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=c2b95d2ac7798fcd13d8f73a3248dddd29077e25;hp=8e60c4f2ea8186eda7c9f1fae34e68a68a8bfafa;hb=ca7327c20c6031829fade8bb84a3a1bb66113f54;hpb=25c634037771dff0138e5e8e3d4378183ff49b86 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 8e60c4f2e..c2b95d2ac 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma @@ -20,101 +20,96 @@ include "ground_2/steps/rtc_max_shift.ma". include "ground_2/steps/rtc_isrt_plus.ma". include "ground_2/steps/rtc_isrt_max_shift.ma". include "basic_2/notation/relations/pred_6.ma". -include "basic_2/notation/relations/pred_5.ma". include "basic_2/rt_transition/cpg.ma". (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************) (* 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)" - 'PRed n h G L T1 T2 = (cpm h G L n T1 T2). - -interpretation - "context-sensitive parallel r-transition (term)" - 'PRed h G L T1 T2 = (cpm h G L O T1 T2). + 'PRed h n G L T1 T2 = (cpm h G L n T1 T2). (* Basic properties *********************************************************) -lemma cpm_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s). +lemma cpm_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ➡[h,1] ⋆(⫯[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. -#n #h #G #K #V1 #V2 #W2 * +lemma cpm_delta: ∀h,n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → + ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ➡[h,n] W2. +#h #n #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. -#n #h #G #K #V1 #V2 #W2 * +lemma cpm_ell: ∀h,n,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → + ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡[h,↑n] W2. +#h #n #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. -#n #h #I #G #K #T #U #i * +lemma cpm_lref: ∀h,n,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T → + ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡[h,n] U. +#h #n #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. -#n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 * +lemma cpm_bind: ∀h,n,p,I,G,L,V1,V2,T1,T2. + ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 → + ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[p,I]V2.T2. +#h #n #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. -#n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 * +lemma cpm_appl: ∀h,n,G,L,V1,V2,T1,T2. + ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 → + ❪G,L❫ ⊢ ⓐV1.T1 ➡[h,n] ⓐV2.T2. +#h #n #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. -#n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 * +lemma cpm_cast: ∀h,n,G,L,U1,U2,T1,T2. + ❪G,L❫ ⊢ U1 ➡[h,n] U2 → ❪G,L❫ ⊢ T1 ➡[h,n] T2 → + ❪G,L❫ ⊢ ⓝU1.T1 ➡[h,n] ⓝU2.T2. +#h #n #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. -#n #h #G #L #T1 #T #HT1 #T2 * +lemma cpm_zeta (h) (n) (G) (L): + ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h,n] T2 → + ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ➡[h,n] T2. +#h #n #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. -#n #h #G #L #V #T1 #T2 * +lemma cpm_eps: ∀h,n,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ ⓝV.T1 ➡[h,n] T2. +#h #n #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. -#n #h #G #L #V1 #V2 #T * +lemma cpm_ee: ∀h,n,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ ⓝV1.T ➡[h,↑n] V2. +#h #n #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. -#n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 * +lemma cpm_beta: ∀h,n,p,G,L,V1,V2,W1,W2,T1,T2. + ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] T2 → + ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡[h,n] ⓓ[p]ⓝW2.V2.T2. +#h #n #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. -#n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 * +lemma cpm_theta: ∀h,n,p,G,L,V1,V,V2,W1,W2,T1,T2. + ❪G,L❫ ⊢ V1 ➡[h,0] V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → + ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] T2 → + ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡[h,n] ⓓ[p]W2.ⓐV2.T2. +#h #n #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. @@ -129,23 +124,23 @@ lemma cpr_refl: ∀h,G,L. reflexive … (cpm h G L 0). (* Advanced properties ******************************************************) lemma cpm_sort (h) (G) (L): - ∀n. n ≤ 1 → ∀s. ❪G,L❫ ⊢ ⋆s ➡[n,h] ⋆((next h)^n s). + ∀n. n ≤ 1 → ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,n] ⋆((next h)^n s). #h #G #L * // #n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/ 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: ∀h,n,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h,n] T2 → ∨∨ T2 = ⓪[J] ∧ n = 0 | ∃∃s. T2 = ⋆(⫯[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 ➡[h,n] 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 ➡[h,m] 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 ➡[h,n] 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 * +#h #n #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/ | #s #H1 #H2 #H3 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/ | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct @@ -158,21 +153,21 @@ 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: ∀h,n,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ➡[h,n] T2 → ∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1. -#n #h #G #L #T2 #s * #c #Hc #H +#h #n #G #L #T2 #s * #c #Hc #H elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct [ lapply (isrt_inv_00 … Hc) | lapply (isrt_inv_01 … Hc) ] -Hc #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: ∀h,n,G,L,T2. ❪G,L❫ ⊢ #0 ➡[h,n] T2 → ∨∨ T2 = #0 ∧ n = 0 - | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[n,h] V2 & ⇧[1] V2 ≘ T2 & + | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] 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 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & n = ↑m. -#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H * +#h #n #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/ | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/ @@ -182,9 +177,9 @@ lemma cpm_inv_zero1: ∀n,h,G,L,T2. ❪G,L❫ ⊢ #0 ➡[n,h] T2 → ] qed-. -lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G): - ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0. -#n #h #I #G #K #X2 #H +lemma cpm_inv_zero1_unit (h) (n) (I) (K) (G): + ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡[h,n] X2 → ∧∧ X2 = #0 & n = 0. +#h #n #I #G #K #X2 #H elim (cpm_inv_zero1 … H) -H * [ #H1 #H2 destruct /2 width=1 by conj/ | #Y #X1 #X2 #_ #_ #H destruct @@ -192,19 +187,19 @@ elim (cpm_inv_zero1 … H) -H * ] qed. -lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[n,h] T2 → +lemma cpm_inv_lref1: ∀h,n,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[h,n] T2 → ∨∨ T2 = #(↑i) ∧ n = 0 - | ∃∃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 * + | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I]. +#h #n #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 /4 width=6 by ex3_3_intro, ex2_intro, or_intror/ ] qed-. -lemma cpm_inv_lref1_ctop (n) (h) (G): - ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0. -#n #h #G #X2 * [| #i ] #H +lemma cpm_inv_lref1_ctop (h) (n) (G): + ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[h,n] X2 → ∧∧ X2 = #i & n = 0. +#h #n #G #X2 * [| #i ] #H [ elim (cpm_inv_zero1 … H) -H * [ #H1 #H2 destruct /2 width=1 by conj/ | #Y #X1 #X2 #_ #_ #H destruct @@ -217,18 +212,18 @@ lemma cpm_inv_lref1_ctop (n) (h) (G): ] qed. -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 +lemma cpm_inv_gref1: ∀h,n,G,L,T2,l. ❪G,L❫ ⊢ §l ➡[h,n] T2 → T2 = §l ∧ n = 0. +#h #n #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: ∀h,n,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] U2 → + ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓑ[p,I]V2.T2 - | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[n,h] U2 & + | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 & p = true & I = Abbr. -#n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H * +#h #n #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 elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct @@ -240,11 +235,11 @@ 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: ∀h,n,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ➡[h,n] U2 → + ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓓ[p]V2.T2 - | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[n,h] U2 & p = true. -#n #h #p #G #L #V1 #T1 #U2 #H + | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ➡[h,n] U2 & p = true. +#h #n #p #G #L #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H [ /3 width=1 by or_introl/ | * /3 width=3 by ex3_intro, or_intror/ @@ -253,35 +248,35 @@ 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: ∀h,n,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ➡[h,n] U2 → + ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 & U2 = ⓛ[p]V2.T2. -#n #h #p #G #L #V1 #T1 #U2 #H +#h #n #p #G #L #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H [ /3 width=1 by or_introl/ | * #T #_ #_ #_ #H destruct ] qed-. -lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]V1.T1 ➡[n,h] ⓛ[p2]V2.T2 → - ∧∧ ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[n,h] T2 & p1 = p2. -#n #h #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H +lemma cpm_inv_abst_bi: ∀h,n,p1,p2,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ ⓛ[p1]V1.T1 ➡[h,n] ⓛ[p2]V2.T2 → + ∧∧ ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ➡[h,n] T2 & p1 = p2. +#h #n #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct /2 width=1 by and3_intro/ 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: ∀h,n,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ➡[h,n] U2 → + ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] 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,0] V2 & ❪G,L❫ ⊢ W1 ➡[h,0] W2 & + ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] 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,0] V & ⇧[1] V ≘ V2 & + ❪G,L❫ ⊢ W1 ➡[h,0] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] 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 * +#h #n #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct @@ -303,12 +298,12 @@ 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: ∀h,n,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ➡[h,n] U2 → + ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,n] V2 & ❪G,L❫ ⊢ U1 ➡[h,n] T2 & U2 = ⓝV2.T2 - | ❪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 * + | ❪G,L❫ ⊢ U1 ➡[h,n] U2 + | ∃∃m. ❪G,L❫ ⊢ V1 ➡[h,m] U2 & n = ↑m. +#h #n #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 lapply (isrt_eq_t_trans … HcV HcVT) -HcVT #H @@ -325,10 +320,10 @@ 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: ∀h,n,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ➡[h,n] T → ∀p. + ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,n] ⓑ[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 +#h #n #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/ qed-. @@ -337,32 +332,32 @@ 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) (⋆(⫯[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 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] 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,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] 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