From c53be14933feb896df2c3c9830b68fe773b2047c Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Fri, 11 May 2018 00:02:57 +0200 Subject: [PATCH] update in basic_2 + first results on cpms and cprs --- .../basic_2/rt_computation/cpms.ma | 44 +++--- .../basic_2/rt_computation/cprs.ma | 127 ++++++++---------- .../basic_2/rt_computation/cprs_etc.ma | 14 ++ .../basic_2/rt_computation/partial.txt | 1 + 4 files changed, 98 insertions(+), 88 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_etc.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma index cbfd28726..edb931591 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma @@ -31,8 +31,27 @@ interpretation "context-sensitive parallel r-computation (term)" 'PRedStar h G L T1 T2 = (cpms h G L O T1 T2). +(* Basic eliminators ********************************************************) + +lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …): + Q 0 T2 → + (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) → + ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1. +#h #G #L #T2 #R @ltc_ind_sn_refl // +qed-. + +lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …): + Q 0 T1 → + (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) → + ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2. +#h #G #L #T1 #R @ltc_ind_dx_refl // +qed-. + (* Basic properties *********************************************************) +(* Basic_1: includes: pr1_pr0 *) +(* Basic_1: uses: pr3_pr2 *) +(* Basic_2A1: includes: cpr_cprs *) lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2. /2 width=1 by ltc_rc/ qed. @@ -44,27 +63,20 @@ lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2. /2 width=3 by ltc_dx/ qed-. +(* Basic_2A1: uses: cprs_bind_dx *) +lemma cpms_bind_dx (n) (h) (G) (L): + ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → + ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 → + ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2. +#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1 +/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed. + (* Basic properties with r-transition ***************************************) +(* Basic_1: was: pr3_refl *) lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0). /2 width=1 by cpm_cpms/ qed. -(* Basic eliminators ********************************************************) - -lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …): - Q 0 T2 → - (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) → - ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1. -#h #G #L #T2 #R @ltc_ind_sn_refl // -qed-. - -lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …): - Q 0 T1 → - (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) → - ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2. -#h #G #L #T1 #R @ltc_ind_dx_refl // -qed-. - (* Basic_2A1: removed theorems 4: sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma index e4abc29d5..af0618bc5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma @@ -12,100 +12,90 @@ (* *) (**************************************************************************) -include "basic_2/notation/relations/predstar_4.ma". -include "basic_2/reduction/cnr.ma". +include "basic_2/rt_transition/cpr.ma". +include "basic_2/rt_computation/cpms.ma". -(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) - -(* Basic_1: includes: pr1_pr0 *) -definition cprs: relation4 genv lenv term term ≝ - λG. CTC … (cpr G). - -interpretation "context-sensitive parallel computation (term)" - 'PRedStar G L T1 T2 = (cprs G L T1 T2). +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION FOR TERMS *************************) (* Basic eliminators ********************************************************) -lemma cprs_ind: ∀G,L,T1. ∀R:predicate term. R T1 → - (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) → - ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2. -#G #L #T1 #R #HT1 #IHT1 #T2 #HT12 -@(TC_star_ind … HT1 IHT1 … HT12) // +(* Basic_2A1: was: cprs_ind_dx *) +lemma cprs_ind_sn (h) (G) (L) (T2) (R:predicate …): + R T2 → + (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → R T → R T1) → + ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T1. +#h #G #L #T2 #R #IH1 #IH2 #T1 +@(insert_eq_0 … 0) #n #H +@(cpms_ind_sn … H) -n -T1 // +#n1 #n2 #T1 #T #HT1 #HT2 #IH #H +elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct +/3 width=4 by/ qed-. -lemma cprs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 → - (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) → - ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1. -#G #L #T2 #R #HT2 #IHT2 #T1 #HT12 -@(TC_star_ind_dx … HT2 IHT2 … HT12) // +(* Basic_2A1: was: cprs_ind *) +lemma cprs_ind_dx (h) (G) (L) (T1) (R:predicate …): + R T1 → + (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → R T → R T2) → + ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T2. +#h #G #L #T1 #R #IH1 #IH2 #T2 +@(insert_eq_0 … 0) #n #H +@(cpms_ind_dx … H) -n -T2 // +#n1 #n2 #T #T2 #HT1 #IH #HT2 #H +elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct +/3 width=4 by/ qed-. (* Basic properties *********************************************************) -(* Basic_1: was: pr3_pr2 *) -lemma cpr_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2. -/2 width=1 by inj/ qed. - -(* Basic_1: was: pr3_refl *) -lemma cprs_refl: ∀G,L,T. ⦃G, L⦄ ⊢ T ➡* T. -/2 width=1 by cpr_cprs/ qed. - -lemma cprs_strap1: ∀G,L,T1,T,T2. - ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2. -normalize /2 width=3 by step/ qed-. - (* Basic_1: was: pr3_step *) -lemma cprs_strap2: ∀G,L,T1,T,T2. - ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2. -normalize /2 width=3 by TC_strap/ qed-. - -lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr. -/3 width=5 by lsubr_cpr_trans, CTC_lsub_trans/ -qed-. - -(* Basic_1: was: pr3_pr1 *) -lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2. -/2 width=3 by lsubr_cprs_trans/ qed. - -lemma cprs_bind_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → - ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2. -#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 -/3 width=3 by cprs_strap2, cpr_cprs, cpr_pair_sn, cpr_bind/ qed. +(* Basic_2A1: was: cprs_strap2 *) +lemma cprs_step_sn (h) (G) (L): + ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → + ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2. +/2 width=3 by cpms_step_sn/ qed-. + +(* Basic_2A1: was: cprs_strap1 *) +lemma cprs_step_dx (h) (G) (L): + ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T → + ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2. +/2 width=3 by cpms_step_dx/ qed-. (* Basic_1: was only: pr3_thin_dx *) -lemma cprs_flat_dx: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → - ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. -#I #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 -/3 width=5 by cprs_strap1, cpr_flat, cpr_cprs, cpr_pair_sn/ +lemma cprs_flat_dx (h) (I) (G) (L): + ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → + ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → + ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2. +#h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1 +/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/ qed. - -lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 → - ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. +(* +lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → + ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2. #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2 /3 width=3 by cprs_strap1, cpr_cprs, cpr_pair_sn, cpr_flat/ qed. lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⬆[0, 1] T2 ≘ T → - ⦃G, L.ⓓV⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* T2. + ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h] T2. #G #L #V #T1 #T #T2 #HT2 #H @(cprs_ind_dx … H) -T1 /3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/ qed. -lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2. +lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h] T2. #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/ qed. lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2. + ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h] T2 → + ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h] ⓓ{a}ⓝW2.V2.T2. #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2 /4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/ qed. lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡ V → ⬆[0, 1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 → - ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2. + ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆[0, 1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h] T2 → + ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h] ⓓ{a}W2.ⓐV2.T2. #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 /4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/ qed. @@ -113,29 +103,22 @@ qed. (* Basic inversion lemmas ***************************************************) (* Basic_1: was: pr3_gen_sort *) -lemma cprs_inv_sort1: ∀G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡* U2 → U2 = ⋆s. +lemma cprs_inv_sort1: ∀G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] U2 → U2 = ⋆s. #G #L #U2 #s #H @(cprs_ind … H) -U2 // #U2 #U #_ #HU2 #IHU2 destruct >(cpr_inv_sort1 … HU2) -HU2 // qed-. (* Basic_1: was: pr3_gen_cast *) -lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨ - ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓝW2.T2. +lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] U2 → ⦃G, L⦄ ⊢ T1 ➡*[h] U2 ∨ + ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & U2 = ⓝW2.T2. #G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_intror/ #U2 #U #_ #HU2 * /3 width=3 by cprs_strap1, or_introl/ * #W #T #HW1 #HT1 #H destruct elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3 by cprs_strap1, or_introl/ * #W2 #T2 #HW2 #HT2 #H destruct /4 width=5 by cprs_strap1, ex3_2_intro, or_intror/ qed-. - -(* Basic_1: was: nf2_pr3_unfold *) -lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U. -#G #L #T #U #H @(cprs_ind_dx … H) -T // -#T0 #T #H1T0 #_ #IHT #H2T0 -lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/ -qed-. - +*) (* Basic_1: removed theorems 13: pr1_head_1 pr1_head_2 pr1_comp clear_pr3_trans pr3_cflat pr3_gen_bind diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_etc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_etc.ma new file mode 100644 index 000000000..254261657 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_etc.ma @@ -0,0 +1,14 @@ +lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr. +/3 width=5 by lsubr_cpr_trans, CTC_lsub_trans/ +qed-. + +(* Basic_1: was: pr3_pr1 *) +lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2. +/2 width=3 by lsubr_cprs_trans/ qed. + +(* Basic_1: was: nf2_pr3_unfold *) +lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U. +#G #L #T #U #H @(cprs_ind_dx … H) -T // +#T0 #T #H1T0 #_ #IHT #H2T0 +lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/partial.txt b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/partial.txt index 24215a065..163c752ff 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/partial.txt +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/partial.txt @@ -10,3 +10,4 @@ fpbs.ma fpbs_fqup.ma fpbs_fqus.ma fpbs_aaa.ma fpbs_cpx.ma fpbs_fpb.ma fpbs_cpxs. fpbg.ma fpbg_fqup.ma fpbg_cpxs.ma fpbg_lfpxs.ma fpbg_fpbs.ma fpbg_fpbg.ma fsb.ma fsb_ffdeq.ma fsb_aaa.ma fsb_csx.ma fsb_fpbg.ma cpms.ma +cprs.ma -- 2.39.2