X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpm.ma;h=d39b9411bf92bfd5cfe73e225fdf77893a158377;hb=d7ff8dcf71f18a17fbf66696f0293cd411c1dbca;hp=ac451b66c580f81de831319b96bbc455013c211c;hpb=3c7b4071a9ac096b02334c1d47468776b948e2de;p=helm.git 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 ac451b66c..d39b9411b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma @@ -16,9 +16,9 @@ include "ground/xoa/ex_4_1.ma". include "ground/xoa/ex_4_3.ma". include "ground/xoa/ex_5_6.ma". include "ground/xoa/ex_6_7.ma". -include "ground/steps/rtc_max_shift.ma". -include "ground/steps/rtc_isrt_plus.ma". -include "ground/steps/rtc_isrt_max_shift.ma". +include "ground/counters/rtc_max_shift.ma". +include "ground/counters/rtc_isrt_plus.ma". +include "ground/counters/rtc_isrt_max_shift.ma". include "static_2/syntax/sh.ma". include "basic_2/notation/relations/pred_6.ma". include "basic_2/rt_transition/cpg.ma". @@ -27,7 +27,7 @@ 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 ⬈[sh_is_next h,rtc_eq_t,c] T2. + λT1,T2. ∃∃c. 𝐑𝐓❨n,c❩ & ❨G,L❩ ⊢ T1 ⬈[sh_is_next h,rtc_eq_t,c] T2. interpretation "t-bound context-sensitive parallel rt-transition (term)" @@ -36,26 +36,26 @@ interpretation (* Basic properties *********************************************************) lemma cpm_ess (h) (G) (L): - ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,1] ⋆(⫯[h]s). + ∀s. ❨G,L❩ ⊢ ⋆s ➡[h,1] ⋆(⫯[h]s). /3 width=3 by cpg_ess, ex2_intro/ qed. 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. + ∀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 (h) (n) (G) (K): - ∀V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 → - ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ➡[h,↑n] W2. + ∀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 (h) (n) (G) (K): - ∀I,T,U,i. ❪G,K❫ ⊢ #i ➡[h,n] T → - ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ➡[h,n] U. + ∀I,T,U,i. ❨G,K❩ ⊢ #i ➡[h,n] T → + ⇧[1] T ≘ U → ❨G,K.ⓘ[I]❩ ⊢ #↑i ➡[h,n] U. #h #n #G #K #I #T #U #i * /3 width=5 by cpg_lref, ex2_intro/ qed. @@ -63,45 +63,45 @@ qed. (* Basic_2A1: includes: cpr_bind *) lemma cpm_bind (h) (n) (G) (L): ∀p,I,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. + ❨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 #G #L #p #I #V1 #V2 #T1 #T2 * #cV #HcV #HV12 * /5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/ qed. 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. + ❨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 (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. + ❨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 (h) (n) (G) (L): - ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h,n] T2 → - ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ➡[h,n] T2. + ∀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 (h) (n) (G) (L): - ∀V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → ❪G,L❫ ⊢ ⓝV.T1 ➡[h,n] T2. + ∀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 (h) (n) (G) (L): - ∀V1,V2,T. ❪G,L❫ ⊢ V1 ➡[h,n] V2 → ❪G,L❫ ⊢ ⓝV1.T ➡[h,↑n] V2. + ∀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. @@ -109,8 +109,8 @@ qed. (* Basic_2A1: includes: cpr_beta *) lemma cpm_beta (h) (n) (G) (L): ∀p,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. + ❨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 #G #L #p #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. @@ -118,8 +118,8 @@ qed. (* Basic_2A1: includes: cpr_theta *) lemma cpm_theta (h) (n) (G) (L): ∀p,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. + ❨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 #G #L #p #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. @@ -135,7 +135,7 @@ lemma cpr_refl (h) (G) (L): reflexive … (cpm h G L 0). (* Advanced properties ******************************************************) lemma cpm_sort (h) (n) (G) (L): n ≤ 1 → - ∀s. ❪G,L❫ ⊢ ⋆s ➡[h,n] ⋆((next h)^n s). + ∀s. ❨G,L❩ ⊢ ⋆s ➡[h,n] ⋆((next h)^n s). #h * // #n #G #L #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/ qed. @@ -143,12 +143,12 @@ qed. (* Basic inversion lemmas ***************************************************) lemma cpm_inv_atom1 (h) (n) (G) (L): - ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h,n] T2 → + ∀J,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 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 - | ∃∃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 ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i). + | ∃∃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 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m + | ∃∃I,K,T,i. ❨G,K❩ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i). #h #n #G #L #J #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H * [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/ | #s1 #s2 #H1 #H2 #H3 #H4 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/ @@ -163,7 +163,7 @@ lemma cpm_inv_atom1 (h) (n) (G) (L): qed-. lemma cpm_inv_sort1 (h) (n) (G) (L): - ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ➡[h,n] T2 → + ∀T2,s1. ❨G,L❩ ⊢ ⋆s1 ➡[h,n] T2 → ∧∧ T2 = ⋆(((next h)^n) s1) & n ≤ 1. #h #n #G #L #T2 #s1 * #c #Hc #H elim (cpg_inv_sort1 … H) -H * @@ -176,10 +176,10 @@ elim (cpg_inv_sort1 … H) -H * qed-. lemma cpm_inv_zero1 (h) (n) (G) (L): - ∀T2. ❪G,L❫ ⊢ #0 ➡[h,n] T2 → + ∀T2. ❨G,L❩ ⊢ #0 ➡[h,n] T2 → ∨∨ ∧∧ T2 = #0 & n = 0 - | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 - | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & n = ↑m. + | ∃∃K,V1,V2. ❨G,K❩ ⊢ V1 ➡[h,n] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 + | ∃∃m,K,V1,V2. ❨G,K❩ ⊢ V1 ➡[h,m] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & n = ↑m. #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 @@ -191,7 +191,7 @@ lemma cpm_inv_zero1 (h) (n) (G) (L): qed-. lemma cpm_inv_zero1_unit (h) (n) (I) (K) (G): - ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ➡[h,n] X2 → ∧∧ X2 = #0 & n = 0. + ∀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/ @@ -201,9 +201,9 @@ elim (cpm_inv_zero1 … H) -H * qed. lemma cpm_inv_lref1 (h) (n) (G) (L): - ∀T2,i. ❪G,L❫ ⊢ #↑i ➡[h,n] T2 → + ∀T2,i. ❨G,L❩ ⊢ #↑i ➡[h,n] T2 → ∨∨ ∧∧ T2 = #(↑i) & n = 0 - | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h,n] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I]. + | ∃∃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 @@ -212,7 +212,7 @@ lemma cpm_inv_lref1 (h) (n) (G) (L): qed-. lemma cpm_inv_lref1_ctop (h) (n) (G): - ∀X2,i. ❪G,⋆❫ ⊢ #i ➡[h,n] X2 → ∧∧ X2 = #i & n = 0. + ∀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/ @@ -227,16 +227,16 @@ lemma cpm_inv_lref1_ctop (h) (n) (G): qed. lemma cpm_inv_gref1 (h) (n) (G) (L): - ∀T2,l. ❪G,L❫ ⊢ §l ➡[h,n] T2 → ∧∧ T2 = §l & n = 0. + ∀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 (h) (n) (G) (L): - ∀p,I,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 ➡[h,n] U2 & p = true & I = Abbr. + ∀p,I,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 ➡[h,n] U2 & p = true & I = Abbr. #h #n #G #L #p #I #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 @@ -250,9 +250,9 @@ qed-. (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *) (* Basic_2A1: includes: cpr_inv_abbr1 *) lemma cpm_inv_abbr1 (h) (n) (G) (L): - ∀p,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 ➡[h,n] U2 & p = true. + ∀p,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 ➡[h,n] U2 & p = true. #h #n #G #L #p #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H [ /3 width=1 by or_introl/ @@ -263,8 +263,8 @@ qed-. (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *) (* Basic_2A1: includes: cpr_inv_abst1 *) lemma cpm_inv_abst1 (h) (n) (G) (L): - ∀p,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. + ∀p,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. #h #n #G #L #p #V1 #T1 #U2 #H elim (cpm_inv_bind1 … H) -H [ /3 width=1 by or_introl/ @@ -273,8 +273,8 @@ elim (cpm_inv_bind1 … H) -H qed-. lemma cpm_inv_abst_bi (h) (n) (G) (L): - ∀p1,p2,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. + ∀p1,p2,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 #G #L #p1 #p2 #V1 #V2 #T1 #T2 #H elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct /2 width=1 by and3_intro/ @@ -283,10 +283,10 @@ qed-. (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *) (* Basic_2A1: includes: cpr_inv_appl1 *) 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,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,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. + ∀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,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,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. #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 @@ -310,10 +310,10 @@ lemma cpm_inv_appl1 (h) (n) (G) (L): qed-. 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 ➡[h,n] U2 - | ∃∃m. ❪G,L❫ ⊢ V1 ➡[h,m] U2 & n = ↑m. + ∀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 ➡[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 @@ -332,8 +332,8 @@ qed-. (* Basic_2A1: includes: cpr_fwd_bind1_minus *) lemma cpm_fwd_bind1_minus (h) (n) (G) (L): - ∀I,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. + ∀I,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. #h #n #G #L #I #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-. @@ -343,32 +343,32 @@ qed-. lemma cpm_ind (h) (Q:relation5 …): (∀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 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] V2 → ❪G,L❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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 ➡[h,n] 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,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,n] 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,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,n] 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 ➡[h,n] 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