X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_computation%2Fcpms.ma;h=11c955108632b31a54a370063d00622b420ca5b3;hp=cbfd28726da4b217c36efd0377120dc984680995;hb=19a25bf176255055193372554437729a6fa1894c;hpb=f129bbbfda0e65a5f92ec086246f6e288376d4f9 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..11c955108 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 +31,35 @@ 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 #Q @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 #Q @ltc_ind_dx_refl // +qed-. + +(* Basic inversion lemmas ***************************************************) + +lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s). +#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 // +#n1 #n2 #X #X2 #_ #IH #HX2 destruct +elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct // +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 +71,83 @@ 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 properties with r-transition ***************************************) +(* 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. -lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0). -/2 width=1 by cpm_cpms/ qed. +lemma cpms_appl_dx (n) (h) (G) (L): + ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → + ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → + ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2. +#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1 +/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/ +qed. -(* Basic eliminators ********************************************************) +lemma cpms_zeta (n) (h) (G) (L): + ∀T1,T. ⬆*[1] T ≘ T1 → + ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2. +#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2 +/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/ +qed. -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-. +(* Basic_2A1: uses: cprs_zeta *) +lemma cpms_zeta_dx (n) (h) (G) (L): + ∀T2,T. ⬆*[1] T2 ≘ T → + ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2. +#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1 +/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/ +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: uses: cprs_eps *) +lemma cpms_eps (n) (h) (G) (L): + ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → + ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2. +#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 +/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/ +qed. + +lemma cpms_ee (n) (h) (G) (L): + ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 → + ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2. +#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n +[ /3 width=1 by cpm_cpms, cpm_ee/ +| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1 + /3 width=3 by cpms_step_sn, cpm_ee/ +] +qed. + +(* Basic_2A1: uses: cprs_beta_dx *) +lemma cpms_beta_dx (n) (h) (G) (L): + ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → + ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 → + ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 → + ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2. +#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2 +/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/ +qed. + +(* Basic_2A1: uses: cprs_theta_dx *) +lemma cpms_theta_dx (n) (h) (G) (L): + ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V → + ∀V2. ⬆*[1] V ≘ V2 → + ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 → + ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 → + ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2. +#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2 +/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/ +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_2A1: removed theorems 4: +(* Basic_2A1: removed theorems 5: sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs + scpds_inv_lstas_eq *)