X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Fcprs_cprs.ma;h=e86df253ee59842817485ddcc2c669d3f7ab671a;hb=a8cd6cc85182245df447a21caf16b6503fa4b3e5;hp=75f74caad725a84a3c556c4b423d3378de6c298b;hpb=bd7183da46c7cb0f389cda40955b270c03b57a4b;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma index 75f74caad..e86df253e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma @@ -12,156 +12,144 @@ (* *) (**************************************************************************) -include "basic_2/reducibility/cpr_lift.ma". -include "basic_2/reducibility/cpr_cpr.ma". -include "basic_2/reducibility/lfpr_cpr.ma". -include "basic_2/computation/cprs_lfpr.ma". +include "basic_2/reduction/lpr_lpr.ma". +include "basic_2/computation/cprs_lift.ma". (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) -(* Advanced properties ******************************************************) +(* Main properties **********************************************************) -lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 → - ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2. -#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2 -[ /3 width=2/ -| /3 width=6 by cprs_strap1, cpr_abst/ (**) (* /3 width=6/ is too slow *) -] +(* Basic_1: was: pr3_t *) +(* Basic_1: includes: pr1_t *) +theorem cprs_trans: ∀G,L. Transitive … (cprs G L). +normalize /2 width=3 by trans_TC/ qed-. + +(* Basic_1: was: pr3_confluence *) +(* Basic_1: includes: pr1_confluence *) +theorem cprs_conf: ∀G,L. confluent2 … (cprs G L) (cprs G L). +normalize /3 width=3 by cpr_conf, TC_confluent2/ qed-. + +theorem cprs_bind: ∀a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 → + ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2. +#a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 +/3 width=5 by cprs_trans, cprs_bind_dx/ qed. -lemma cprs_abbr1_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 -[ /3 width=5/ -| #T1 #T #HT1 #_ #IHT1 - @(cprs_strap2 … IHT1) -IHT1 /2 width=1/ -] +(* Basic_1: was: pr3_flat *) +theorem cprs_flat: ∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 → + ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡* ⓕ{I}V2.T2. +#I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 +/3 width=3 by cprs_flat_dx, cprs_strap1, cpr_pair_sn/ qed. -lemma cpr_abbr1: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -/3 width=1/ qed. +theorem cprs_beta_rc: ∀a,G,L,V1,V2,W1,W2,T1,T2. + ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → + ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2. +#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1 by cprs_beta_dx/ +#W #W2 #_ #HW2 #IHW1 (**) (* fulla uto too slow 14s *) +@(cprs_trans … IHW1) -IHW1 /3 width=1 by cprs_flat_dx, cprs_bind/ +qed. -lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 -lapply (lfpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/ +theorem cprs_beta: ∀a,G,L,V1,V2,W1,W2,T1,T2. + ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 → + ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2. +#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1 by cprs_beta_rc/ +#V #V2 #_ #HV2 #IHV1 +@(cprs_trans … IHV1) -IHV1 /3 width=1 by cprs_flat_sn, cprs_bind/ qed. -(* Basic_1: was: pr3_strip *) -lemma cprs_strip: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡ T2 → - ∃∃T0. L ⊢ T1 ➡ T0 & L ⊢ T2 ➡* T0. -/3 width=3/ qed. +theorem cprs_theta_rc: ∀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. +#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cprs_ind … H) -W2 +/3 width=5 by cprs_trans, cprs_theta_dx, cprs_bind_dx/ +qed. + +theorem cprs_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2. + ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 → + ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2. +#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(cprs_ind_dx … H) -V1 +/3 width=3 by cprs_trans, cprs_theta_rc, cprs_flat_dx/ +qed. (* Advanced inversion lemmas ************************************************) (* Basic_1: was pr3_gen_appl *) -lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → - ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 & +lemma cprs_inv_appl1: ∀G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 → + ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓐV2. T2 - | ∃∃a,V2,W,T. L ⊢ V1 ➡* V2 & - L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ➡* U2 - | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 & - L ⊢ T1 ➡* ⓓ{a}V. T & L ⊢ ⓓ{a}V. ⓐV2. T ➡* U2. -#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ + | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T & + ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2 + | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⬆[0,1] V0 ≡ V2 & + ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T & + ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2. +#G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/ #U #U2 #_ #HU2 * * [ #V0 #T0 #HV10 #HT10 #H destruct elim (cpr_inv_appl1 … HU2) -HU2 * - [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ - | #a #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/ - | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct - @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) + [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cprs_strap1, or3_intro0, ex3_2_intro/ + | #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct + lapply (cprs_strap1 … HV10 … HV02) -V0 #HV12 + lapply (lsubr_cpr_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2 + /5 width=5 by cprs_bind, cprs_flat_dx, cpr_cprs, lsubr_beta, ex2_3_intro, or3_intro1/ + | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct + /5 width=10 by cprs_flat_sn, cprs_bind_dx, cprs_strap1, ex4_5_intro, or3_intro2/ ] -| /4 width=9/ -| /4 width=11/ +| /4 width=9 by cprs_strap1, or3_intro1, ex2_3_intro/ +| /4 width=11 by cprs_strap1, or3_intro2, ex4_5_intro/ ] qed-. -(* Main propertis ***********************************************************) - -(* Basic_1: was: pr3_confluence *) -theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 → - ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0. -/3 width=3/ qed. - -(* Basic_1: was: pr3_t *) -theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2. -/2 width=3/ qed. - -(* Basic_1: was: pr3_flat *) -lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 → - L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2. -#I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 -@(cprs_trans … IHV1) -IHV1 /2 width=1/ -qed. - -lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 → - ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2. -#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a #I @(cprs_ind … HV12) -V2 -[ lapply (cprs_lsubr_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/ -| #V0 #V2 #_ #HV02 #IHV01 - @(cprs_trans … IHV01) -V1 /2 width=2/ +(* Properties concerning sn parallel reduction on local environments ********) + +(* Basic_1: was just: pr3_pr2_pr2_t *) +(* Basic_1: includes: pr3_pr0_pr2_t *) +lemma lpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lpr G). +#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2 +[ /2 width=3 by/ +| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12 + elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H + elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct + /4 width=6 by cprs_strap2, cprs_delta/ +|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/ +|4,6: /3 width=1 by cprs_flat, cprs_eps/ +|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/ ] -qed. - -lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → - ∀a.L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #T1 #T2 #HT12 #V2 #HV12 #a @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 -@(cprs_trans … IHV1) -IHV1 /2 width=1/ -qed. - -lemma cprs_bind1: ∀I,L,V1,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 → - ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2. -* /2 width=1/ /2 width=2/ -qed. +qed-. -lemma cprs_abbr2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1 -[ /2 width=1/ -| #T1 #T #HT1 #_ #IHT1 - lapply (lfpr_cpr_trans (L. ⓓV1) … HT1) -HT1 /2 width=1/ #HT1 - @(cprs_trans … IHT1) -IHT1 /2 width=1/ -] -qed. +lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 → + ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2. +/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed. -lemma cprs_abbr2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. -#L #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/ -#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 #a -lapply (IHV1 T1 T1 ? a) -IHV1 // #HV1 -@(cprs_trans … HV1) -HV1 /2 width=1/ -qed. +(* Advanced properties ******************************************************) -lemma cprs_bind2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 → - ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2. -#L #V1 #V2 #HV12 * /2 width=1/ /2 width=2/ -qed. +(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) +lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G). +#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *) +qed-. -lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2. - L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 → - ∀a.L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2. -#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #a @(cprs_ind … HT12) -T2 -[ /3 width=1/ -| -HV12 #T #T2 #_ #HT2 #IHT1 - lapply (cpr_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 - @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/ -] -qed. +(* Basic_1: was: pr3_strip *) +(* Basic_1: includes: pr1_strip *) +lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L). +normalize /4 width=3 by cpr_conf, TC_strip1/ qed-. + +lemma cprs_lpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → + ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T. +#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/ +#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01) +#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0 +#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T +/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/ +qed-. -lemma ltpr_cprs_trans: ∀L1,L2. L1 ➡ L2 → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT2 -@(cprs_trans … IHT2) /2 width=3/ -qed. +lemma cprs_lpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → + ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → + ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T. +#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 +/3 width=3 by lpr_cprs_trans, ex2_intro/ +qed-. -(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) -lemma lcpr_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → - ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. -#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 // -#T #T2 #_ #HT2 #IHT2 -@(cprs_trans … IHT2) /2 width=3/ -qed. +lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → + ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 → + ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2. +/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.