From: Ferruccio Guidi Date: Wed, 19 Sep 2018 18:31:44 +0000 (+0200) Subject: update in ground_2, static_2, basic_2 X-Git-Tag: make_still_working~279 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=945b096d007f70e8336847f07b174c48f26467e0;p=helm.git update in ground_2, static_2, basic_2 + confluence for restricted rt_transition under hyps + positive terms wrt the system of reference parked --- diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_conf.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_conf.ma new file mode 100644 index 000000000..f45bf4d42 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_conf.ma @@ -0,0 +1,136 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/xoa/ex_4_1_props.ma". +include "basic_2/dynamic/cnv_cpm_tdeq.ma". + +(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) + +definition IH_cnv_cpm_tdeq_conf_lpr (a) (h) (o): relation3 genv lenv term ≝ + λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → + ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → + ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. + +(* Diamond propery with restricted rt-transition for terms ******************) + +fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (o) (G0) (L1) (L2) (I): + ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛[h,o] T & ⦃G0, L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛[h,o] T. +#h #o #G0 #L1 #L2 #I +/2 width=5 by ex4_intro/ +qed-. + +fact cnv_cpm_tdeq_conf_lpr_atom_ess_aux (h) (o) (G0) (L1) (L2) (s): + deg h o s 0 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⋆(next h s) ➡[h] T & ⋆(next h s) ≛[h,o] T. +#h #o #G0 #L1 #L2 #s #Hs +/4 width=5 by tdeq_sort, deg_next, ex4_intro/ +qed-. + +fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (a) (h) (o) (p) (I) (G0) (L0) (V0) (T0): + (∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h o G L T) → + ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![a,h] → + ∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → + ∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛[h,o] T. +#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0 +#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02 +#L1 #HL01 #L2 #HL02 +elim (cnv_inv_bind … H0) -H0 #_ #HT0 +elim (IH … H1T01 H2T01 … H1T02 H2T02 (L1.ⓑ{I}V0) … (L2.ⓑ{I}V0)) [|*: /2 width=1 by lpr_bind_refl_dx/ ] +#T #H1T1 #H2T1 #H1T2 #H2T2 -L0 -T0 +/3 width=7 by cpm_bind, tdeq_pair, ex4_intro/ +qed-. + +fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (a) (h) (o) (G0) (L0) (V0) (T0): + (∀G,L,T. ⦃G0,L0,ⓐV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h o G L T) → + ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![a,h] → + ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → + ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛[h,o] T. +#a #h #o #G0 #L0 #V0 #T0 #IH #H0 +#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02 +#L1 #HL01 #L2 #HL02 +elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #_ #HT0 #_ #_ -n0 -p0 -X01 -X02 +elim (IH … H1T01 H2T01 … H1T02 H2T02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] +#T #H1T1 #H2T1 #H1T2 #H2T2 -L0 -T0 +/3 width=7 by cpm_appl, tdeq_pair, ex4_intro/ +qed-. + +fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (a) (h) (o) (G0) (L0) (V0) (T0): + (∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h o G L T) → + ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![a,h] → + ∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛[h,o] V1 → + ∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛[h,o] V2 → + ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → + ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1≛[h,o]T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2≛[h,o]T. +#a #h #o #G0 #L0 #V0 #T0 #IH #H0 +#n1 #V1 #H1V01 #H2V01 #n2 #V2 #H1V02 #H2V02 #T1 #H1T01 #H2T01 #T2 #H1T02 #H2T02 +#L1 #HL01 #L2 #HL02 +elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0 +elim (IH … H1V01 H2V01 … H1V02 H2V02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] +elim (IH … H1T01 H2T01 … H1T02 H2T02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] +#T #H1T1 #H2T1 #H1T2 #H2T2 #V #H1V1 #H2V1 #H1V2 #H2V2 -L0 -V0 -T0 +/3 width=7 by cpm_cast, tdeq_pair, ex4_intro/ +qed-. + +fact cnv_cpm_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0): + (∀G,L,T. ⦃G0,L0,T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_conf_lpr a h o G L T) → + ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr a h o G L T. +#a #h #o #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]] +[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct + elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 * + elim (cpm_tdeq_inv_atom_sn … H1X2 H2X2) -H1X2 -H2X2 * + [ #H21 #H22 #H11 #H12 destruct -a -L + [h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → + (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) → + ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 → + ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T. +#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01 +@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0 +[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02 + arith_l3 + /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/ +] +qed-. + +fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0): + (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → + (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) → + ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 → + ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T. +#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01 +generalize in match IH1; generalize in match IH2; +@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0 +[ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02 + arith_l3 + /3 width=7 by cpms_step_sn, tdeq_trans, ex4_intro/ +] +qed-. + +(* fact cnv_cpms_conf_lpr_refl_refl_aux (h) (G0) (L1) (L2) (T0:term): ∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[h] T & ⦃G0,L2⦄ ⊢ T0 ➡*[h] T. /2 width=3 by ex2_intro/ qed-. @@ -94,3 +149,4 @@ elim (cpms_fwd_tdpos_sn … HT0 H0 … HT02) * @(cnv_cpms_conf_lpr_step_step_aux … IH2 IH1) -IH2 -IH1 /2 width=4 by/ ] qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdeq.ma index 3950479e1..b46d12d54 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdeq.ma @@ -31,7 +31,7 @@ fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (o) (G) (L) (T1): [ elim (tdeq_dec h o T T2) #H2T2 [ -IH -IH2 -IH1 -H0T1 /4 width=7 by tdeq_trans, ex4_3_intro/ | lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] -H1T2 -H2T12 #H0T - elim (IH H0T H2T2) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -IH -IH2 -H0T -H2T2 + elim (IH H0T H2T2) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -IH -IH2 -H0T -H2T2 (**) #m1 #m2 #T0 #H1T0 #H2T0 #H1T02 #H destruct elim (cnv_cpm_tdeq_cpm_trans_aux … IH1 … H0T1 … H1T1 H2T1 … H1T0) -IH1 -H0T1 -H1T1 -H1T0 #T3 #H1T13 #H1T30 #H2T30 @@ -56,7 +56,7 @@ fact cpms_tdeq_ind_sn (a) (h) (o) (G) (L) (T2) (Q:relation2 …): elim (tdeq_dec h o T1 T) #H2T1 [ lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] #H0T lapply (tdeq_canc_sn … H2T1 … H2T12) -H2T12 #H2T2 - /6 width=7 by cpm_fpbq, fpbq_fpbg_trans/ + /6 width=7 by cpm_fpbq, fpbq_fpbg_trans/ (**) | -IB2 -IH -IH2 -IH1 elim (cnv_fpbg_refl_false … o … H0T1) -a -Q /3 width=8 by cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdpos.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdpos.ma deleted file mode 100644 index 049e0727f..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_tdpos.ma +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/dynamic/cnv_cpm_tdpos.ma". - -(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) - -(* Forward lemmas with positive rt-computation for terms ********************) - -lemma cpms_fwd_tdpos_sn (a) (h) (o) (n) (G) (L): - ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → 𝐏[h,o]⦃T1⦄ → - ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → - ∨∨ ∧∧ T1 = T2 & 0 = n - | ∃∃n1,n2,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & (T1 ≛[h,o] T → ⊥) & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 & n1+n2=n. -#a #h #o #n #G #L #T1 #H1T1 #H2T1 #T2 #H -@(cpms_ind_dx … H) -n -T2 -[ /3 width=1 by or_introl, conj/ -| #n1 #n2 #T #T2 #HT1 * * - [ #H1 #H2 #HT2 destruct -HT1 - elim (cpm_fwd_tdpos_sn … H1T1 H2T1 … HT2) -H1T1 -H2T1 - [ * #H1 #H2 destruct -HT2 /3 width=1 by or_introl, conj/ - | #HnT2 >commutative_plus in ⊢ (??%); /4 width=7 by ex4_3_intro, or_intror/ - ] - | #m1 #m2 #T0 #H1T10 #H2T10 #HT0 #H #HT2 destruct -H1T1 -H2T1 -HT1 - >associative_plus in ⊢ (??%); /4 width=7 by cpms_step_dx, ex4_3_intro, or_intror/ - ] -] -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve.ma new file mode 100644 index 000000000..9e372bb3e --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve.ma @@ -0,0 +1,104 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/dynamic/cnv_cpms_conf.ma". + +(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) + +(* Main preservation properties *********************************************) +(* +(* Basic_2A1: uses: snv_preserve *) +lemma cnv_preserve (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → + ∧∧ IH_cnv_cpms_conf_lpr a h G L T + & IH_cnv_cpm_trans_lpr a h G L T. +#a #h #G #L #T #HT +elim (tdpos_total h … T) #o +lapply (cnv_fwd_fsb … o … HT) -HT #H +@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH #Ho +@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ] +@(aux … o … G L T) // #G0 #L0 #T0 #H +elim (IH … H) -IH -H // + + +lemma snv_preserve: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → + ∧∧ IH_da_cpr_lpr h o G L T + & IH_snv_cpr_lpr h o G L T + & IH_snv_lstas h o G L T + & IH_lstas_cpr_lpr h o G L T. +#h #o #G #L #T #HT elim (snv_fwd_aaa … HT) -HT +#A #HT @(aaa_ind_fpbg h o … HT) -G -L -T -A +#G #L #T #A #_ #IH -A @and4_intro +[ letin aux ≝ da_cpr_lpr_aux | letin aux ≝ snv_cpr_lpr_aux +| letin aux ≝ snv_lstas_aux | letin aux ≝ lstas_cpr_lpr_aux +] +@(aux … G L T) // #G0 #L0 #T0 #H elim (IH … H) -IH -H // +qed-. + +theorem da_cpr_lpr: ∀h,o,G,L,T. IH_da_cpr_lpr h o G L T. +#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/ +qed-. + +theorem snv_cpr_lpr: ∀h,o,G,L,T. IH_snv_cpr_lpr h o G L T. +#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/ +qed-. + +theorem snv_lstas: ∀h,o,G,L,T. IH_snv_lstas h o G L T. +#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/ +qed-. + +theorem lstas_cpr_lpr: ∀h,o,G,L,T. IH_lstas_cpr_lpr h o G L T. +#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/ +qed-. + +(* Advanced preservation properties *****************************************) + +lemma snv_cprs_lpr: ∀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 #G #L1 #T1 #HT1 #T2 #H +@(cprs_ind … H) -T2 /3 width=5 by snv_cpr_lpr/ +qed-. + +lemma da_cprs_lpr: ∀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 #G #L1 #T1 #HT1 #d #Hd #T2 #H +@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/ +qed-. + +lemma lstas_cprs_lpr: ∀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 #o #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H +@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ] +#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12 +elim (IHT1 L1) // -IHT1 #U #HTU #HU1 +elim (lstas_cpr_lpr … o … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2 +[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1 +/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/ +qed-. + +lemma lstas_cpcs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] → + ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 → + ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, o] → + ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, o] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 → + ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2. +#h #o #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12 +elim (cpcs_inv_cprs … H) -H #T #H1 #H2 +elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1 +elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2 +lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/ +qed-. +*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.etc b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.etc deleted file mode 100644 index 482b94451..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.etc +++ /dev/null @@ -1,94 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/computation/fsb_aaa.ma". -include "basic_2/dynamic/snv_da_lpr.ma". -include "basic_2/dynamic/snv_lstas.ma". -include "basic_2/dynamic/snv_lstas_lpr.ma". -include "basic_2/dynamic/snv_lpr.ma". - -(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************) - -(* Main preservation properties *********************************************) - -lemma snv_preserve: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, o] → - ∧∧ IH_da_cpr_lpr h o G L T - & IH_snv_cpr_lpr h o G L T - & IH_snv_lstas h o G L T - & IH_lstas_cpr_lpr h o G L T. -#h #o #G #L #T #HT elim (snv_fwd_aaa … HT) -HT -#A #HT @(aaa_ind_fpbg h o … HT) -G -L -T -A -#G #L #T #A #_ #IH -A @and4_intro -[ letin aux ≝ da_cpr_lpr_aux | letin aux ≝ snv_cpr_lpr_aux -| letin aux ≝ snv_lstas_aux | letin aux ≝ lstas_cpr_lpr_aux -] -@(aux … G L T) // #G0 #L0 #T0 #H elim (IH … H) -IH -H // -qed-. - -theorem da_cpr_lpr: ∀h,o,G,L,T. IH_da_cpr_lpr h o G L T. -#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/ -qed-. - -theorem snv_cpr_lpr: ∀h,o,G,L,T. IH_snv_cpr_lpr h o G L T. -#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=1 by/ -qed-. - -theorem snv_lstas: ∀h,o,G,L,T. IH_snv_lstas h o G L T. -#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=5 by/ -qed-. - -theorem lstas_cpr_lpr: ∀h,o,G,L,T. IH_lstas_cpr_lpr h o G L T. -#h #o #G #L #T #HT elim (snv_preserve … HT) /2 width=3 by/ -qed-. - -(* Advanced preservation properties *****************************************) - -lemma snv_cprs_lpr: ∀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 #G #L1 #T1 #HT1 #T2 #H -@(cprs_ind … H) -T2 /3 width=5 by snv_cpr_lpr/ -qed-. - -lemma da_cprs_lpr: ∀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 #G #L1 #T1 #HT1 #d #Hd #T2 #H -@(cprs_ind … H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/ -qed-. - -lemma lstas_cprs_lpr: ∀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 #o #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H -@(cprs_ind … H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ] -#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12 -elim (IHT1 L1) // -IHT1 #U #HTU #HU1 -elim (lstas_cpr_lpr … o … Hd21 … HTU … HTT2 … HL12) -HTU -HTT2 -[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1 -/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/ -qed-. - -lemma lstas_cpcs_lpr: ∀h,o,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, o] → - ∀d,d1. d ≤ d1 → ⦃G, L1⦄ ⊢ T1 ▪[h, o] d1 → ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, d] U1 → - ∀T2. ⦃G, L1⦄ ⊢ T2 ¡[h, o] → - ∀d2. d ≤ d2 → ⦃G, L1⦄ ⊢ T2 ▪[h, o] d2 → ∀U2. ⦃G, L1⦄ ⊢ T2 •*[h, d] U2 → - ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U1 ⬌* U2. -#h #o #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12 -elim (cpcs_inv_cprs … H) -H #T #H1 #H2 -elim (lstas_cprs_lpr … HT1 … Hd1 HTd1 … HTU1 … H1 … HL12) -T1 #W1 #H1 #HUW1 -elim (lstas_cprs_lpr … HT2 … Hd2 HTd2 … HTU2 … H2 … HL12) -T2 #W2 #H2 #HUW2 -lapply (lstas_mono … H1 … H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cnv/cnv_cpm_tdeq1_conf.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cnv/cnv_cpm_tdeq1_conf.etc deleted file mode 100644 index 01de9e4d1..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/cnv/cnv_cpm_tdeq1_conf.etc +++ /dev/null @@ -1,377 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/rt_transition/cpm_lsubr.ma". -include "basic_2/rt_transition/lpr_drops.ma". -include "basic_2/dynamic/cnv_drops.ma". -include "basic_2/dynamic/cnv_cpm_tdeq.ma". -include "basic_2/dynamic/cnv_preserve_sub.ma". - -(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) - -definition IH_cnv_cpm_tdeq1_conf_lpr (a) (h) (o): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → - ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → L0 ≛[h, o, T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. - -definition IH_cnv_cpms_tdeq1_strip_lpr (a) (h) (o): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → - ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → L0 ≛[h, o, T0] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. - -definition IH_cnv_cpm_tdeq2_conf_lpr (a) (h) (o): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → - ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. - -definition IH_cnv_cpm_tdeq2_strip_lpr (a) (h) (o): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 → - ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. - -include "ground_2/lib/arith_2b.ma". -include "basic_2/rt_computation/cpms_rdeq.ma". -include "basic_2/rt_computation/cpms_fpbg.ma". - -fact cnv_cpms_tdeq1_strip_lpr_sub (a) (h) (o): - ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G1 L1 T1) → - ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_tdeq1_strip_lpr a h o G1 L1 T1. -#a #h #o #G #L #T #IH2 #IH1 #G0 #L0 #T0 #H0 #HT0 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #H @(cpms_ind_dx … H) -n2 -T2 -[ #L1 #H1L01 #H2L01 #L2 #HL02 -IH2 - elim (IH1 … H1T01 H2T01 0 T0 … L1 … L2) // -L0 -IH1 -H2T01 #T3 #HT13 #H1T03 #H2T03 - /3 width=4 by cpm_cpms, ex3_intro/ -| #n #n2 #X2 #T2 #HX2 #IHX2 #HXT2 #L1 #H1L01 #H2L01 #L2 #HL02 -H1T01 -H2T01 - elim (IHX2 L1 … L0) -IHX2 // #X1 #HTX1 #H1X21 #H2X21 - elim (IH1 … H1X21 H2X21 … HXT2 L1 … L2) - [|*: /3 width=12 by cnv_cpms_trans_lpr_sub, fpbg_cpms_trans, cpms_rdeq_conf_sn/ ] -X2 -IH1 -IH2 #X2 #HX12 #H1TX2 #H2TX2 - >arith_l3 @(ex3_intro … H2TX2) /2 width=3 by cpms_step_dx/ (**) (* explict constructor *) -] -qed-. - -(* Sub diamond propery with restricted rt-transition for terms **************) - -fact cnv_cpm_tdeq1_conf_lpr_atom_atom_aux (h) (o) (G) (L1) (L2) (I): - ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[O,h] T & ⓪{I} ≛[h,o] T. -/2 width=4 by ex3_intro/ qed-. - -fact cnv_cpm_tdeq1_conf_lpr_atom_ess_aux (h) (o) (G) (L1) (L2) (s): - ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡[1,h] T & ⦃G,L2⦄ ⊢ ⋆(next h s) ➡[h] T & ⋆(next h s) ≛[h,o] T. -/2 width=4 by ex3_intro/ qed-. - -fact cnv_cpm_tdeq1_conf_lpr_ess_atom_aux (h) (o) (G) (L1) (L2) (s): - deg h o s 0 → - ∃∃T. ⦃G,L1⦄ ⊢ ⋆(next h s) ➡[h] T & ⦃G,L2⦄ ⊢⋆s ➡[1,h] T & ⋆s ≛[h,o] T. -/4 width=6 by tdeq_sort, deg_next, ex3_intro/ qed-. - -fact cnv_cpm_tdeq1_conf_lpr_atom_delta_aux (a) (h) (o) (G) (L) (i): - (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ #i ![a,h] → - ∀K,V. ⬇*[i]L ≘ K.ⓓV → - ∀n,XV. ⦃G,K⦄ ⊢ V ➡[n,h] XV → - ∀X. ⬆*[↑i]XV ≘ X → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,#i] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ #i ➡[n,h] T & ⦃G,L2⦄ ⊢ X ➡[h] T & X ≛[h,o] T. -#a #h #o #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #H1L1 #H2L1 #L2 #HL2 -lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV -elim (lpr_drops_conf … HLK … H1L1) -H1L1 // #Y1 #H1 #HLK1 -elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #H1K1 #H1V1 #H destruct -elim (rdeq_inv_lref_pair_bi … H2L1 … HLK … HLK1) -H2L1 #H2K1 #H2V1 #_ -elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2 -elim (lpr_inv_pair_sn … H2) -H2 #K2 #V2 #HK2 #_ #H destruct -lapply (drops_isuni_fwd_drop2 … HLK2) -V2 // #HLK2 -lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK -elim (IH … H1V1 H2V1 … HVX … H1K1 H2K1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -V -[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄⊢#i![a,h] → - ∀K,W. ⬇*[i]L ≘ K.ⓛW → - ∀n,XW. ⦃G,K⦄ ⊢ W ➡[n,h] XW → - ∀X. ⬆*[↑i]XW ≘ X → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,#i] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ #i ➡[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡[h] T & X ≛[h,o] T. -#a #h #o #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #H1L1 #H2L1 #L2 #HL2 -lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW -elim (lpr_drops_conf … HLK … H1L1) -H1L1 // #Y1 #H1 #HLK1 -elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #H1K1 #H1W1 #H destruct -elim (rdeq_inv_lref_pair_bi … H2L1 … HLK … HLK1) -H2L1 #H2K1 #H2W1 #_ -elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2 -elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK2 #_ #H destruct -lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2 -lapply (fqup_lref (Ⓣ) … G … HLK) -HLK #HLK -elim (IH … H1W1 H2W1 … HWX … H1K1 H2K1 … HK2) [|*: /2 width=1 by fqup_fpbg/ ] -L -K -W -[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] → - ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → - ∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 → - ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓑ{p,I}V.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[n1-n2,h] T & ⓑ{p,I}V2.T2 ≛[h,o] T. -#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0 -#V2 #HV02 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #HT02 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_bind … H0) -H0 #HV0 #HT0 -elim (rdeq_inv_bind … H2L01) -H2L01 #H0L01 #H2L01 -elim (IH … V0 … 0 V0 … HV02 … H1L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] --H0L01 [h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] → - ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 → - ∀T2. ⬆*[1]T2 ≘ T → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,+ⓓV.T]L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ +ⓓV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡[n1-n2,h] T & XT2 ≛[h,o] T. -#a #h #o #G0 #L0 #V0 #T0 #IH #H0 -#n1 #T1 #H1T01 #H2T01 #T2 #HT20 #n2 #XT2 #HXT2 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_bind … H0) -H0 #_ #HT0 -elim (rdeq_inv_bind … H2L01) -H2L01 #_ #H2L01 -lapply (cnv_inv_lifts … HT0 (Ⓣ) … L0 … HT20) -HT0 -[ /3 width=3 by drops_refl, drops_drop/ ] #HT2 -elim (cpm_inv_lifts_sn … H1T01 (Ⓣ) … L0 … HT20) -H1T01 -[| /3 width=1 by drops_refl, drops_drop/ ] #XT1 #HXT1 #H1XT12 -lapply (tdeq_inv_lifts_bi … H2T01 … HT20 … HXT1) -H2T01 #H2XT12 -lapply (rdeq_inv_lifts_bi … H2L01 (Ⓣ) … L0 L1 … HT20) -H2L01 -[4:|*: /3 width=1 by drops_refl, drops_drop/ ] #H2L01 -elim (IH … H1XT12 H2XT12 … HXT2 … H1L01 … HL02) -[|*: /3 width=1 by fqup_fpbg, fqup_zeta/ ] -L0 -T0 #T #HT1 #H1T2 #H2T2 -/3 width=4 by cpm_zeta, ex3_intro/ -qed-. - -fact cnv_cpm_tdeq1_conf_lpr_appl_appl_aux (a) (h) (o) (G) (L) (V) (T): - (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓐV.T ![a,h] → - ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → - ∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 → - ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡[n1-n2,h] T & ⓐV2.T2 ≛[h,o] T. -#a #h #o #G0 #L0 #V0 #T0 #IH #H0 -#V2 #HV02 #n1 #T1 #H1T01 #H2T01 #n2 #T2 #HT02 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02 -elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01 -elim (IH … V0 … 0 V0 … HV02 … H1L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] --H0L01 [h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] → - ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 → - ∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ⓛ{p}W.T ≛[h,o] T1 → - ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.ⓛ{p}W.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[n1-n2,h] T & ⓓ{p}ⓝW2.V2.T2 ≛[h,o] T. -#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0 -#V2 #HV02 #W2 #HW02 #n1 #X #H1X #H2X #n2 #T2 #HT02 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02 -elim (cnv_inv_bind … H0) -H0 #HW0 #HT0 -elim (cpm_inv_abst1 … H1X) -H1X #W1 #T1 #H1W01 #H1T01 #H destruct -elim (tdeq_inv_pair … H2X) -H2X #_ #H2W01 #H2T01 -elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01 -elim (rdeq_inv_bind … H2L01) -H2L01 #H3L01 #H2L01 -elim (IH … V0 … 0 V0 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] --HV0 -HV02 -H0L01 [h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] → - ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 → - ∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ⓓ{p}W.T ≛[h,o] T1 → - ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 → ∀U2. ⬆*[1]V2 ≘ U2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓐV.ⓓ{p}W.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓐV.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[n1-n2,h] T & ⓓ{p}W2.ⓐU2.T2 ≛[h,o] T. -#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0 -#V2 #HV02 #W2 #HW02 #n1 #X #H1X #H2X #n2 #T2 #HT02 #U2 #HVU2 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02 -elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T1 #HW0 #HT0 #H1T01 #H2T01 #H destruct -elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01 -elim (rdeq_inv_bind … H2L01) -H2L01 #H3L01 #H2L01 -elim (IH … V0 … 0 V0 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] --HV0 -HV02 -H0L01 [h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓝV.T ![a,h] → - ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 → - ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 → - ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 → - ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2≛[h,o]T. -#a #h #o #G0 #L0 #V0 #T0 #IH #H0 -#n1 #V1 #H1V01 #H2V01 #n2 #V2 #HV02 #T1 #H1T01 #H2T01 #T2 #HT02 -#L1 #H1L01 #H2L01 #L2 #HL02 -elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0 -elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01 -elim (IH … H1V01 H2V01 … HV02 … H1L01 H0L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] -elim (IH … H1T01 H2T01 … HT02 … H1L01 H2L01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] -#T #HT1 #H1T2 #H2T2 #V #HV1 #H1V2 #H2V2 -L0 -V0 -T0 -/3 width=6 by cpm_cast, tdeq_pair, ex3_intro/ -qed-. -(* -fact cnv_cpm_tdeq1_conf_lpr_cast_epsilon_aux (a) (h) (o) (G) (L) (V) (T): - (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓝV.T ![a,h] → - ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 → - ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o] T1 → - ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T. -#a #h #o #G0 #L0 #V0 #T0 #IH #H0 -#n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02 -#L1 #HL01 #L2 #HL02 -elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0 -elim (cnv_cpm_conf_lpr_sub … IH … HT01 … HT02 … HL01 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] -#T #HT1 #HT2 -L0 -V0 -T0 -/3 width=3 by cpms_eps, ex2_intro/ -qed-. -*) -fact cnv_cpm_tdeq1_conf_lpr_cast_ee_aux (a) (h) (o) (G) (L) (V) (T): - (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) → - (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G0 L0 T0) → - ⦃G,L⦄ ⊢ ⓝV.T ![a,h] → - ∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → V ≛[h,o] V1 → - ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 → - ∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → T ≛[h,o]T1 → - ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → L ≛[h,o,ⓝV.T]L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡[n1-↑n2,h] T & V2 ≛[h,o] T. -#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0 -#n1 #V1 #H1V01 #H2V01 #n2 #V2 #HV02 #T1 #H1T01 #H2T01 -#L1 #H1L01 #H2L01 #L2 #HL02 -(* n2=0 n1=1 *) - -H1V01 -H2V01 -elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0 -elim (rdeq_inv_flat … H2L01) -H2L01 #H0L01 #H2L01 -lapply (cnv_cpms_trans_lpr_sub … IH2 … HVX0 … L0 ?) [4:|*: /2 width=1 by fqup_fpbg/ ] #HX0 -elim (cnv_cpms_strip_lpr_sub … IH1 … HVX0 … HV02 … L0 … HL02) [|*: /2 width=1 by fqup_fpbg/ ] -elim (cnv_cpms_strip_lpr_sub … IH1 … HTX0 … HT01 … L0 … HL01) [|*: /2 width=1 by fqup_fpbg/ ] --HV02 -HTX0 -HT01 minus_plus #H2 -lapply (cpms_trans … HT1 … HTU) -T [h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_tdeq1_conf_lpr a h o G1 L1 T1) → - ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_tdeq1_conf_lpr a h o G1 L1 T1. -#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]] -[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #HX2 #L1 #H1L1 #H2L1 #L2 #HL2 destruct - elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 * - elim (cpm_inv_atom1_drops … HX2) -HX2 * - [ #H21 #H22 #H11 #H12 destruct -a -L - iter_n_Sm - /5 width=6 by cpm_sort_iter, tdeq_sort, deg_iter, deg_next, ex3_intro/ - ] -| #p #I #V1 #T1 #HG #HL #HT #H0 #n1 #X1 #H1X #H2X #n2 #X2 #HX2 destruct - elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T #H1T1 #H2T1 #H3T1 #H destruct - elim (cpm_inv_bind1 … HX2) -HX2 * - [ #V2 #T2 #HV12 #HT2 #H destruct - elim (IH … H1T1 … H2T1 H3T1 … HT2) -IH -H1T1 -H2T1 -H3T1 -HT2 [2: // ] #T0 #HT10 #H1T02 #H2T02 - @(ex3_intro … (ⓑ{p,I}V2.T0)) - [ /2 width=1 by cpm_bind/ - | - | /2 width=1 by tdeq_pair/ - -*) diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl index e10340776..e34970bbe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl @@ -31,7 +31,7 @@ table { *) [ { "context-sensitive native validity" * } { [ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ] - [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tfeq_trans" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_preserve_sub" + "cnv_preserve" * ] + [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_preserve_sub" + "cnv_preserve" * ] } ] } diff --git a/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_4_1_props.ma b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_4_1_props.ma new file mode 100644 index 000000000..fe081386b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2/xoa/ex_4_1_props.ma @@ -0,0 +1,22 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground_2/xoa/xoa.ma". + +(* Properties with multiple existental quantifier (5, 1) ********************) + +lemma ex4_commute (A0) (P0,P1,P2,P3:A0→Prop): + (∃∃x0. P0 x0 & P1 x0 & P2 x0 & P3 x0) → ∃∃x0. P2 x0 & P3 x0 & P0 x0 & P1 x0. +#A0 #P0 #P1 #P2 #P3 * /2 width=5 by ex4_intro/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/relations/positive_3.ma b/matita/matita/contribs/lambdadelta/static_2/notation/relations/positive_3.ma deleted file mode 100644 index e73de4930..000000000 --- a/matita/matita/contribs/lambdadelta/static_2/notation/relations/positive_3.ma +++ /dev/null @@ -1,19 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) - -notation "hvbox( 𝐏 [ term 46 h, term 46 o ] ⦃ term 46 T ⦄ )" - non associative with precedence 45 - for @{ 'Positive $h $o $T }. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma deleted file mode 100644 index 1b6b1197e..000000000 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma +++ /dev/null @@ -1,63 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "static_2/notation/relations/positive_3.ma". -include "static_2/syntax/item_sd.ma". -include "static_2/syntax/term.ma". - -(* DEGREE POSITIVITY ON TERMS ***********************************************) - -inductive tdpos (h) (o): predicate term ≝ -| tdpos_sort: ∀s,d. deg h o s (↑d) → tdpos h o (⋆s) -| tdpos_lref: ∀i. tdpos h o (#i) -| tdpos_gref: ∀l. tdpos h o (§l) -| tdpos_pair: ∀I,V,T. tdpos h o V → tdpos h o T → tdpos h o (②{I}V.T) -. - -interpretation - "context-free degree positivity (term)" - 'Positive h o T = (tdpos h o T). - -(* Basic inversion lemmas ***************************************************) - -fact tdpos_inv_sort_aux (h) (o): - ∀X. 𝐏[h,o]⦃X⦄ → ∀s. X = ⋆s → ∃d. deg h o s (↑d). -#h #o #H * -[ #s #d #Hsd #x #H destruct /2 width=2 by ex_intro/ -| #i #x #H destruct -| #l #x #H destruct -| #I #V #T #_ #_ #x #H destruct -] -qed-. - -lemma tdpos_inv_sort (h) (o): ∀s. 𝐏[h,o]⦃⋆s⦄ → ∃d. deg h o s (↑d). -/2 width=3 by tdpos_inv_sort_aux/ qed-. - -fact tdpos_inv_pair_aux (h) (o): ∀X. 𝐏[h,o]⦃X⦄ → ∀I,V,T. X = ②{I}V.T → - ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄. -#h #o #H * -[ #s #d #_ #Z #X1 #X2 #H destruct -| #i #Z #X1 #X2 #H destruct -| #l #Z #X1 #X2 #H destruct -| #I #V #T #HV #HT #Z #X1 #X2 #H destruct /2 width=1 by conj/ -] -qed-. - -lemma tdpos_inv_pair (h) (o): ∀I,V,T. 𝐏[h,o]⦃②{I}V.T⦄ → - ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄. -/2 width=4 by tdpos_inv_pair_aux/ qed-. - -(* Basic properties *********************************************************) - -axiom tdpos_total (h): ∀T. ∃o. 𝐏[h,o]⦃T⦄.