X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Fcpxs_cpxs.ma;h=8958323296e1c68538e3f96df123682204ebdaef;hb=29973426e0227ee48368d1c24dc0c17bf2baef77;hp=2399ca6654550ca63f1fa5c0cf3fa81f1377abd9;hpb=f95f6cb21b86f3dad114b21f687aa5df36088064;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma index 2399ca665..895832329 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma @@ -22,51 +22,51 @@ include "basic_2/computation/cpxs_lift.ma". theorem cpxs_trans: ∀h,g,L. Transitive … (cpxs h g L). #h #g #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *) -theorem cpxs_bind: ∀h,g,a,I,L,V1,V2,T1,T2. ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[g] T2 → - ⦃h, L⦄ ⊢ V1 ➡*[g] V2 → - ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2. +theorem cpxs_bind: ∀h,g,a,I,L,V1,V2,T1,T2. ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 → + ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → + ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2. #h #g #a #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/ #V #V2 #_ #HV2 #IHV1 @(cpxs_trans … IHV1) -V1 /2 width=1/ qed. -theorem cpxs_flat: ∀h,g,I,L,V1,V2,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → - ⦃h, L⦄ ⊢ V1 ➡*[g] V2 → - ⦃h, L⦄ ⊢ ⓕ{I} V1.T1 ➡*[g] ⓕ{I} V2.T2. +theorem cpxs_flat: ∀h,g,I,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → + ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → + ⦃G, L⦄ ⊢ ⓕ{I} V1.T1 ➡*[h, g] ⓕ{I} V2.T2. #h #g #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/ #V #V2 #_ #HV2 #IHV1 @(cpxs_trans … IHV1) -IHV1 /2 width=1/ qed. theorem cpxs_beta_rc: ∀h,g,a,L,V1,V2,W1,W2,T1,T2. - ⦃h, L⦄ ⊢ V1 ➡[g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 → - ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2. + ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃h, 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 #HT12 #H @(cpxs_ind … H) -W2 /2 width=1/ #W #W2 #_ #HW2 #IHW1 @(cpxs_trans … IHW1) -IHW1 /3 width=1/ qed. theorem cpxs_beta: ∀h,g,a,L,V1,V2,W1,W2,T1,T2. - ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V2 → - ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2. + ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → + ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2. #h #g #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2 /2 width=1/ #V #V2 #_ #HV2 #IHV1 @(cpxs_trans … IHV1) -IHV1 /3 width=1/ qed. theorem cpxs_theta_rc: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2. - ⦃h, L⦄ ⊢ V1 ➡[g] V → ⇧[0, 1] V ≡ V2 → - ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 → - ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2. + ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → + ⦃h, 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 #HT12 #H elim H -W2 /2 width=3/ #W #W2 #_ #HW2 #IHW1 @(cpxs_trans … IHW1) -IHW1 /2 width=1/ qed. theorem cpxs_theta: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2. - ⇧[0, 1] V ≡ V2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 → - ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V → - ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2. + ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 → + ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V → + ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2. #h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/ #V1 #V0 #HV10 #_ #IHV0 @(cpxs_trans … IHV0) -IHV0 /2 width=1/ @@ -74,12 +74,12 @@ qed. (* Advanced inversion lemmas ************************************************) -lemma cpxs_inv_appl1: ∀h,g,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓐV1.T1 ➡*[g] U2 → - ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L⦄ ⊢ T1 ➡*[g] T2 & +lemma cpxs_inv_appl1: ∀h,g,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h, g] U2 → + ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓐV2. T2 - | ∃∃a,W,T. ⦃h, L⦄ ⊢ T1 ➡*[g] ⓛ{a}W.T & ⦃h, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[g] U2 - | ∃∃a,V0,V2,V,T. ⦃h, L⦄ ⊢ V1 ➡*[g] V0 & ⇧[0,1] V0 ≡ V2 & - ⦃h, L⦄ ⊢ T1 ➡*[g] ⓓ{a}V.T & ⦃h, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[g] U2. + | ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, g] U2 + | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V0 & ⇧[0,1] V0 ≡ V2 & + ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U2. #h #g #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5/ ] #U #U2 #_ #HU2 * * [ #V0 #T0 #HV10 #HT10 #H destruct @@ -120,9 +120,9 @@ lemma lpx_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpx h g). ] qed-. -lemma cpx_bind2: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 → - ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡[g] T2 → - ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2. +lemma cpx_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → + ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ 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 lapply (lpx_cpx_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/ qed. @@ -132,9 +132,9 @@ qed. lemma lpx_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpx h g). /3 width=5 by s_r_trans_TC1, lpx_cpx_trans/ qed-. -lemma cpxs_bind2_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 → - ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[g] T2 → - ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2. +lemma cpxs_bind2_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → + ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ 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 lapply (lpx_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/ qed.