rtmap)" 'DropPreds L K f = (ltls f L K). - -lemma ltls_refl: ∀f,L1,L2. |L1| ≤ |L2| → ⫱*[L1, L2] f = f. -#f #L1 #L2 #HL12 whd in ⊢ (??%?); >(eq_minus_O … HL12) -HL12 // -qed. - -lemma ltls_pair2: ∀f,I,L1,L2,V. |L2| < |L1| → ⫱⫱*[L1, L2.ⓑ{I}V] f = ⫱*[L1, L2] f. -#f #I #L1 #L2 #V #HL12 whd in ⊢ (??(?%)%); minus_Sn_m // -qed. - -lemma ltls_pair1_next: ∀f,I,L1,L2,V. |L2| ≤ |L1| → ⫱*[L1.ⓑ{I}V, L2] ⫯f = ⫱*[L1, L2] f. -#f #I #L1 #L2 #V #HL12 whd in ⊢ (??%%); >minus_Sn_m // -qed. - -lemma ltls_sle_pair: ∀f1,f2,L1,L2. ⫱*[L2, L1] f2 ⊆ ⫱*[L1, L2] f1 → - ∀I,V1. ⫱*[L2, L1.ⓑ{I}V1] f2 ⊆ ⫱*[L1.ⓑ{I}V1, L2] ⫯f1. -#f1 #f2 #L1 #L2 elim (lt_or_ge (|L1|) (|L2|)) -[ #HL12 >ltls_refl in ⊢ (??%→?); /2 width=1 by lt_to_le/ - #Hf21 #I #V1 >ltls_refl in ⊢ (??%); // - <(ltls_pair2 … I … V1 HL12) in Hf21; -HL12 /2 width=1 by sle_inv_tl1/ -| #HL21 >ltls_refl // #Hf21 #I #V1 >ltls_refl /2 width=1 by le_S/ - >ltls_pair1_next // -] -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/droppreds_3.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/droppreds_3.etc new file mode 100644 index 000000000..b21fe5117 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/droppreds_3.etc @@ -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( ⫱ * [ term 46 L , break term 46 K ] term 46 f )" + non associative with precedence 46 + for @{ 'DropPreds $L $K $f }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/ltls.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/ltls.etc new file mode 100644 index 000000000..c68bf4bf2 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/ltls/ltls.etc @@ -0,0 +1,47 @@ +include "basic_2/notation/functions/droppreds_3.ma". +include "basic_2/grammar/lenv_length.ma". + +axiom pred_minus: ∀x,y. y < x → ⫰(x - y) = x - ⫯y. + +(* +axiom drops_T_isuni_inv_refl: ∀n,L. ⬇*[n] L ≡ L → n = 0. + +lemma le_succ_trans: ∀m,n. ⫯m ≤ n → m ≤ n. +/2 width=1 by lt_to_le/ qed-. +*) + +lemma tls_pred: ∀f,n. 0 < n → ⫱*[n] f = ⫱ ⫱*[⫰n] f. +#f #n #Hn >tls_S >S_pred // +qed-. + +definition ltls (f): lenv → lenv → rtmap ≝ λL,K. ⫱*[|L|-|K|] f. + +interpretation "ltls (rtmap)" 'DropPreds L K f = (ltls f L K). + +lemma ltls_refl: ∀f,L1,L2. |L1| ≤ |L2| → ⫱*[L1, L2] f = f. +#f #L1 #L2 #HL12 whd in ⊢ (??%?); >(eq_minus_O … HL12) -HL12 // +qed. + +lemma ltls_pair2: ∀f,I,L1,L2,V. |L2| < |L1| → ⫱⫱*[L1, L2.ⓑ{I}V] f = ⫱*[L1, L2] f. +#f #I #L1 #L2 #V #HL12 whd in ⊢ (??(?%)%); minus_Sn_m // +qed. + +lemma ltls_pair1_next: ∀f,I,L1,L2,V. |L2| ≤ |L1| → ⫱*[L1.ⓑ{I}V, L2] ⫯f = ⫱*[L1, L2] f. +#f #I #L1 #L2 #V #HL12 whd in ⊢ (??%%); >minus_Sn_m // +qed. + +lemma ltls_sle_pair: ∀f1,f2,L1,L2. ⫱*[L2, L1] f2 ⊆ ⫱*[L1, L2] f1 → + ∀I,V1. ⫱*[L2, L1.ⓑ{I}V1] f2 ⊆ ⫱*[L1.ⓑ{I}V1, L2] ⫯f1. +#f1 #f2 #L1 #L2 elim (lt_or_ge (|L1|) (|L2|)) +[ #HL12 >ltls_refl in ⊢ (??%→?); /2 width=1 by lt_to_le/ + #Hf21 #I #V1 >ltls_refl in ⊢ (??%); // + <(ltls_pair2 … I … V1 HL12) in Hf21; -HL12 /2 width=1 by sle_inv_tl1/ +| #HL21 >ltls_refl // #Hf21 #I #V1 >ltls_refl /2 width=1 by le_S/ + >ltls_pair1_next // +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/notation/lazyor_5.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/notation/lazyor_5.etc new file mode 100644 index 000000000..70d3a1c66 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/notation/lazyor_5.etc @@ -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 T , break term 46 f ] break term 46 L2 ≡ break term 46 L )" + non associative with precedence 45 + for @{ 'LazyOr $L1 $T $f $L2 $L }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/notation/notation.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/notation/notation.etc new file mode 100644 index 000000000..e60c75d42 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/notation/notation.etc @@ -0,0 +1,47 @@ +(**************************************************************************) +(* ___ *) +(* ||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( ⦃G, L⦄ ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )" + non associative with precedence 45 + for @{ 'ICM $L $T $s }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )" + non associative with precedence 45 + for @{ 'BinaryArity $h $L $T $A }. + +notation "hvbox( h ⊢ break term 46 L1 ÷ ⫃ break term 46 L2 )" + non associative with precedence 45 + for @{ 'LRSubEqB $h $L1 $L2 }. + +notation "hvbox( L1 ⊢ ⬌ break term 46 L2 )" + non associative with precedence 45 + for @{ 'PConvSn $L1 $L2 }. + +notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )" + non associative with precedence 45 + for @{ 'PConvSnStar $L1 $L2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeType $h $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeAlt $h $L $T1 $T2 }. + +notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeStar $h $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc deleted file mode 100644 index 56cb72ec4..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/psubst_6.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubst $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc deleted file mode 100644 index 90362edb6..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststar_6.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStar $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc deleted file mode 100644 index 80a29a841..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/psubststaralt_6.etc +++ /dev/null @@ -1,19 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 break ▶ ▶ * [ term 46 l , break term 46 m ] break term 46 T2 )" - non associative with precedence 45 - for @{ 'PSubstStarAlt $G $L $T1 $l $k $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma deleted file mode 100644 index e60c75d42..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma +++ /dev/null @@ -1,47 +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 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( ⦃G, L⦄ ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )" - non associative with precedence 45 - for @{ 'ICM $L $T $s }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )" - non associative with precedence 45 - for @{ 'BinaryArity $h $L $T $A }. - -notation "hvbox( h ⊢ break term 46 L1 ÷ ⫃ break term 46 L2 )" - non associative with precedence 45 - for @{ 'LRSubEqB $h $L1 $L2 }. - -notation "hvbox( L1 ⊢ ⬌ break term 46 L2 )" - non associative with precedence 45 - for @{ 'PConvSn $L1 $L2 }. - -notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )" - non associative with precedence 45 - for @{ 'PConvSnStar $L1 $L2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeType $h $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeAlt $h $L $T1 $T2 }. - -notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )" - non associative with precedence 45 - for @{ 'NativeTypeStar $h $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lfxs.ma new file mode 100644 index 000000000..4ce37d910 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lfxs.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||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/rt_transition/cpx_lfxs.ma". +include "basic_2/rt_transition/cpm_cpx.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************) + +(* Properties with generic extension on referred entries ********************) + +(* Basic_2A1: was just: cpr_llpx_sn_conf *) +lemma cpm_lfxs_conf: ∀R,n,h,G. s_r_confluent1 … (cpm n h G) (lfxs R). +/3 width=5 by cpm_fwd_cpx, cpx_lfxs_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_llpx_sn.ma deleted file mode 100644 index 8395d7501..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_llpx_sn.ma +++ /dev/null @@ -1,47 +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 "basic_2/multiple/llpx_sn_drop.ma". -include "basic_2/reduction/cpr.ma". - -(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************) - -(* Properties on lazy sn pointwise extensions *******************************) - -lemma cpr_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R → - ∀G. b_c_confluent1 … (cpr G) (llpx_sn R 0). -#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2 -[ // -| #G #Ls #Ks #V1b #V2b #W2b #i #HLKs #_ #HVW2b #IHV12b #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H - #Kd #V1l #HLKd #HV1b #HV1sd - lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs - lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd - @(llpx_sn_lift_le … HLKs HLKd … HVW2b) -HLKs -HLKd -HVW2b /2 width=1 by/ (**) (* full auto too slow *) -| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H - /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/ -| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - /3 width=1 by llpx_sn_flat/ -| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H - /3 width=10 by llpx_sn_inv_lift_le, drop_drop/ -| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/ -| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - #HV1 #H elim (llpx_sn_inv_bind_O … H) -H - /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/ -| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - #HV1 #H elim (llpx_sn_inv_bind_O … H) -H // - #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *) - [ /3 width=10 by llpx_sn_lift_le, drop_drop/ - | /3 width=4 by llpx_sn_bind_repl_O/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma index 7d20bc47f..0241664db 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma @@ -230,3 +230,34 @@ lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H /3 width=4 by ex2_2_intro, ex_intro/ qed-. + +(* Basic eliminators ********************************************************) + +lemma cpx_ind: ∀h. ∀R:relation4 genv lenv term term. + (∀I,G,L. R G L (⓪{I}) (⓪{I})) → + (∀G,L,s. R G L (⋆s) (⋆(next h s))) → + (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → R G K V1 V2 → + ⬆*[1] V2 ≡ W2 → R G (K.ⓑ{I}V1) (#0) W2 + ) → (∀I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → R G K (#i) T → + ⬆*[1] T ≡ U → R G (K.ⓑ{I}V) (#⫯i) (U) + ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 → + R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) + ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → + R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) + ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T → R G (L.ⓓV) T1 T → + ⬆*[1] T2 ≡ T → R G L (+ⓓV.T1) T2 + ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2 → + R G L (ⓝV.T1) T2 + ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → R G L V1 V2 → + R G L (ⓝV1.T) V2 + ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 → + R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 → + R G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2) + ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 → + R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 → + ⬆*[1] V ≡ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2) + ) → + ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2. +#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2 +* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfdeq.ma new file mode 100644 index 000000000..1a4d722e7 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfdeq.ma @@ -0,0 +1,114 @@ +(**************************************************************************) +(* ___ *) +(* ||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/relocation/lifts_tdeq.ma". +include "basic_2/static/lfdeq.ma". +include "basic_2/rt_transition/lfpx.ma". + +(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) + +(* Properties with degree-based equivalence for terms ***********************) + +(* Basic_2A1: was just: cpx_lleq_conf *) +lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o). +#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/ +[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02 + elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct + /4 width=3 by tdeq_sort, deg_next, ex2_intro/ +| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2 + >(tdeq_inv_lref1 … H0) -H0 + elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct + elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct + elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2 + elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/ +| #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2 + >(tdeq_inv_lref1 … H0) -H0 + elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct + elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct + elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2 + elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/ +| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct + elim (lfpx_inv_bind … H1) -H1 #HL01 #H1 + elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2 + lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2 + elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02 + elim (IHT … HT02 … H1 … H2) -L0 -T0 + /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/ +| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct + elim (lfpx_inv_flat … H1) -H1 #HL01 #H1 + elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2 + elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02 + elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0 + /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/ +| #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct + elim (lfpx_inv_bind … H1) -H1 #HL01 #H1 + elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2 + lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2 + elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1 + elim (tdeq_inv_lifts … HT1 … HUT1) -T1 + /3 width=5 by cpx_zeta, ex2_intro/ +| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct + elim (lfpx_inv_flat … H1) -H1 #HL01 #H1 + elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2 + elim (IH … HT02 … H1 … H2) -L0 -V0 -T0 + /3 width=3 by cpx_eps, ex2_intro/ +| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct + elim (lfpx_inv_flat … H1) -H1 #HL01 #H1 + elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2 + elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1 + /3 width=3 by cpx_ee, ex2_intro/ +| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct + elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct + elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1 + elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0 + elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2 + elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0 + lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0 + elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 + elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0 + elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0 + /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *) +| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2 + elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct + elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct + elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1 + elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0 + elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2 + elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0 + lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0 + elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1 + elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0 + elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0 + elim (tdeq_lifts … HV1 … HVU1) -V1 + /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *) +] +qed-. +(* +lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 → + ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2. +/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-. + +lemma cpx_lleq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h o G) (lleq 0). +/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-. + +lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 → + ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2. +/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-. +*) \ No newline at end of file diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfxs.ma new file mode 100644 index 000000000..b47cd88bf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lfxs.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||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/rt_transition/lfpx_frees.ma". + +(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) + +(* Properties with generic extension on referred entries ********************) + +(* Note: lemma 1000 *) +(* Basic_2A1: was just: cpx_llpx_sn_conf *) +lemma cpx_lfxs_conf: ∀R,h,G. s_r_confluent1 … (cpx h G) (lfxs R). +#R #h #G #L1 #T1 #T2 #H #L2 * #f1 #Hf1 elim (cpx_frees_conf … Hf1 … H) -T1 +/3 width=5 by sle_lexs_trans, ex2_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma deleted file mode 100644 index 946975957..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lleq.ma +++ /dev/null @@ -1,55 +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 "basic_2/multiple/lleq_drop.ma". -include "basic_2/reduction/cpx_llpx_sn.ma". - -(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) - -(* Properties on lazy equivalence for local environments ********************) - -lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 → - ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2. -#h #o #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/ -[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2 - [ #H elim (ylt_yle_false … H) // - | * /3 width=7 by cpx_delta/ - ] -| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H - /3 width=1 by cpx_bind/ -| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H - /3 width=1 by cpx_flat/ -| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #H elim (lleq_inv_bind_O … H) -H - /3 width=3 by cpx_zeta/ -| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H - /3 width=1 by cpx_eps/ -| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H - /3 width=1 by cpx_ct/ -| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H - #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=1 by cpx_beta/ -| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H - #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=3 by cpx_theta/ -] -qed-. - -lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 → - ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2. -/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-. - -lemma cpx_lleq_conf_sn: ∀h,o,G. b_c_confluent1 … (cpx h o G) (lleq 0). -/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-. - -lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 → - ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2. -/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma deleted file mode 100644 index ec6255dfc..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_llpx_sn.ma +++ /dev/null @@ -1,50 +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 "basic_2/multiple/llpx_sn_drop.ma". -include "basic_2/reduction/cpx.ma". - -(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) - -(* Properties on lazy sn pointwise extensions *******************************) - -(* Note: lemma 1000 *) -lemma cpx_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R → - ∀h,o,G. b_c_confluent1 … (cpx h o G) (llpx_sn R 0). -#R #H1R #H2R #H3R #h #o #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2 -[ // -| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/ -| #I #G #Ls #Ks #V1b #V2b #W2b #i #HLKs #_ #HVW2b #IHV12b #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H - #Kd #V1l #HLKd #HV1b #HV1sd - lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs - lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd - @(llpx_sn_lift_le … HLKs HLKd … HVW2b) -HLKs -HLKd -HVW2b /2 width=1 by/ (**) (* full auto too slow *) -| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H - /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/ -| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - /3 width=1 by llpx_sn_flat/ -| #G #Ls #V #T1 #T2 #T #_ #HT2 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H - /3 width=10 by llpx_sn_inv_lift_le, drop_drop/ -| #G #Ls #V #T1 #T2 #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/ -| #G #Ls #V1 #V2 #T #_ #IHV12 #Ld #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/ -| #a #G #Ls #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - #HV1 #H elim (llpx_sn_inv_bind_O … H) -H - /4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_flat, llpx_sn_bind/ -| #a #G #Ls #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H - #HV1 #H elim (llpx_sn_inv_bind_O … H) -H // - #HW1 #HT1 @llpx_sn_bind_O /2 width=1 by/ @llpx_sn_flat (**) (* full auto too slow *) - [ /3 width=10 by llpx_sn_lift_le, drop_drop/ - | /3 width=4 by llpx_sn_bind_repl_O/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma deleted file mode 100644 index 9c41e252a..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lreq.ma +++ /dev/null @@ -1,32 +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 "basic_2/substitution/drop_lreq.ma". -include "basic_2/reduction/cpx.ma". - -(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************) - -(* Properties on equivalence for local environments *************************) - -lemma lreq_cpx_trans: ∀h,o,G. lsub_trans … (cpx h o G) (lreq 0 (∞)). -#h #o #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -[ // -| /2 width=2 by cpx_st/ -| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (lreq_drop_trans_be … HL12 … HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/ -|4,9: /4 width=1 by cpx_bind, cpx_beta, lreq_pair_O_Y/ -|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/ -|6,10: /4 width=3 by cpx_zeta, cpx_theta, lreq_pair_O_Y/ -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma index cce7d39ff..f0e7362b2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma @@ -13,28 +13,29 @@ (**************************************************************************) include "basic_2/notation/relations/btpredproper_8.ma". -include "basic_2/substitution/fqu.ma". -include "basic_2/multiple/lleq.ma". -include "basic_2/reduction/lpx.ma". +include "basic_2/s_transition/fqu.ma". +include "basic_2/static/lfdeq.ma". +include "basic_2/rt_transition/lfpr_lfpx.ma". -(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************) +(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************) inductive fpb (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝ | fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h o G1 L1 T1 G2 L2 T2 -| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2 -| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1 +| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≡[h, o] T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2 +| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h, T1] L2 → (L1 ≡[h, o, T1] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1 . interpretation - "'rst' proper parallel reduction (closure)" + "proper parallel rst-transition (closure)" 'BTPRedProper h o G1 L1 T1 G2 L2 T2 = (fpb h o G1 L1 T1 G2 L2 T2). (* Basic properties *********************************************************) -lemma cpr_fpb: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → +(* Basic_2A1: includes: cpr_fpb *) +lemma cpm_fpb: ∀n,h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≡[h, o] T2 → ⊥) → ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄. -/3 width=1 by fpb_cpx, cpr_cpx/ qed. +/3 width=2 by fpb_cpx, cpm_fwd_cpx/ qed. -lemma lpr_fpb: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) → +lemma lpr_fpb: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → ⦃G, L1, T⦄ ≻[h, o] ⦃G, L2, T⦄. -/3 width=1 by fpb_lpx, lpr_lpx/ qed. +/3 width=1 by fpb_lpx, lfpr_fwd_lfpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_lift.ma deleted file mode 100644 index 6fae399d9..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_lift.ma +++ /dev/null @@ -1,28 +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 "basic_2/unfold/lstas_da.ma". -include "basic_2/reduction/cpx_lift.ma". -include "basic_2/reduction/fpb.ma". - -(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************) - -(* Advanced properties ******************************************************) - -lemma sta_fpb: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → - ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄. -#h #o #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2) -/3 width=2 by fpb_cpx, sta_cpx/ #H destruct -elim (lstas_inv_refl_pos h G L T2 0) // -qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_lift.ma deleted file mode 100644 index 4029d9417..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_lift.ma +++ /dev/null @@ -1,25 +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 "basic_2/reduction/cpx_lift.ma". -include "basic_2/reduction/fpbq.ma". - -(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************) - -(* Advanced properties ******************************************************) - -lemma sta_fpbq: ∀h,o,G,L,T1,T2,d. - ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → - ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄. -/3 width=4 by fpbq_cpx, sta_cpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt index f2ca8d42a..d00ba2bea 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt @@ -1,6 +1,7 @@ cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma -cpx.ma cpx_simple.ma cpx_drops.ma cpx_fqus.ma cpx_lsubr.ma +cpx.ma cpx_simple.ma cpx_drops.ma cpx_fqus.ma cpx_lsubr.ma cpx_lfxs.ma cpx_lfdeq.ma lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma lfpx_frees.ma lfpx_aaa.ma -cpm.ma cpm_simple.ma cpm_drops.ma cpm_lsubr.ma cpm_cpx.ma +cpm.ma cpm_simple.ma cpm_drops.ma cpm_lsubr.ma cpm_lfxs.ma cpm_cpx.ma cpr.ma cpr_drops.ma lfpr.ma lfpr_length.ma lfpr_drops.ma lfpr_fqup.ma lfpr_frees.ma lfpr_aaa.ma lfpr_lfpx.ma lfpr_lfpr.ma +fpb.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma index 207e1e2d3..a5c11db63 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/ffdeq.ma @@ -14,7 +14,7 @@ include "basic_2/notation/relations/lazyeq_8.ma". include "basic_2/syntax/genv.ma". -include "basic_2/static/lfdeq_fqup.ma". +include "basic_2/static/lfdeq.ma". (* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************) diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl index d98e42c5c..e920f2c0d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl @@ -131,27 +131,27 @@ table { *) class "water" [ { "rt-transition" * } { -(* - [ { "parallel qrst-reduction" * } { - [ "fpbq ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_alt ( ⦃?,?,?⦄ ≽≽[?,?] ⦃?,?,?⦄ )" "fpbq_lift" + "fpbq_aaa" * ] - [ "fpb ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpb_lift" + "fpb_lleq" + "fpb_fleq" * ] + [ { "parallel qrst-rtransition" * } { +(* [ "fpbq ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_alt ( ⦃?,?,?⦄ ≽≽[?,?] ⦃?,?,?⦄ )" + "fpbq_aaa" * ] *) + [ "fpb ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" (* "fpb_lleq" + "fpb_fleq" *) * ] } ] +(* [ { "context-sensitive rt-reduction" * } { [ "lpx_lleq" * ] - [ "cpx_lreq" + "cpx_llpx_sn" + "cpx_lleq" * ] + [ "cpx_lleq" * ] } ] *) [ { "t-bound context-sensitive rt-transition" * } { [ "lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fqup" + "lfpr_frees" + "lfpr_aaa" + "lfpr_lfpx" + "lfpr_lfpr" * ] - [ "cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" (* + "cpr_llpx_sn" + "cpr_cir" *) * ] - [ "cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_drops" + "cpm_lsubr" + "cpm_cpx" * ] + [ "cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" * ] + [ "cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_drops" + "cpm_lsubr" + "cpm_lfxs" + "cpm_cpx" * ] } ] [ { "uncounted context-sensitive rt-transition" * } { [ "lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fqup" + "lfpx_frees" + "lfpx_aaa" * ] - [ "cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" * ] + [ "cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_lfxs" + "cpx_lfdeq" * ] } ] [ { "counted context-sensitive rt-transition" * } { @@ -360,5 +360,6 @@ class "italic" { 1 } [ "lpx_sn" "lpx_sn_alt" + "lpx_sn_tc" + "lpx_sn_drop" + "lpx_sn_lpx_sn" * ] } ] + [ "cpx_lreq" + "cpr_cir" + "fpb_lift" + "fpbq_lift" ] [ "lleq ( ? ≡[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_lreq" + "lleq_drop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ] *)