X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpm.ma;h=c8f746351252b275d6e2728642b8f3fe2adc824c;hp=341ecee8cfde5b23e00b615b04c4d3ef40673cc7;hb=f308429a0fde273605a2330efc63268b4ac36c99;hpb=87f57ddc367303c33e19c83cd8989cd561f3185b diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma index 341ecee8c..c8f746351 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma @@ -20,7 +20,7 @@ include "basic_2/rt_transition/cpg.ma". (* Basic_2A1: includes: cpr *) definition cpm (h) (G) (L) (n): relation2 term term ≝ - λT1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2. + λT1,T2. ∃∃c. 𝐑𝐓⦃n,c⦄ & ⦃G,L⦄ ⊢ T1 ⬈[eq_t,c,h] T2. interpretation "t-bound context-sensitive parallel rt-transition (term)" @@ -32,81 +32,81 @@ interpretation (* Basic properties *********************************************************) -lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s). +lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(next h s). /2 width=3 by cpg_ess, ex2_intro/ qed. -lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → - ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2. +lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → + ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2. #n #h #G #K #V1 #V2 #W2 * /3 width=5 by cpg_delta, ex2_intro/ qed. -lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → - ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[↑n, h] W2. +lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → + ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2. #n #h #G #K #V1 #V2 #W2 * /3 width=5 by cpg_ell, ex2_intro, isrt_succ/ qed. -lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → - ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡[n, h] U. +lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → + ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U. #n #h #I #G #K #T #U #i * /3 width=5 by cpg_lref, ex2_intro/ qed. (* Basic_2A1: includes: cpr_bind *) lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 → - ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2. + ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 → + ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2. #n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 * /5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/ qed. lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] ⓐV2.T2. + ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2. #n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 * /5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/ qed. lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2. - ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → - ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n, h] ⓝU2.T2. + ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → + ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2. #n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 * /4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/ qed. (* Basic_2A1: includes: cpr_zeta *) lemma cpm_zeta (n) (h) (G) (L): - ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[n,h] T2 → - ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2. + ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 → + ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[n,h] T2. #n #h #G #L #T1 #T #HT1 #T2 * /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/ qed. (* Basic_2A1: includes: cpr_eps *) -lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2. +lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ➡[n,h] T2. #n #h #G #L #V #T1 #T2 * /3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/ qed. -lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[↑n, h] V2. +lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ➡[↑n,h] V2. #n #h #G #L #V1 #V2 #T * /3 width=3 by cpg_ee, isrt_succ, ex2_intro/ qed. (* Basic_2A1: includes: cpr_beta *) lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2. + ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n,h] ⓓ{p}ⓝW2.V2.T2. #n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 * /6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/ qed. (* Basic_2A1: includes: cpr_theta *) lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → - ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2. + ⦃G,L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → + ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{p}W2.ⓐV2.T2. #n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 * /6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/ qed. @@ -129,14 +129,14 @@ qed. (* Basic inversion lemmas ***************************************************) -lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 → +lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 → ∨∨ T2 = ⓪{J} ∧ n = 0 | ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1 - | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 - | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m - | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & + | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I} & J = LRef (↑i). #n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H * [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/ @@ -151,7 +151,7 @@ lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 → ] qed-. -lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 → +lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] T2 → ∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1. #n #h #G #L #T2 #s * #c #Hc #H elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct @@ -159,11 +159,11 @@ elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct #H destruct /2 width=1 by conj/ qed-. -lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 → +lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 → ∨∨ T2 = #0 ∧ n = 0 - | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 - | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓛV1 & n = ↑m. #n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H * [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/ @@ -175,9 +175,9 @@ lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 → ] qed-. -lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 → +lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 → ∨∨ T2 = #(↑i) ∧ n = 0 - | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. + | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H * [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/ | #I #K #V2 #HV2 #HVT2 #H destruct @@ -185,16 +185,16 @@ lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 → ] qed-. -lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0. +lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0. #n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H #H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/ qed-. (* Basic_2A1: includes: cpr_inv_bind1 *) -lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 & +lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 & U2 = ⓑ{p,I}V2.T2 - | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & + | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true & I = Abbr. #n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct @@ -208,10 +208,10 @@ qed-. (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *) (* Basic_2A1: includes: cpr_inv_abbr1 *) -lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 & +lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 & U2 = ⓓ{p}V2.T2 - | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & p = true. + | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true. #n #h #p #G #L #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H [ /3 width=1 by or_introl/ @@ -221,8 +221,8 @@ qed-. (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *) (* Basic_2A1: includes: cpr_inv_abst1 *) -lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 & +lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U2 → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 & U2 = ⓛ{p}V2.T2. #n #h #p #G #L #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H @@ -240,14 +240,14 @@ qed-. (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *) (* Basic_2A1: includes: cpr_inv_appl1 *) -lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 & +lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 & U2 = ⓐV2.T2 - | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 & - ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 & + | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 & + ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 - | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 & - ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 & + | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 & + ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2. #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct @@ -271,11 +271,11 @@ lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 ] qed-. -lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 & +lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ➡[n,h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 & U2 = ⓝV2.T2 - | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 - | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ↑m. + | ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 + | ∃∃m. ⦃G,L⦄ ⊢ V1 ➡[m,h] U2 & n = ↑m. #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct @@ -293,8 +293,8 @@ qed-. (* Basic forward lemmas *****************************************************) (* Basic_2A1: includes: cpr_fwd_bind1_minus *) -lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p. - ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 & +lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n,h] T → ∀p. + ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2 & T = -ⓑ{I}V2.T2. #n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H /3 width=4 by ex2_2_intro, ex2_intro/ @@ -305,32 +305,32 @@ qed-. lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term. (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) → (∀G,L,s. Q 1 G L (⋆s) (⋆(next h s))) → - (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 → + (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 → ⬆*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2 - ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 → + ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 → ⬆*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2 - ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → Q n G K (#i) T → + ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T → ⬆*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U) - ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 → Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) - ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2) - ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2) - ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[n, h] T2 → + ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 → Q n G L T T2 → Q n G L (+ⓓV.T1) T2 - ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2 → Q n G L (ⓝV.T1) T2 - ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → + ) → (∀n,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2 - ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 → Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 → Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2) - ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 → + ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 → Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 → ⬆*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2) ) → - ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → Q n G L T1 T2. + ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2. #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2 * #c #HC #H generalize in match HC; -HC generalize in match n; -n elim H -c -G -L -T1 -T2