X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fdynamic%2Fsnv_scpes.ma;h=61d7fbdea82172e8188d0ed1752ae5323897c5b4;hb=1083ac3b1acac5f1ac1fa40a9a417dd9d268dced;hp=185f0d52c7b3a03f6cf9f523a1e596e90d345143;hpb=c60524dec7ace912c416a90d6b926bee8553250b;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma index 185f0d52c..61d7fbdea 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma @@ -22,87 +22,87 @@ include "basic_2/dynamic/snv.ma". (* Inductive premises for the preservation results **************************) definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝ - λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g]. + λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o]. definition IH_da_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝ - λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d → + λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d → ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ⦃G, L2⦄ ⊢ T2 ▪[h, g] d. + ⦃G, L2⦄ ⊢ T2 ▪[h, o] d. definition IH_lstas_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝ - λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 → + λh,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 → ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2. definition IH_snv_lstas: ∀h:sh. sd h → relation3 genv lenv term ≝ - λh,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → - ∀d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, g] d1 → - ∀U. ⦃G, L⦄ ⊢ T •*[h, d2] U → ⦃G, L⦄ ⊢ U ¡[h, g]. + λh,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → + ∀d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 → + ∀U. ⦃G, L⦄ ⊢ T •*[h, d2] U → ⦃G, L⦄ ⊢ U ¡[h, o]. (* Properties for the preservation results **********************************) -fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g]. -#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H +fact snv_cprs_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, o]. +#h #o #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H @(cprs_ind … H) -T2 /4 width=6 by fpbg_fpbs_trans, cprs_fpbs/ qed-. -fact da_cprs_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d → - ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, g] d. -#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H +fact da_cprs_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d → + ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ▪[h, o] d. +#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H @(cprs_ind … H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/ qed-. -fact da_scpds_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀d1. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 → - ∀T2,d2. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, d2] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → - d2 ≤ d1 ∧ ⦃G, L2⦄ ⊢ T2 ▪[h, g] d1-d2. -#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12 +fact da_scpds_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d1. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → + ∀T2,d2. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d2] T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → + d2 ≤ d1 ∧ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d1-d2. +#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12 lapply (da_mono … H … Hd1) -H #H destruct lapply (lstas_da_conf … HT1 … Hd1) #Hd12 lapply (da_cprs_lpr_aux … IH2 IH1 … Hd12 … HT2 … HL12) -IH2 -IH1 -HT2 -HL12 /3 width=8 by fpbg_fpbs_trans, lstas_fpbs, conj/ qed-. -fact da_scpes_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] → - ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] → - ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, g] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, g] d12 → - ∀d21,d22. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21, d22] T2 → +fact da_scpes_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] → + ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] → + ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 → + ∀d21,d22. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 → ∧∧ d21 ≤ d11 & d22 ≤ d12 & d11 - d21 = d12 - d22. -#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2 +#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2 elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd11 … HT1 … L) -Hd11 -HT1 // elim (da_scpds_lpr_aux … IH3 IH2 IH1 … Hd12 … HT2 … L) -Hd12 -HT2 // /3 width=7 by da_mono, and3_intro/ qed-. -fact lstas_cprs_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, g] d1 → +fact lstas_cprs_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d1,d2. d2 ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d2] U1 → ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, d2] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2. -#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H +#h #o #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H @(cprs_ind … H) -T2 [ /2 width=10 by/ ] #T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12 elim (IHT1 L1) // -IHT1 #U #HTU #HU1 @@ -114,14 +114,14 @@ elim (IH1 … Hd21 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2 /4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/ qed-. -fact scpds_cpr_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀U1,d. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g, d] U1 → +fact scpds_cpr_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀U1,d. ⦃G, L1⦄ ⊢ T1 •*➡*[h, o, d] U1 → ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g, d] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2. -#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12 + ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, o, d] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2. +#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12 elim (IH1 … H01 … HTW1 … HT12 … HL12) -IH1 // #W2 #HTW2 #HW12 lapply (IH2 … H01 … HTd1 … HT12 … HL12) -L0 -T0 // -T1 lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1 @@ -129,30 +129,30 @@ lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H elim (cpcs_inv_cprs … H) -H /3 width=6 by ex4_2_intro, ex2_intro/ qed-. -fact scpes_cpr_lpr_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) → - ∀G,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] → - ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, g] → - ∀d1,d2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, g, d1, d2] T2 → +fact scpes_cpr_lpr_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) → + ∀G,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L1, T2⦄ → ⦃G, L1⦄ ⊢ T2 ¡[h, o] → + ∀d1,d2. ⦃G, L1⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 → ∀U1. ⦃G, L1⦄ ⊢ T1 ➡ U1 → ∀U2. ⦃G, L1⦄ ⊢ T2 ➡ U2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → - ⦃G, L2⦄ ⊢ U1 •*⬌*[h, g, d1, d2] U2. -#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12 + ⦃G, L2⦄ ⊢ U1 •*⬌*[h, o, d1, d2] U2. +#h #o #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12 elim (scpds_cpr_lpr_aux … IH2 IH1 … HT10 … HTU1 … HL12) -HT10 -HTU1 // #X1 #HUX1 #H1 elim (scpds_cpr_lpr_aux … IH2 IH1 … HT20 … HTU2 … HL12) -HT20 -HTU2 // #X2 #HUX2 #H2 elim (cprs_conf … H1 … H2) -T0 /3 width=5 by scpds_div, scpds_cprs_trans/ qed-. -fact lstas_scpds_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) → - ∀G,L,T. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] → - ∀d,d1. d1 ≤ d → ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀T1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 → - ∀T2,d2. ⦃G, L⦄ ⊢ T •*➡*[h, g, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d2-d1, d1-d2] T2. -#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2 +fact lstas_scpds_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) → + ∀G,L,T. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, o] → + ∀d,d1. d1 ≤ d → ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀T1. ⦃G, L⦄ ⊢ T •*[h, d1] T1 → + ∀T2,d2. ⦃G, L⦄ ⊢ T •*➡*[h, o, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d2-d1, d1-d2] T2. +#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2 lapply (da_mono … H … HTd) -H #H destruct lapply (lstas_da_conf … HT1 … HTd) #HTd1 lapply (lstas_da_conf … HTX … HTd) #HXd @@ -168,23 +168,23 @@ elim (le_or_ge d1 d2) #Hd12 >(eq_minus_O … Hd12) ] qed-. -fact scpes_le_aux: ∀h,g,G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_lstas h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h g G1 L1 T1) → - ∀G,L,T1. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] → - ∀T2. ⦃G0, L0, T0⦄ >≡[h, g] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, g] → - ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, g] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, g] d12 → +fact scpes_le_aux: ∀h,o,G0,L0,T0. + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) → + (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) → + ∀G,L,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, o] → + ∀T2. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G, L, T2⦄ → ⦃G, L⦄ ⊢ T2 ¡[h, o] → + ∀d11. ⦃G, L⦄ ⊢ T1 ▪[h, o] d11 → ∀d12. ⦃G, L⦄ ⊢ T2 ▪[h, o] d12 → ∀d21,d22,d. d21 + d ≤ d11 → d22 + d ≤ d12 → - ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21, d22] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, d21+d, d22+d] T2. -#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20 + ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21, d22] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d21+d, d22+d] T2. +#h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20 elim (da_lstas … Hd11 (d21+d)) #X1 #HTX1 #_ elim (da_lstas … Hd12 (d22+d)) #X2 #HTX2 #_ lapply (lstas_scpds_aux … IH4 IH3 IH2 IH1 … Hd11 … HTX1 … HT10) -HT10 -[1,2,3: // | >eq_minus_O [2: // ] eq_minus_O [2: // ] eq_minus_O [2: // ] eq_minus_O [2: // ]