X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Fcpxs.ma;h=a11bb4892f9749cf0b10f133e02f3da220a0d856;hb=82500a9ceb53e1af0263c22afbd5954fa3a83190;hp=ba5fd7c5afae197944e2a3ee8303d3bb181d52b4;hpb=8ed01fd6a38bea715ceb449bb7b72a46bad87851;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma index ba5fd7c5a..a11bb4892 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma @@ -12,115 +12,117 @@ (* *) (**************************************************************************) -include "basic_2/notation/relations/predstar_5.ma". +include "basic_2/notation/relations/predstar_6.ma". include "basic_2/unfold/sstas.ma". include "basic_2/reduction/cnx.ma". include "basic_2/computation/cprs.ma". (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************) -definition cpxs: ∀h. sd h → lenv → relation term ≝ - λh,g. LTC … (cpx h g). +definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝ + λh,g,G. LTC … (cpx h g G). interpretation "extended context-sensitive parallel computation (term)" - 'PRedStar h g L T1 T2 = (cpxs h g L T1 T2). + 'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2). (* Basic eliminators ********************************************************) -lemma cpxs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 → +lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 → (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) → ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2. -#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12 +#h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) // qed-. -lemma cpxs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 → +lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 → (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) → ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1. -#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12 +#h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12 @(TC_star_ind_dx … HT2 IHT2 … HT12) // qed-. (* Basic properties *********************************************************) -lemma cpxs_refl: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T. +lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T. /2 width=1/ qed. -lemma cpx_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. +lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. /2 width=1/ qed. -lemma cpxs_strap1: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → +lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. normalize /2 width=3/ qed. -lemma cpxs_strap2: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → +lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. normalize /2 width=3/ qed. -lemma lsubr_cpxs_trans: ∀h,g. lsub_trans … (cpxs h g) lsubr. +lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr. /3 width=5 by lsubr_cpx_trans, TC_lsub_trans/ qed-. -lemma sstas_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •* [h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. -#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // +lemma sstas_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •* [h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. +#h #g #G #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by cpxs_strap1, ssta_cpx/ qed. -lemma cprs_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. -#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/ +lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2. +#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/ qed. -lemma cpxs_bind_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → - ∀I,T1,T2. ⦃h, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 → +lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → + ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 → ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2. -#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1 +#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1 /3 width=1/ /3 width=3/ qed. -lemma cpxs_flat_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → +lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h, g] ⓕ{I} V2. T2. -#h #g #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/ + ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2. +#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/ qed. -lemma cpxs_flat_sn: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → +lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h, g] ⓕ{I} V2. T2. -#h #g #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/ + ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2. +#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/ qed. -lemma cpxs_zeta: ∀h,g,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T → - ⦃h, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2. -#h #g #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/ +lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T → + ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2. +#h #g #G #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/ qed. -lemma cpxs_tau: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2. -#h #g #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/ +lemma cpxs_tau: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → + ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2. +#h #g #G #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/ qed. -lemma cpxs_ti: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2. -#h #g #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/ +lemma cpxs_ti: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → + ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2. +#h #g #G #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/ qed. -lemma cpxs_beta_dx: ∀h,g,a,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → +lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2. + ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2. -#h #g #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/ +#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/ /4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *) qed. -lemma cpxs_theta_dx: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → +lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2. + ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2. -#h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ] +#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ] /4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *) qed. (* Basic inversion lemmas ***************************************************) -lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 → +lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 → ∃∃n,l. deg h g k (n+l) & U2 = ⋆((next h)^n k). -#h #g #L #U2 #k #H @(cpxs_ind … H) -U2 +#h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2 [ elim (deg_total h g k) #l #Hkl @(ex2_2_intro … 0 … Hkl) -Hkl // | #U #U2 #_ #HU2 * #n #l #Hknl #H destruct @@ -132,11 +134,11 @@ lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 → ] qed-. -lemma cpxs_inv_cast1: ∀h,g,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 → +lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 → ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2 | ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2 | ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2. -#h #g #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/ +#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/ #U2 #U #_ #HU2 * /3 width=3/ * #W #T #HW1 #HT1 #H destruct elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3/ * @@ -145,8 +147,8 @@ lapply (cpxs_strap1 … HW1 … HW2) -W lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/ qed-. -lemma cpxs_inv_cnx1: ∀h,g,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U. -#h #g #L #T #U #H @(cpxs_ind_dx … H) -T // +lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U. +#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T // #T0 #T #H1T0 #_ #IHT #H2T0 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/ qed-.