X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2Fbasic_2%2Fcomputation%2Fcprs_cprs.ma;h=d350f0977dd18dacd0fc1ce8ca0cefa5c5ac23ae;hb=a2144f09d1bd7022c1f2dfd4909a1fb9772c8d56;hp=47ec400c88519ddb8d60765fa96755dc22d7d576;hpb=a8c166f1e1baeeae04553058bd179420ada8bbe7;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma index 47ec400c8..d350f0977 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma @@ -21,46 +21,59 @@ include "basic_2/computation/cprs_lcpr.ma". (* Advanced properties ******************************************************) -lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. -#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1 +lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. + L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}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 *) +] +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/ ] qed. -lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. +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. lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 → - L ⊢ ⓓV1. T1 ➡* ⓓV2. T2. + ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2. #L #V1 #V2 #HV12 #T1 #T2 #HT12 lapply (lcpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/ 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. + (* 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 & - U2 = ⓐV2. T2 - | ∃∃V2,W,T. L ⊢ V1 ➡* V2 & - L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ➡* U2 - | ∃∃V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 & - L ⊢ T1 ➡* ⓓV. T & L ⊢ ⓓV. ⓐV2. T ➡* U2. + ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & 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/ #U #U2 #_ #HU2 * * [ #V0 #T0 #HV10 #HT10 #H destruct elim (cpr_inv_appl1 … HU2) -HU2 * [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ - | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/ - | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct - @or3_intro2 @(ex4_4_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) + | #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 *) ] -| /4 width=8/ -| /4 width=10/ +| /4 width=9/ +| /4 width=11/ ] qed-. @@ -75,11 +88,57 @@ theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 → 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/ +@(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. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2. +#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HV12) -V2 +[ lapply (cprs_lsubs_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/ +| #V0 #V2 #_ #HV02 #IHV01 + @(cprs_trans … IHV01) -V1 /2 width=2/ +] +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_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 (lcpr_cpr_trans (L. ⓓV1) … HT1) -HT1 /2 width=1/ #HT1 + @(cprs_trans … IHT1) -IHT1 /2 width=1/ +] +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. + +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_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 + @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/ +] qed. (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)