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=d1817d6ff7479f56220a26507a47fc36024f41e8;hp=454ee83127c6499287d18df4e8decbff5ce1ae03;hb=f308429a0fde273605a2330efc63268b4ac36c99;hpb=87f57ddc367303c33e19c83cd8989cd561f3185b 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 454ee8312..d1817d6ff 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma @@ -35,15 +35,15 @@ interpretation 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. + (∀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. + (∀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-. @@ -52,36 +52,36 @@ qed-. (* 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. +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. -lemma cpms_step_sn (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. +lemma cpms_step_sn (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_sn/ qed-. -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. +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_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. + ∀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 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. + ∀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. 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. + ∀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. @@ -89,22 +89,22 @@ 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. + ∀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. (* 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. + ∀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. + ∀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 @@ -114,21 +114,21 @@ 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. + ∀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 → + ∀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. + ∀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. @@ -150,17 +150,17 @@ 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). +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-. lemma cpms_inv_cast1 (h) (n) (G) (L): - ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 → - ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2 - | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2 - | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m. + ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 → + ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2 + | ⦃G,L⦄ ⊢ T1 ➡*[n,h] X2 + | ∃∃m. ⦃G,L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m. #h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2 [ /3 width=5 by or3_intro0, ex3_2_intro/ | #n1 #n2 #X #X2 #_ * [ * || * ]