From f308429a0fde273605a2330efc63268b4ac36c99 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Thu, 13 Jun 2019 17:07:51 +0200 Subject: [PATCH] some restyling ... + fixing some notations + fixing some spaces --- .../lambdadelta/apps_2/examples/ex_cnv_eta.ma | 4 +- .../apps_2/examples/ex_cpr_omega.ma | 12 +- .../apps_2/examples/ex_fpbg_refl.ma | 8 +- .../apps_2/functional/flifts_flifts_basic.ma | 2 +- .../lambdadelta/apps_2/functional/mf_cpr.ma | 4 +- .../contribs/lambdadelta/apps_2/models/deq.ma | 2 +- .../lambdadelta/apps_2/models/deq_cpr.ma | 2 +- .../contribs/lambdadelta/apps_2/models/li.ma | 2 +- .../lambdadelta/apps_2/models/model_props.ma | 26 ++-- .../contribs/lambdadelta/apps_2/models/tm.ma | 2 +- .../contribs/lambdadelta/apps_2/models/veq.ma | 2 +- .../lambdadelta/apps_2/models/veq_lifts.ma | 4 +- .../lambdadelta/apps_2/models/vpushs.ma | 6 +- .../lambdadelta/apps_2/models/vpushs_fold.ma | 8 +- .../lambdadelta/basic_2/dynamic/cnv.ma | 64 ++++---- .../lambdadelta/basic_2/dynamic/cnv_aaa.ma | 12 +- .../lambdadelta/basic_2/dynamic/cnv_cpes.ma | 26 ++-- .../basic_2/dynamic/cnv_cpm_conf.ma | 6 +- .../basic_2/dynamic/cnv_cpm_tdeq.ma | 50 +++--- .../basic_2/dynamic/cnv_cpm_tdeq_conf.ma | 12 +- .../basic_2/dynamic/cnv_cpm_tdeq_trans.ma | 6 +- .../basic_2/dynamic/cnv_cpm_trans.ma | 4 +- .../basic_2/dynamic/cnv_cpms_conf.ma | 4 +- .../basic_2/dynamic/cnv_cpms_tdeq.ma | 10 +- .../lambdadelta/basic_2/dynamic/cnv_drops.ma | 16 +- .../lambdadelta/basic_2/dynamic/cnv_fqus.ma | 16 +- .../lambdadelta/basic_2/dynamic/cnv_fsb.ma | 6 +- .../basic_2/dynamic/cnv_preserve_sub.ma | 48 +++--- .../lambdadelta/basic_2/dynamic/lsubv.ma | 10 +- .../lambdadelta/basic_2/dynamic/lsubv_cnv.ma | 4 +- .../basic_2/dynamic/nta_preserve.ma | 4 +- .../lambdadelta/basic_2/i_dynamic/ntas.ma | 2 +- .../lambdadelta/basic_2/i_dynamic/ntas_etc.ma | 24 +-- .../lambdadelta/basic_2/i_dynamic/ntas_nta.ma | 8 +- .../notation/relations/prediteval_5.ma | 3 + .../basic_2/rt_computation/cpme.ma | 2 +- .../basic_2/rt_computation/cpms.ma | 68 ++++----- .../basic_2/rt_computation/cpms_aaa.ma | 2 +- .../basic_2/rt_computation/cpms_cpms.ma | 72 ++++----- .../basic_2/rt_computation/cpms_cpxs.ma | 2 +- .../basic_2/rt_computation/cpms_drops.ma | 36 ++--- .../basic_2/rt_computation/cpms_fpbg.ma | 14 +- .../basic_2/rt_computation/cpms_fpbs.ma | 2 +- .../basic_2/rt_computation/cpms_lpr.ma | 20 +-- .../basic_2/rt_computation/cpms_rdeq.ma | 4 +- .../basic_2/rt_computation/cpre_cpre.ma | 2 +- .../basic_2/rt_computation/cpre_csx.ma | 2 +- .../basic_2/rt_computation/cprs.ma | 34 ++--- .../basic_2/rt_computation/cprs_cnr.ma | 2 +- .../basic_2/rt_computation/cprs_cprs.ma | 22 +-- .../basic_2/rt_computation/cprs_drops.ma | 4 +- .../basic_2/rt_computation/cprs_lpr.ma | 10 +- .../basic_2/rt_computation/cpue.ma | 2 +- .../basic_2/rt_computation/cpue_csx.ma | 2 +- .../basic_2/rt_computation/cpxs.ma | 74 ++++----- .../basic_2/rt_computation/cpxs_cnx.ma | 2 +- .../basic_2/rt_computation/cpxs_cpxs.ma | 42 ++--- .../basic_2/rt_computation/cpxs_drops.ma | 24 +-- .../basic_2/rt_computation/cpxs_fdeq.ma | 6 +- .../basic_2/rt_computation/cpxs_fqus.ma | 48 +++--- .../basic_2/rt_computation/cpxs_lpx.ma | 24 +-- .../basic_2/rt_computation/cpxs_rdeq.ma | 12 +- .../basic_2/rt_computation/cpxs_tdeq.ma | 8 +- .../basic_2/rt_computation/cpxs_theq.ma | 22 +-- .../rt_computation/cpxs_theq_vector.ma | 24 +-- .../lambdadelta/basic_2/rt_computation/csx.ma | 38 ++--- .../basic_2/rt_computation/csx_aaa.ma | 26 ++-- .../basic_2/rt_computation/csx_cnx.ma | 4 +- .../basic_2/rt_computation/csx_cnx_vector.ma | 6 +- .../basic_2/rt_computation/csx_cpxs.ma | 22 +-- .../basic_2/rt_computation/csx_csx.ma | 22 +-- .../basic_2/rt_computation/csx_csx_vector.ma | 14 +- .../basic_2/rt_computation/csx_fdeq.ma | 4 +- .../basic_2/rt_computation/csx_fpbq.ma | 4 +- .../basic_2/rt_computation/csx_fqus.ma | 16 +- .../basic_2/rt_computation/csx_lpx.ma | 20 +-- .../basic_2/rt_computation/csx_lpxs.ma | 4 +- .../basic_2/rt_computation/csx_lsubr.ma | 20 +-- .../basic_2/rt_computation/csx_rdeq.ma | 6 +- .../basic_2/rt_computation/csx_simple.ma | 6 +- .../basic_2/rt_computation/csx_simple_theq.ma | 6 +- .../basic_2/rt_computation/csx_vector.ma | 8 +- .../basic_2/rt_computation/fpbg.ma | 26 ++-- .../basic_2/rt_computation/fpbg_cpxs.ma | 8 +- .../basic_2/rt_computation/fpbg_fpbs.ma | 30 ++-- .../basic_2/rt_computation/fpbg_fqup.ma | 6 +- .../basic_2/rt_computation/fpbg_lpxs.ma | 4 +- .../basic_2/rt_computation/fpbs.ma | 32 ++-- .../basic_2/rt_computation/fpbs_aaa.ma | 4 +- .../basic_2/rt_computation/fpbs_cpx.ma | 6 +- .../basic_2/rt_computation/fpbs_cpxs.ma | 26 ++-- .../basic_2/rt_computation/fpbs_csx.ma | 4 +- .../basic_2/rt_computation/fpbs_fpb.ma | 4 +- .../basic_2/rt_computation/fpbs_fqup.ma | 18 +-- .../basic_2/rt_computation/fpbs_fqus.ma | 12 +- .../basic_2/rt_computation/fpbs_lpxs.ma | 50 +++--- .../lambdadelta/basic_2/rt_computation/fsb.ma | 8 +- .../basic_2/rt_computation/fsb_aaa.ma | 26 ++-- .../basic_2/rt_computation/fsb_csx.ma | 20 +-- .../basic_2/rt_computation/fsb_fdeq.ma | 4 +- .../basic_2/rt_computation/fsb_fpbg.ma | 24 +-- .../basic_2/rt_computation/lprs.ma | 30 ++-- .../basic_2/rt_computation/lprs_cpms.ma | 32 ++-- .../basic_2/rt_computation/lprs_cprs.ma | 30 ++-- .../basic_2/rt_computation/lprs_ctc.ma | 4 +- .../basic_2/rt_computation/lprs_length.ma | 2 +- .../basic_2/rt_computation/lprs_lpr.ma | 18 +-- .../basic_2/rt_computation/lprs_lpxs.ma | 2 +- .../basic_2/rt_computation/lprs_tc.ma | 4 +- .../basic_2/rt_computation/lpxs.ma | 32 ++-- .../basic_2/rt_computation/lpxs_cpxs.ma | 16 +- .../basic_2/rt_computation/lpxs_fdeq.ma | 6 +- .../basic_2/rt_computation/lpxs_length.ma | 2 +- .../basic_2/rt_computation/lpxs_lpx.ma | 24 +-- .../basic_2/rt_computation/lpxs_rdeq.ma | 10 +- .../basic_2/rt_computation/lsubsx.ma | 50 +++--- .../basic_2/rt_computation/lsubsx_lsubsx.ma | 4 +- .../basic_2/rt_computation/lsubsx_rdsx.ma | 14 +- .../basic_2/rt_computation/rdsx.ma | 28 ++-- .../basic_2/rt_computation/rdsx_csx.ma | 12 +- .../basic_2/rt_computation/rdsx_drops.ma | 12 +- .../basic_2/rt_computation/rdsx_fqup.ma | 10 +- .../basic_2/rt_computation/rdsx_length.ma | 6 +- .../basic_2/rt_computation/rdsx_lpxs.ma | 60 ++++---- .../basic_2/rt_computation/rdsx_rdsx.ma | 8 +- .../lambdadelta/basic_2/rt_conversion/cpc.ma | 6 +- .../basic_2/rt_conversion/cpc_cpc.ma | 4 +- .../lambdadelta/basic_2/rt_conversion/lpce.ma | 2 +- .../basic_2/rt_equivalence/cpcs.ma | 40 ++--- .../basic_2/rt_equivalence/cpcs_aaa.ma | 4 +- .../basic_2/rt_equivalence/cpcs_cpcs.ma | 16 +- .../basic_2/rt_equivalence/cpcs_cprs.ma | 80 +++++----- .../basic_2/rt_equivalence/cpcs_lprs.ma | 28 ++-- .../basic_2/rt_equivalence/cpes.ma | 8 +- .../basic_2/rt_equivalence/cpes_aaa.ma | 6 +- .../lambdadelta/basic_2/rt_transition/cnr.ma | 14 +- .../basic_2/rt_transition/cnr_drops.ma | 8 +- .../basic_2/rt_transition/cnr_simple.ma | 4 +- .../basic_2/rt_transition/cnr_tdeq.ma | 4 +- .../basic_2/rt_transition/cnu_drops.ma | 4 +- .../basic_2/rt_transition/cnu_tdeq.ma | 4 +- .../lambdadelta/basic_2/rt_transition/cnx.ma | 16 +- .../basic_2/rt_transition/cnx_cnx.ma | 4 +- .../basic_2/rt_transition/cnx_drops.ma | 6 +- .../basic_2/rt_transition/cnx_simple.ma | 8 +- .../lambdadelta/basic_2/rt_transition/cpg.ma | 102 ++++++------- .../basic_2/rt_transition/cpg_drops.ma | 20 +-- .../basic_2/rt_transition/cpg_simple.ma | 4 +- .../lambdadelta/basic_2/rt_transition/cpm.ma | 132 ++++++++-------- .../basic_2/rt_transition/cpm_aaa.ma | 2 +- .../basic_2/rt_transition/cpm_cpx.ma | 2 +- .../basic_2/rt_transition/cpm_drops.ma | 28 ++-- .../basic_2/rt_transition/cpm_lsubr.ma | 6 +- .../basic_2/rt_transition/cpm_simple.ma | 4 +- .../lambdadelta/basic_2/rt_transition/cpr.ma | 64 ++++---- .../basic_2/rt_transition/cpr_drops.ma | 8 +- .../basic_2/rt_transition/cpr_drops_basic.ma | 2 +- .../lambdadelta/basic_2/rt_transition/cpx.ma | 144 +++++++++--------- .../basic_2/rt_transition/cpx_drops.ma | 12 +- .../basic_2/rt_transition/cpx_drops_basic.ma | 2 +- .../basic_2/rt_transition/cpx_fdeq.ma | 6 +- .../basic_2/rt_transition/cpx_fqus.ma | 48 +++--- .../basic_2/rt_transition/cpx_lsubr.ma | 6 +- .../basic_2/rt_transition/cpx_rdeq.ma | 2 +- .../basic_2/rt_transition/cpx_req.ma | 6 +- .../basic_2/rt_transition/cpx_simple.ma | 4 +- .../lambdadelta/basic_2/rt_transition/fpb.ma | 14 +- .../basic_2/rt_transition/fpb_fdeq.ma | 10 +- .../basic_2/rt_transition/fpb_rdeq.ma | 8 +- .../lambdadelta/basic_2/rt_transition/fpbq.ma | 12 +- .../basic_2/rt_transition/fpbq_aaa.ma | 4 +- .../basic_2/rt_transition/fpbq_fpb.ma | 18 +-- .../lambdadelta/basic_2/rt_transition/lpr.ma | 44 +++--- .../basic_2/rt_transition/lpr_fquq.ma | 48 +++--- .../basic_2/rt_transition/lpr_length.ma | 2 +- .../basic_2/rt_transition/lpr_lpr.ma | 136 ++++++++--------- .../basic_2/rt_transition/lpr_lpx.ma | 2 +- .../lambdadelta/basic_2/rt_transition/lpx.ma | 44 +++--- .../basic_2/rt_transition/lpx_aaa.ma | 6 +- .../basic_2/rt_transition/lpx_fquq.ma | 24 +-- .../basic_2/rt_transition/lpx_fsle.ma | 6 +- .../basic_2/rt_transition/lpx_length.ma | 2 +- .../basic_2/rt_transition/lpx_rdeq.ma | 4 +- .../lambdadelta/basic_2/rt_transition/rpx.ma | 78 +++++----- .../basic_2/rt_transition/rpx_fqup.ma | 10 +- .../basic_2/rt_transition/rpx_fsle.ma | 10 +- .../basic_2/rt_transition/rpx_length.ma | 8 +- .../basic_2/rt_transition/rpx_lpx.ma | 8 +- .../basic_2/rt_transition/rpx_rdeq.ma | 38 ++--- .../basic_2/rt_transition/rpx_rpx.ma | 16 +- .../ground_2/notation/functions/basic_2.ma | 2 +- .../notation/functions/cocompose_2.ma | 2 +- .../notation/functions/droppreds_2.ma | 2 +- .../notation/functions/uparrowstar_2.ma | 2 +- .../notation/functions/upspoonstar_2.ma | 2 +- .../ground_2/relocation/mr2_plus.ma | 8 +- .../ground_2/relocation/rtmap_at.ma | 108 ++++++------- .../ground_2/relocation/rtmap_istot.ma | 14 +- .../lambdadelta/ground_2/steps/rtc.ma | 2 +- .../lambdadelta/ground_2/steps/rtc_isrt.ma | 20 +-- .../lambdadelta/ground_2/steps/rtc_max.ma | 28 ++-- .../lambdadelta/ground_2/steps/rtc_plus.ma | 22 +-- .../lambdadelta/ground_2/steps/rtc_shift.ma | 14 +- matita/matita/contribs/lambdadelta/replace.sh | 4 +- .../lambdadelta/static_2/etc/sh_lt.etc | 6 + .../lambdadelta/static_2/i_static/rexs.ma | 70 ++++----- .../static_2/i_static/rexs_drops.ma | 18 +-- .../static_2/i_static/rexs_fqup.ma | 20 +-- .../static_2/i_static/rexs_length.ma | 2 +- .../lambdadelta/static_2/i_static/rexs_lex.ma | 6 +- .../static_2/notation/functions/dxabbr_2.ma | 2 +- .../static_2/notation/functions/dxabst_2.ma | 2 +- .../static_2/notation/functions/dxbind1_2.ma | 2 +- .../static_2/notation/functions/dxbind2_3.ma | 2 +- .../static_2/notation/functions/upspoon_2.ma | 19 +++ .../notation/relations/suptermplus_6.ma | 2 +- .../notation/relations/suptermplus_7.ma | 2 +- .../notation/relations/suptermstar_6.ma | 2 +- .../notation/relations/suptermstar_7.ma | 2 +- .../lambdadelta/static_2/relocation/drops.ma | 132 ++++++++-------- .../static_2/relocation/drops_drops.ma | 32 ++-- .../static_2/relocation/drops_length.ma | 28 ++-- .../static_2/relocation/drops_lex.ma | 20 +-- .../static_2/relocation/drops_seq.ma | 12 +- .../static_2/relocation/drops_sex.ma | 54 +++---- .../static_2/relocation/drops_vector.ma | 2 +- .../static_2/relocation/drops_weight.ma | 8 +- .../lambdadelta/static_2/relocation/lex.ma | 2 +- .../lambdadelta/static_2/relocation/lifts.ma | 14 +- .../lambdadelta/static_2/relocation/sex.ma | 96 ++++++------ .../static_2/relocation/sex_length.ma | 6 +- .../static_2/relocation/sex_sex.ma | 22 +-- .../lambdadelta/static_2/relocation/sex_tc.ma | 18 +-- .../static_2/s_computation/fqup.ma | 38 ++--- .../static_2/s_computation/fqup_drops.ma | 6 +- .../static_2/s_computation/fqup_weight.ma | 8 +- .../static_2/s_computation/fqus.ma | 82 +++++----- .../static_2/s_computation/fqus_drops.ma | 2 +- .../static_2/s_computation/fqus_fqup.ma | 26 ++-- .../static_2/s_computation/fqus_weight.ma | 6 +- .../lambdadelta/static_2/s_transition/fqu.ma | 38 ++--- .../static_2/s_transition/fqu_length.ma | 4 +- .../static_2/s_transition/fqu_tdeq.ma | 4 +- .../static_2/s_transition/fqu_weight.ma | 6 +- .../lambdadelta/static_2/s_transition/fquq.ma | 2 +- .../static_2/s_transition/fquq_length.ma | 2 +- .../static_2/s_transition/fquq_weight.ma | 4 +- .../lambdadelta/static_2/static/aaa.ma | 56 +++---- .../lambdadelta/static_2/static/aaa_aaa.ma | 2 +- .../lambdadelta/static_2/static/aaa_drops.ma | 14 +- .../lambdadelta/static_2/static/aaa_fdeq.ma | 4 +- .../lambdadelta/static_2/static/aaa_fqus.ma | 16 +- .../lambdadelta/static_2/static/aaa_rdeq.ma | 4 +- .../lambdadelta/static_2/static/fdeq.ma | 6 +- .../lambdadelta/static_2/static/fdeq_fdeq.ma | 12 +- .../lambdadelta/static_2/static/fdeq_fqup.ma | 2 +- .../lambdadelta/static_2/static/fdeq_fqus.ma | 6 +- .../lambdadelta/static_2/static/fdeq_req.ma | 2 +- .../static_2/static/frees_drops.ma | 16 +- .../lambdadelta/static_2/static/fsle.ma | 6 +- .../lambdadelta/static_2/static/fsle_drops.ma | 14 +- .../lambdadelta/static_2/static/fsle_fqup.ma | 24 +-- .../lambdadelta/static_2/static/fsle_fsle.ma | 46 +++--- .../static_2/static/fsle_length.ma | 10 +- .../lambdadelta/static_2/static/gcp_aaa.ma | 8 +- .../lambdadelta/static_2/static/gcp_cr.ma | 10 +- .../lambdadelta/static_2/static/lsuba.ma | 10 +- .../lambdadelta/static_2/static/lsuba_aaa.ma | 8 +- .../static_2/static/lsuba_drops.ma | 8 +- .../lambdadelta/static_2/static/lsubc.ma | 10 +- .../static_2/static/lsubc_drops.ma | 8 +- .../lambdadelta/static_2/static/lsubf.ma | 120 +++++++-------- .../static_2/static/lsubf_frees.ma | 2 +- .../static_2/static/lsubf_lsubf.ma | 6 +- .../static_2/static/lsubf_lsubr.ma | 6 +- .../static_2/static/lsubr_drops.ma | 12 +- .../lambdadelta/static_2/static/rdeq.ma | 6 +- .../lambdadelta/static_2/static/rdeq_drops.ma | 2 +- .../lambdadelta/static_2/static/rdeq_fqus.ma | 36 ++--- .../static_2/static/rdeq_length.ma | 2 +- .../lambdadelta/static_2/static/req.ma | 2 +- .../lambdadelta/static_2/static/req_drops.ma | 2 +- .../lambdadelta/static_2/static/req_fsle.ma | 2 +- .../lambdadelta/static_2/static/rex.ma | 118 +++++++------- .../lambdadelta/static_2/static/rex_drops.ma | 44 +++--- .../lambdadelta/static_2/static/rex_fqup.ma | 12 +- .../lambdadelta/static_2/static/rex_fsle.ma | 34 ++--- .../lambdadelta/static_2/static/rex_length.ma | 18 +-- .../lambdadelta/static_2/static/rex_lex.ma | 4 +- .../lambdadelta/static_2/static/rex_rex.ma | 30 ++-- .../static_2/syntax/cl_restricted_weight.ma | 12 +- .../lambdadelta/static_2/syntax/cl_weight.ma | 10 +- .../lambdadelta/static_2/syntax/lveq.ma | 56 +++---- .../static_2/syntax/lveq_length.ma | 32 ++-- .../lambdadelta/static_2/syntax/lveq_lveq.ma | 16 +- .../static_2/syntax/{item_sd.ma => sd.ma} | 0 .../lambdadelta/static_2/syntax/sh.ma | 26 ++++ .../static_2/syntax/{item_sh.ma => sh_lt.ma} | 30 ++-- 298 files changed, 2716 insertions(+), 2670 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/static_2/etc/sh_lt.etc create mode 100644 matita/matita/contribs/lambdadelta/static_2/notation/functions/upspoon_2.ma rename matita/matita/contribs/lambdadelta/static_2/syntax/{item_sd.ma => sd.ma} (100%) create mode 100644 matita/matita/contribs/lambdadelta/static_2/syntax/sh.ma rename matita/matita/contribs/lambdadelta/static_2/syntax/{item_sh.ma => sh_lt.ma} (65%) diff --git a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cnv_eta.ma b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cnv_eta.ma index 2b3dfcd4c..c3c661085 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cnv_eta.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cnv_eta.ma @@ -20,7 +20,7 @@ include "basic_2/dynamic/cnv.ma". (* Extended validy (basic_2B) vs. restricted validity (basic_1A) ************) (* Note: extended validity of a closure, height of cnv_appl > 1 *) -lemma cnv_extended (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 !*[h]. +lemma cnv_extended (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 !*[h]. #h #p #G #L #s @(cnv_appl … 2 p … (⋆s) … (⋆s)) [ // @@ -32,7 +32,7 @@ lemma cnv_extended (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0 qed. (* Note: restricted validity of the η-expanded closure, height of cnv_appl = 1 **) -lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h]. +lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h]. #h #p #G #L #s @(cnv_appl … 1 p … (⋆s) … (ⓐ#0.#2)) [ /2 width=1 by ylt_inj/ diff --git a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.ma b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.ma index bb4edfd67..72b4b347c 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_cpr_omega.ma @@ -51,7 +51,7 @@ lapply (cpm_inv_appl1 … H) -H * * qed-. lemma cpr_inv_Delta_sn (h) (G) (L) (s): - ∀X. ⦃G, L⦄ ⊢ Delta s ➡[h] X → Delta s = X. + ∀X. ⦃G,L⦄ ⊢ Delta s ➡[h] X → Delta s = X. #h #G #L #s #X #H elim (cpm_inv_abst1 … H) -H #X1 #X2 #H1 #H2 #H destruct lapply (cpr_inv_sort1 … H1) -H1 #H destruct @@ -60,19 +60,19 @@ qed-. (* Main properties **********************************************************) -theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega1 s ➡[h] Omega2 s. +theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega1 s ➡[h] Omega2 s. /2 width=1 by cpm_beta/ qed. -theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega3 s. +theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega3 s. /5 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, Delta_lifts/ qed. -theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega3 s ➡[h] Omega1 s. +theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega3 s ➡[h] Omega1 s. /4 width=3 by cpm_zeta, Delta_lifts, lifts_flat/ qed. (* Main inversion properties ************************************************) theorem cpr_inv_Omega1_sn (h) (G) (L) (s): - ∀X. ⦃G, L⦄ ⊢ Omega1 s ➡[h] X → + ∀X. ⦃G,L⦄ ⊢ Omega1 s ➡[h] X → ∨∨ Omega1 s = X | Omega2 s = X. #h #G #L #s #X #H elim (cpm_inv_appl1 … H) -H * [ #W2 #T2 #HW2 #HT2 #H destruct @@ -87,7 +87,7 @@ theorem cpr_inv_Omega1_sn (h) (G) (L) (s): ] qed-. -theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥. +theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥. #h #G #L #s #H elim (cpm_inv_bind1 … H) -H * [ #W #T #_ #_ whd in ⊢ (??%?→?); #H destruct | #X #H #_ #_ #_ diff --git a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma index 6ee55d8d0..f75a20a29 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma @@ -36,16 +36,16 @@ lemma ApplDelta_lifts (f:rtmap) (s0) (s): ⬆*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s). /5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed. -lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s. +lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s. /2 width=1 by cpm_beta/ qed. -lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s. +lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s. /6 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, ApplDelta_lifts/ qed. -lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s. +lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s. /4 width=3 by cpm_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed. -lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s. +lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s. /5 width=4 by cpxs_strap1, cpm_fwd_cpx/ qed. lemma fqup_ApplOmega_41 (G) (L) (s0) (s): ⦃G,L,ApplOmega4 s0 s⦄ ⊐+ ⦃G,L,ApplOmega1 s0 s⦄. diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/flifts_flifts_basic.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/flifts_flifts_basic.ma index 9b7f6fa2b..7ebdc5dab 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/functional/flifts_flifts_basic.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/flifts_flifts_basic.ma @@ -23,7 +23,7 @@ theorem flifts_basic_swap (T) (d1) (d2) (h1) (h2): d2 ≤ d1 → ↑[d2,h2]↑[d1,h1]T = ↑[h2+d1,h1]↑[d2,h2]T. /3 width=1 by flifts_comp, basic_swap/ qed-. (* -lemma flift_join: ∀e1,e2,T. ⬆[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T. +lemma flift_join: ∀e1,e2,T. ⬆[e1,e2] ↑[0,e1] T ≡ ↑[0,e1 + e2] T. #e1 #e2 #T lapply (flift_lift T 0 (e1+e2)) #H elim (lift_split … H e1 e1) -H // #U #H diff --git a/matita/matita/contribs/lambdadelta/apps_2/functional/mf_cpr.ma b/matita/matita/contribs/lambdadelta/apps_2/functional/mf_cpr.ma index 2e7f3d5ef..059a1f66f 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/functional/mf_cpr.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/functional/mf_cpr.ma @@ -20,9 +20,9 @@ include "apps_2/functional/mf_exteq.ma". (* Properties with relocation ***********************************************) -lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → +lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 → ∀T,L,l. ⬇*[l] L ≘ K.ⓓV1 → - ∀gv,lv. ⦃G, L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T. + ∀gv,lv. ⦃G,L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T. #h #G #K #V1 #V2 #HV #T elim T -T * // [ #i #L #l #HKL #gv #lv >mf_lref >mf_lref diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/deq.ma b/matita/matita/contribs/lambdadelta/apps_2/models/deq.ma index 5d0f2d324..849ad9dd4 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/deq.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/deq.ma @@ -19,7 +19,7 @@ include "static_2/syntax/genv.ma". (* DENOTATIONAL EQUIVALENCE ************************************************) definition deq (M): relation4 genv lenv term term ≝ - λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, lv]. + λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,lv]. interpretation "denotational equivalence (model)" 'RingEq M G L T1 T2 = (deq M G L T1 T2). diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/deq_cpr.ma b/matita/matita/contribs/lambdadelta/apps_2/models/deq_cpr.ma index 181acf2e6..3e085aeea 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/deq_cpr.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/deq_cpr.ma @@ -21,7 +21,7 @@ include "apps_2/models/deq.ma". (* Forward lemmas with context-sensitive parallel reduction for terms *******) lemma cpr_fwd_deq (h) (M): is_model M → is_extensional M → - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ≗{M} T2. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ≗{M} T2. #h #M #H1M #H2M #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2 [ /2 width=2 by deq_refl/ | #G #K #V1 #V2 #W2 #_ #IH #HVW2 #gv #v #H diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/li.ma b/matita/matita/contribs/lambdadelta/apps_2/models/li.ma index 4d36f1427..b3ad39157 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/li.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/li.ma @@ -20,7 +20,7 @@ include "apps_2/notation/models/inwbrackets_4.ma". inductive li (M) (gv): relation2 lenv (evaluation M) ≝ | li_atom: ∀lv. li M gv (⋆) lv -| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv, lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv) +| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv,lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv) | li_abst: ∀lv,d,L,W. li M gv L lv → li M gv (L.ⓛW) (⫯[0←d]lv) | li_unit: ∀lv,d,I,L. li M gv L lv → li M gv (L.ⓤ{I}) (⫯[0←d]lv) | li_veq : ∀lv1,lv2,L. li M gv L lv1 → lv1 ≗ lv2 → li M gv L lv2 diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma b/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma index b3acfc262..d06b4efda 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma @@ -26,34 +26,34 @@ record is_model (M): Prop ≝ { (* Note: application: compatibility *) mp: compatible_3 … (ap M) (sq M) (sq M) (sq M); (* Note: interpretation: sort *) - ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv, lv] ≗ sv M s; + ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv,lv] ≗ sv M s; (* Note: interpretation: local reference *) - ml: ∀gv,lv,i. ⟦#i⟧{M}[gv, lv] ≗ lv i; + ml: ∀gv,lv,i. ⟦#i⟧{M}[gv,lv] ≗ lv i; (* Note: interpretation: global reference *) - mg: ∀gv,lv,l. ⟦§l⟧{M}[gv, lv] ≗ gv l; + mg: ∀gv,lv,l. ⟦§l⟧{M}[gv,lv] ≗ gv l; (* Note: interpretation: intensional binder *) - mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1, lv1] ≗ ⟦W⟧{M}[gv2, lv2] → - (∀d. ⟦T⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2, ⫯[0←d]lv2]) → - ⟦ⓛ{p}W.T⟧[gv1, lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2, lv2]; + mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1,lv1] ≗ ⟦W⟧{M}[gv2,lv2] → + (∀d. ⟦T⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2,⫯[0←d]lv2]) → + ⟦ⓛ{p}W.T⟧[gv1,lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2,lv2]; (* Note: interpretation: abbreviation *) - md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] ⊕[p] ⟦T⟧[gv, ⫯[0←⟦V⟧[gv, lv]]lv]; + md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] ⊕[p] ⟦T⟧[gv,⫯[0←⟦V⟧[gv,lv]]lv]; (* Note: interpretation: application *) - ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] @ ⟦T⟧[gv, lv]; + ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] @ ⟦T⟧[gv,lv]; (* Note: interpretation: ζ-equivalence *) mz: ∀d1,d2. d1 ⊕{M}[Ⓣ] d2 ≗ d2; (* Note: interpretation: ϵ-equivalence *) - me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, lv]; + me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv,lv] ≗ ⟦T⟧[gv,lv]; (* Note: interpretation: β-requivalence *) - mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv, lv] ≗ d ⊕[p] ⟦T⟧[gv, ⫯[0←d]lv]; + mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv,lv] ≗ d ⊕[p] ⟦T⟧[gv,⫯[0←d]lv]; (* Note: interpretation: θ-requivalence *) mh: ∀p,d1,d2,d3. d1 @ (d2 ⊕{M}[p] d3) ≗ d2 ⊕[p] (d1 @ d3) }. record is_extensional (M): Prop ≝ { (* Note: interpretation: extensional abstraction *) - mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1, lv1] ≗ ⟦W2⟧{M}[gv2, lv2] → - (∀d. ⟦T1⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2, ⫯[0←d]lv2]) → - ⟦ⓛ{p}W1.T1⟧[gv1, lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2, lv2] + mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1,lv1] ≗ ⟦W2⟧{M}[gv2,lv2] → + (∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2]) → + ⟦ⓛ{p}W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2,lv2] }. record is_injective (M): Prop ≝ { diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/tm.ma b/matita/matita/contribs/lambdadelta/apps_2/models/tm.ma index a0f7b84d1..00358c9a2 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/tm.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/tm.ma @@ -20,7 +20,7 @@ include "apps_2/models/model.ma". definition tm_dd ≝ term. -definition tm_sq (h) (T1) (T2) ≝ ⦃⋆, ⋆⦄ ⊢ T1 ⬌*[h] T2. +definition tm_sq (h) (T1) (T2) ≝ ⦃⋆,⋆⦄ ⊢ T1 ⬌*[h] T2. definition tm_sv (s) ≝ ⋆s. diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma b/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma index 4d3d16c1e..3d9e440a7 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/veq.ma @@ -83,7 +83,7 @@ qed-. lemma ti_comp (M): is_model M → ∀T,gv1,gv2. gv1 ≗ gv2 → ∀lv1,lv2. lv1 ≗ lv2 → - ⟦T⟧[gv1, lv1] ≗{M} ⟦T⟧[gv2, lv2]. + ⟦T⟧[gv1,lv1] ≗{M} ⟦T⟧[gv2,lv2]. #M #HM #T elim T -T * [||| #p * | * ] [ /4 width=5 by seq_trans, seq_sym, ms/ | /4 width=5 by seq_sym, ml, mq/ diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/veq_lifts.ma b/matita/matita/contribs/lambdadelta/apps_2/models/veq_lifts.ma index d7a225140..13a0f9113 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/veq_lifts.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/veq_lifts.ma @@ -22,7 +22,7 @@ include "apps_2/models/veq.ma". fact lifts_fwd_vpush_aux (M): is_model M → is_extensional M → ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀m. 𝐁❴m,1❵ = f → - ∀gv,lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[m←d]lv]. + ∀gv,lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[m←d]lv]. #M #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2 [ #f #s #m #Hf #gv #lv #d @(mq … H1M) [4,5: /3 width=2 by seq_sym, ms/ |1,2: skip ] @@ -67,5 +67,5 @@ qed-. lemma lifts_SO_fwd_vpush (M) (gv): is_model M → is_extensional M → ∀T1,T2. ⬆*[1] T1 ≘ T2 → - ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[0←d]lv]. + ∀lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[0←d]lv]. /2 width=3 by lifts_fwd_vpush_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/vpushs.ma b/matita/matita/contribs/lambdadelta/apps_2/models/vpushs.ma index dabd2c431..34e9de237 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/vpushs.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/vpushs.ma @@ -20,7 +20,7 @@ include "apps_2/models/veq.ma". inductive vpushs (M) (gv) (lv): relation2 lenv (evaluation M) ≝ | vpushs_atom: vpushs M gv lv (⋆) lv -| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv, v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v) +| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv,v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v) | vpushs_abst: ∀v,d,K,V. vpushs M gv lv K v → vpushs M gv lv (K.ⓛV) (⫯[0←d]v) | vpushs_unit: ∀v,d,I,K. vpushs M gv lv K v → vpushs M gv lv (K.ⓤ{I}) (⫯[0←d]v) | vpushs_repl: ∀v1,v2,L. vpushs M gv lv L v1 → v1 ≗ v2 → vpushs M gv lv L v2 @@ -51,7 +51,7 @@ lemma vpushs_inv_atom (M) (gv) (lv): is_model M → fact vpushs_inv_abbr_aux (M) (gv) (lv): is_model M → ∀y,L. L ⨁{M}[gv] lv ≘ y → ∀K,V. K.ⓓV = L → - ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y. + ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y. #M #gv #lv #HM #y #L #H elim H -y -L [ #Y #X #H destruct | #v #d #K #V #Hv #Hd #_ #Y #X #H destruct @@ -66,7 +66,7 @@ qed-. lemma vpushs_inv_abbr (M) (gv) (lv): is_model M → ∀y,K,V. K.ⓓV ⨁{M}[gv] lv ≘ y → - ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y. + ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y. /2 width=3 by vpushs_inv_abbr_aux/ qed-. fact vpushs_inv_abst_aux (M) (gv) (lv): is_model M → diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/vpushs_fold.ma b/matita/matita/contribs/lambdadelta/apps_2/models/vpushs_fold.ma index 59f9b3e84..8ac169986 100644 --- a/matita/matita/contribs/lambdadelta/apps_2/models/vpushs_fold.ma +++ b/matita/matita/contribs/lambdadelta/apps_2/models/vpushs_fold.ma @@ -21,8 +21,8 @@ include "apps_2/models/vpushs.ma". lemma vpushs_fold (M): is_model M → is_extensional M → ∀L,T1,T2,gv,lv. - (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v]) → - ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv]. + (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v]) → + ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv]. #M #H1M #H2M #L elim L -L [| #K * [| * ]] [ #T1 #T2 #gv #lv #H12 >fold_atom >fold_atom @@ -44,8 +44,8 @@ qed. (* Inversion lemmas with fold for restricted closures ***********************) lemma vpushs_inv_fold (M): is_model M → is_injective M → - ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv] → - ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v]. + ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv] → + ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v]. #M #H1M #H2M #L elim L -L [| #K * [| * ]] [ #T1 #T2 #gv #lv >fold_atom >fold_atom #H12 #v #H diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma index fe0b19cf3..0fa415ae4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma @@ -28,9 +28,9 @@ inductive cnv (a:ynat) (h): relation3 genv lenv term ≝ | cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i) | cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T) | cnv_appl: ∀n,p,G,L,V,W0,T,U0. yinj n < a → cnv a h G L V → cnv a h G L T → - ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T) + ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T) | cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T → - ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T) + ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv a h G L (ⓝU.T) . interpretation "context-sensitive native validity (term)" @@ -44,8 +44,8 @@ interpretation "context-sensitive extended native validity (term)" (* Basic inversion lemmas ***************************************************) -fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 → - ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V. +fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → X = #0 → + ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V. #a #h #G #L #X * -G -L -X [ #G #L #s #H destruct | #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/ @@ -56,12 +56,12 @@ fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 → ] qed-. -lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] → - ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V. +lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G,L⦄ ⊢ #0 ![a,h] → + ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V. /2 width=3 by cnv_inv_zero_aux/ qed-. -fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) → - ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}. +fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀i. X = #(↑i) → + ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}. #a #h #G #L #X * -G -L -X [ #G #L #s #j #H destruct | #I #G #K #V #_ #j #H destruct @@ -72,11 +72,11 @@ fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = ] qed-. -lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] → - ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}. +lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![a,h] → + ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}. /2 width=3 by cnv_inv_lref_aux/ qed-. -fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥. +fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀l. X = §l → ⊥. #a #h #G #L #X * -G -L -X [ #G #L #s #l #H destruct | #I #G #K #V #_ #l #H destruct @@ -88,13 +88,13 @@ fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = qed-. (* Basic_2A1: uses: snv_inv_gref *) -lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥. +lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![a,h] → ⊥. /2 width=8 by cnv_inv_gref_aux/ qed-. -fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → +fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀p,I,V,T. X = ⓑ{p,I}V.T → - ∧∧ ⦃G, L⦄ ⊢ V ![a, h] - & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h]. + ∧∧ ⦃G,L⦄ ⊢ V ![a,h] + & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h]. #a #h #G #L #X * -G -L -X [ #G #L #s #q #Z #X1 #X2 #H destruct | #I #G #K #V #_ #q #Z #X1 #X2 #H destruct @@ -106,14 +106,14 @@ fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → qed-. (* Basic_2A1: uses: snv_inv_bind *) -lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] → - ∧∧ ⦃G, L⦄ ⊢ V ![a, h] - & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h]. +lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] → + ∧∧ ⦃G,L⦄ ⊢ V ![a,h] + & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h]. /2 width=4 by cnv_inv_bind_aux/ qed-. -fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T → - ∃∃n,p,W0,U0. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0. +fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀V,T. X = ⓐV.T → + ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0. #a #h #G #L #X * -L -X [ #G #L #s #X1 #X2 #H destruct | #I #G #K #V #_ #X1 #X2 #H destruct @@ -125,14 +125,14 @@ fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X qed-. (* Basic_2A1: uses: snv_inv_appl *) -lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] → - ∃∃n,p,W0,U0. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0. +lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] → + ∃∃n,p,W0,U0. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0. /2 width=3 by cnv_inv_appl_aux/ qed-. -fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T → - ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0. +fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀U,T. X = ⓝU.T → + ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0. #a #h #G #L #X * -G -L -X [ #G #L #s #X1 #X2 #H destruct | #I #G #K #V #_ #X1 #X2 #H destruct @@ -144,16 +144,16 @@ fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X qed-. (* Basic_2A1: uses: snv_inv_appl *) -lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] → - ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0. +lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] → + ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0. /2 width=3 by cnv_inv_cast_aux/ qed-. (* Basic forward lemmas *****************************************************) lemma cnv_fwd_flat (a) (h) (I) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ⓕ{I}V.T ![a,h] → - ∧∧ ⦃G, L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ T ![a,h]. + ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![a,h] → + ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h]. #a #h * #G #L #V #T #H [ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_ | elim (cnv_inv_cast … H) #U #HV #HT #_ #_ diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma index c63a1aa12..aa0c10865 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_aaa.ma @@ -22,7 +22,7 @@ include "basic_2/dynamic/cnv.ma". (* Forward lemmas on atomic arity assignment for terms **********************) (* Basic_2A1: uses: snv_fwd_aaa *) -lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A. +lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A. #a #h #G #L #T #H elim H -G -L -T [ /2 width=2 by aaa_sort, ex_intro/ | #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/ @@ -46,7 +46,7 @@ qed-. (* Forward lemmas with t_bound rt_transition for terms **********************) lemma cnv_fwd_cpm_SO (a) (h) (G) (L): - ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U. + ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U. #a #h #G #L #T #H elim (cnv_fwd_aaa … H) -H #A #HA /2 width=2 by aaa_cpm_SO/ @@ -55,7 +55,7 @@ qed-. (* Forward lemmas with t_bound rt_computation for terms *********************) lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L): - ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U. + ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U. #a #h #n #G #L #T #H elim (cnv_fwd_aaa … H) -H #A #HA /2 width=2 by cpms_total_aaa/ @@ -64,9 +64,9 @@ qed-. (* Advanced inversion lemmas ************************************************) lemma cnv_inv_appl_pred (a) (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![yinj a, h] → - ∃∃p,W0,U0. ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[↓a, h] ⓛ{p}W0.U0. + ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![yinj a,h] → + ∃∃p,W0,U0. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[↓a,h] ⓛ{p}W0.U0. #a #h #G #L #V #T #H elim (cnv_inv_appl … H) -H #n #p #W #U #Ha #HV #HT #HVW #HTU lapply (ylt_inv_inj … Ha) -Ha #Ha diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpes.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpes.ma index e2eca5cdf..190165fe6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpes.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpes.ma @@ -21,42 +21,42 @@ include "basic_2/dynamic/cnv_aaa.ma". lemma cnv_appl_cpes (a) (h) (G) (L): ∀n. yinj n < a → - ∀V. ⦃G, L⦄ ⊢ V ![a, h] → ∀T. ⦃G, L⦄ ⊢ T ![a, h] → - ∀W. ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W → - ∀p,U. ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U → ⦃G, L⦄ ⊢ ⓐV.T ![a, h]. + ∀V. ⦃G,L⦄ ⊢ V ![a,h] → ∀T. ⦃G,L⦄ ⊢ T ![a,h] → + ∀W. ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W → + ∀p,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![a,h]. #a #h #G #L #n #Hn #V #HV #T #HT #W * /4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/ qed. lemma cnv_cast_cpes (a) (h) (G) (L): - ∀U. ⦃G, L⦄ ⊢ U ![a, h] → - ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G, L⦄ ⊢ ⓝU.T ![a, h]. + ∀U. ⦃G,L⦄ ⊢ U ![a,h] → + ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G,L⦄ ⊢ ⓝU.T ![a,h]. #a #h #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/ qed. (* Inversion lemmas with t-bound rt-equivalence for terms *******************) lemma cnv_inv_appl_cpes (a) (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] → - ∃∃n,p,W,U. yinj n < a & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W.U. + ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] → + ∃∃n,p,W,U. yinj n < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U. #a #h #G #L #V #T #H elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU /3 width=7 by cpms_div, ex5_4_intro/ qed-. lemma cnv_inv_appl_pred_cpes (a) (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ⓐV.T ![yinj a, h] → - ∃∃p,W,U. ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & - ⦃G, L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G, L⦄ ⊢ T ➡*[↓a, h] ⓛ{p}W.U. + ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![yinj a,h] → + ∃∃p,W,U. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & + ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[↓a,h] ⓛ{p}W.U. #a #h #G #L #V #T #H elim (cnv_inv_appl_pred … H) -H #p #W #U #HV #HT #HVW #HTU /3 width=7 by cpms_div, ex4_3_intro/ qed-. lemma cnv_inv_cast_cpes (a) (h) (G) (L): - ∀U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] → - ∧∧ ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] & ⦃G, L⦄ ⊢ U ⬌*[h,0,1] T. + ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] → + ∧∧ ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T. #a #h #G #L #U #T #H elim (cnv_inv_cast … H) -H /3 width=3 by cpms_div, and3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma index f0eb6355b..141a733c2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_conf.ma @@ -24,7 +24,7 @@ include "basic_2/dynamic/cnv_preserve_sub.ma". (* Sub diamond propery with t-bound rt-transition for terms *****************) fact cnv_cpm_conf_lpr_atom_atom_aux (h) (G) (L1) (L2) (I): - ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡*[O,h] T. + ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡*[O,h] T. /2 width=3 by ex2_intro/ qed-. fact cnv_cpm_conf_lpr_atom_ess_aux (h) (G) (L1) (L2) (s): @@ -388,8 +388,8 @@ qed-. fact cnv_cpm_conf_lpr_aux (a) (h): ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1. #a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]] [ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq.ma index d1f41252b..c10cdba20 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq.ma @@ -23,8 +23,8 @@ include "basic_2/dynamic/cnv_fsb.ma". (* Inversion lemmas with restricted rt-transition for terms *****************) lemma cnv_cpr_tdeq_fwd_refl (a) (h) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → - ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → + ⦃G,L⦄ ⊢ T1 ![a,h] → T1 = T2. #a #h #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2 [ // | #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2 @@ -55,9 +55,9 @@ lemma cnv_cpr_tdeq_fwd_refl (a) (h) (G) (L): qed-. lemma cpm_tdeq_inv_bind_sn (a) (h) (n) (p) (I) (G) (L): - ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] → - ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X → - ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2. + ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] → + ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X → + ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2. #a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2 elim (cpm_inv_bind1 … H1) -H1 * [ #XV #T2 #HXV #HT12 #H destruct @@ -75,8 +75,8 @@ qed-. lemma cpm_tdeq_inv_appl_sn (a) (h) (n) (G) (L): ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![a,h] → ∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X → - ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ V ➡*[1,h] W & ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1 - & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2. + ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1 + & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2. #a #h #n #G #L #V #T1 #H0 #X #H1 #H2 elim (cpm_inv_appl1 … H1) -H1 * [ #XV #T2 #HXV #HT12 #H destruct @@ -92,11 +92,11 @@ elim (cpm_inv_appl1 … H1) -H1 * qed-. lemma cpm_tdeq_inv_cast_sn (a) (h) (n) (G) (L): - ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] → - ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X → + ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![a,h] → + ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X → ∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 - & ⦃G, L⦄ ⊢ U1 ![a,h] & ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2 - & ⦃G, L⦄ ⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2. + & ⦃G,L⦄ ⊢ U1 ![a,h] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2 + & ⦃G,L⦄ ⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2. #a #h #n #G #L #U1 #T1 #H0 #X #H1 #H2 elim (cpm_inv_cast1 … H1) -H1 [ * || * ] [ #U2 #T2 #HU12 #HT12 #H destruct @@ -115,9 +115,9 @@ elim (cpm_inv_cast1 … H1) -H1 [ * || * ] qed-. lemma cpm_tdeq_inv_bind_dx (a) (h) (n) (p) (I) (G) (L): - ∀X. ⦃G, L⦄ ⊢ X ![a,h] → - ∀V,T2. ⦃G, L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 → - ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1. + ∀X. ⦃G,L⦄ ⊢ X ![a,h] → + ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 → + ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1. #a #h #n #p #I #G #L #X #H0 #V #T2 #H1 #H2 elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct @@ -134,14 +134,14 @@ lemma cpm_tdeq_ind (a) (h) (n) (G) (Q:relation3 …): Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2) ) → (∀m. yinj m < a → - ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G, L⦄ ⊢ V ➡*[1,h] W → - ∀p,T1,U1. ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] → - ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → + ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W → + ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] → + ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2) ) → (∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 → - ∀U2. ⦃G, L⦄ ⊢ U1 ![a,h] → ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 → - ∀T2. ⦃G, L⦄ ⊢ T1 ![a,h] → ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → + ∀U2. ⦃G,L⦄ ⊢ U1 ![a,h] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 → + ∀T2. ⦃G,L⦄ ⊢ T1 ![a,h] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2) ) → ∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] → @@ -170,9 +170,9 @@ qed-. (* Advanced properties with restricted rt-transition for terms **************) lemma cpm_tdeq_free (a) (h) (n) (G) (L): - ∀T1. ⦃G, L⦄ ⊢ T1 ![a,h] → - ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → - ∀F,K. ⦃F, K⦄ ⊢ T1 ➡[n,h] T2. + ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → + ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → + ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2. #a #h #n #G #L #T1 #H0 #T2 #H1 #H2 @(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2 [ #I #L #H #F #K destruct // @@ -189,9 +189,9 @@ qed-. (* Advanced inversion lemmas with restricted rt-transition for terms ********) lemma cpm_tdeq_inv_bind_sn_void (a) (h) (n) (p) (I) (G) (L): - ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] → - ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X → - ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2. + ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] → + ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X → + ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2. #a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2 elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H /3 width=5 by ex5_intro, cpm_tdeq_free/ 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 index 8ea98673b..3dbea60d6 100644 --- 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 @@ -18,16 +18,16 @@ include "basic_2/dynamic/cnv_cpm_tdeq.ma". (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) definition IH_cnv_cpm_tdeq_conf_lpr (a) (h): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 → - ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T. + λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] → + ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 → + ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 → + ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T. (* Diamond propery with restricted rt-transition for terms ******************) fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (G0) (L1) (L2) (I): - ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0, L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T. + ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0,L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T. #h #G0 #L1 #L2 #I /2 width=5 by ex4_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_trans.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_trans.ma index 3a3919fc4..44b8fe219 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_trans.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_tdeq_trans.ma @@ -18,7 +18,7 @@ include "basic_2/dynamic/cnv_cpm_tdeq.ma". (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************) definition IH_cnv_cpm_tdeq_cpm_trans (a) (h): relation3 genv lenv term ≝ - λG,L,T1. ⦃G, L⦄ ⊢ T1 ![a,h] → + λG,L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛ T → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡[n1,h] T2 & T0 ≛ T2. @@ -26,7 +26,7 @@ definition IH_cnv_cpm_tdeq_cpm_trans (a) (h): relation3 genv lenv term ≝ (* Transitive properties restricted rt-transition for terms *****************) fact cnv_cpm_tdeq_cpm_trans_sub (a) (h) (G0) (L0) (T0): - (∀G,L,T. ⦃G0, L0, T0⦄ >[h] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → + (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → (∀G,L,T. ⦃G0,L0,T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cnv_cpm_tdeq_cpm_trans a h G L T) → ∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans a h G L T1. #a #h #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]] @@ -89,7 +89,7 @@ fact cnv_cpm_tdeq_cpm_trans_sub (a) (h) (G0) (L0) (T0): qed-. fact cnv_cpm_tdeq_cpm_trans_aux (a) (h) (G0) (L0) (T0): - (∀G,L,T. ⦃G0, L0, T0⦄ >[h] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → + (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) → IH_cnv_cpm_tdeq_cpm_trans a h G0 L0 T0. #a #h #G0 #L0 #T0 @(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #IH0 diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_trans.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_trans.ma index 8c720f4ca..21f0e500c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_trans.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpm_trans.ma @@ -25,8 +25,8 @@ include "basic_2/dynamic/lsubv_cnv.ma". fact cnv_cpm_trans_lpr_aux (a) (h): ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr a h G1 L1 T1. #a #h #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ] [ #s #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #_ destruct -IH2 -IH1 -H1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_conf.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_conf.ma index d1d60870d..11d2b3652 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_conf.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_cpms_conf.ma @@ -78,7 +78,7 @@ fact cnv_cpms_conf_lpr_step_tdneq_sub (a) (h) (G0) (L0) (T0) (m11) (m12) (m21) ( lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X02 … L0 ?) // #HX2 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … H1X01 … H1X02 … L0 … L0) // #Z0 #HXZ10 #HXZ20 -cut (⦃G0,L0,T0⦄ >[h] ⦃G0,L0,X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *) +cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *) lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ20 … L0 ?) // #HZ0 elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02 @@ -135,7 +135,7 @@ fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (a) (h) (G0) (L0) (T0) (m11) (m12) (m21) lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1 lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2 elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX01 … HX02 … L0 … L0) // #Z0 #HXZ10 #HXZ20 -cut (⦃G0,L0,T0⦄ >[h] ⦃G0,L0,X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *) +cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *) lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ10 … L0 ?) // #HZ0 elim (IH1 … HXT1 … HXZ10 … L1 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT1 -HXZ10 #Z1 #HTZ1 #HZ01 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 44790ef38..8394bfa8d 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 @@ -19,10 +19,10 @@ include "basic_2/dynamic/cnv_cpm_tdeq_trans.ma". (* Properties with restricted rt-computation for terms **********************) fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (G) (L) (T1): - ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛ T2 → ⊥) → + ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → (T1 ≛ T2 → ⊥) → (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) → (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) → - ∃∃n1,n2,T0. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G, L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n. + ∃∃n1,n2,T0. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n. #a #h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -n -T1 [ #_ #H2T2 elim H2T2 -H2T2 // @@ -43,9 +43,9 @@ fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (G) (L) (T1): qed-. fact cpms_tdeq_ind_sn (a) (h) (G) (L) (T2) (Q:relation2 …): - (⦃G, L⦄ ⊢ T2 ![a,h] → Q 0 T2) → - (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛ T → ⦃G, L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G, L⦄ ⊢ T ![a,h] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) → - ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛ T2 → + (⦃G,L⦄ ⊢ T2 ![a,h] → Q 0 T2) → + (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T ![a,h] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) → + ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![a,h] → T1 ≛ T2 → (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) → (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) → Q n T1. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_drops.ma index 7bcc3f1fc..9483346c4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_drops.ma @@ -20,8 +20,8 @@ include "basic_2/dynamic/cnv.ma". (* Advanced dproperties *****************************************************) (* Basic_2A1: uses: snv_lref *) -lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G, K⦄ ⊢ V ![a, h] → - ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ #i ![a, h]. +lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G,K⦄ ⊢ V ![a,h] → + ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ #i ![a,h]. #a #h #G #I #K #V #i elim i -i [ #L #HV #H lapply (drops_fwd_isid … H ?) -H // #H destruct @@ -36,8 +36,8 @@ qed. (* Basic_2A1: uses: snv_inv_lref *) lemma cnv_inv_lref_drops (a) (h) (G): - ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] → - ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ![a, h]. + ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] → + ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ![a,h]. #a #h #G #i elim i -i [ #L #H elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct @@ -50,15 +50,15 @@ lemma cnv_inv_lref_drops (a) (h) (G): qed-. lemma cnv_inv_lref_pair (a) (h) (G): - ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] → - ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ![a, h]. + ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] → + ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ![a,h]. #a #h #G #i #L #H #I #K #V #HLK elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX lapply (drops_mono … HLY … HLK) -L #H destruct // qed-. lemma cnv_inv_lref_atom (a) (h) (b) (G): - ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] → + ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] → ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥. #a #h #b #G #i #L #H #Hi elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_ @@ -67,7 +67,7 @@ lapply (drops_mono … HLY … Hi) -L #H destruct qed-. lemma cnv_inv_lref_unit (a) (h) (G): - ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] → + ∀i,L. ⦃G,L⦄ ⊢ #i ![a,h] → ∀I,K. ⬇*[i] L ≘ K.ⓤ{I} → ⊥. #a #h #G #i #L #H #I #K #HLK elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_ diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fqus.ma index 113ffe922..678495b8e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fqus.ma @@ -20,8 +20,8 @@ include "basic_2/dynamic/cnv_drops.ma". (* Properties with supclosure ***********************************************) (* Basic_2A1: uses: snv_fqu_conf *) -lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h]. +lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h]. #a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I1 #G1 #L1 #V1 #H elim (cnv_inv_zero … H) -H #I2 #L2 #V2 #HV2 #H destruct // @@ -43,22 +43,22 @@ lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2 qed-. (* Basic_2A1: uses: snv_fquq_conf *) -lemma cnv_fquq_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h]. +lemma cnv_fquq_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h]. #a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*] /2 width=5 by cnv_fqu_conf/ qed-. (* Basic_2A1: uses: snv_fqup_conf *) -lemma cnv_fqup_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h]. +lemma cnv_fqup_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h]. #a #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 /3 width=5 by fqup_strap1, cnv_fqu_conf/ qed-. (* Basic_2A1: uses: snv_fqus_conf *) -lemma cnv_fqus_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h]. +lemma cnv_fqus_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ T1 ![a,h] → ⦃G2,L2⦄ ⊢ T2 ![a,h]. #a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*] /2 width=5 by cnv_fqup_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma index 763faf1d9..bf6732dab 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_fsb.ma @@ -20,13 +20,13 @@ include "basic_2/dynamic/cnv_aaa.ma". (* Forward lemmas with strongly rst-normalizing closures ********************) (* Basic_2A1: uses: snv_fwd_fsb *) -lemma cnv_fwd_fsb (a) (h): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ≥[h] 𝐒⦃G, L, T⦄. +lemma cnv_fwd_fsb (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ≥[h] 𝐒⦃G,L,T⦄. #a #h #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/ qed-. (* Forward lemmas with strongly rt-normalizing terms ************************) -lemma cnv_fwd_csx (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma cnv_fwd_csx (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #a #h #G #L #T #H /3 width=2 by cnv_fwd_fsb, fsb_inv_csx/ qed-. @@ -34,5 +34,5 @@ qed-. (* Inversion lemmas with proper parallel rst-computation for closures *******) lemma cnv_fpbg_refl_false (a) (h) (G) (L) (T): - ⦃G, L⦄ ⊢ T ![a,h] → ⦃G, L, T⦄ >[h] ⦃G, L, T⦄ → ⊥. + ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥. /3 width=7 by cnv_fwd_fsb, fsb_fpbg_refl_false/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_sub.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_sub.ma index ce9c46f7e..2eedc2889 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_sub.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_sub.ma @@ -20,39 +20,39 @@ include "basic_2/dynamic/cnv.ma". (* Inductive premises for the preservation results **************************) definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝ - λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] → - ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 → - ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h]. + λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] → + ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡[n,h] T2 → + ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h]. definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝ - λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] → - ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 → - ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h]. + λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![a,h] → + ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![a,h]. definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T. + λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] → + ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → + ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T. definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T. + λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] → + ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → + ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T. definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝ - λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] → - ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T. + λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![a,h] → + ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡*[n2,h] T2 → + ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T. (* Auxiliary properties for preservation ************************************) fact cnv_cpms_trans_lpr_sub (a) (h): ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → - ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1. + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) → + ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1. #a #h #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H @(cpms_ind_dx … H) -n -T2 /3 width=7 by fpbg_cpms_trans/ @@ -60,12 +60,12 @@ qed-. fact cnv_cpm_conf_lpr_sub (a) (h): ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → - ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1. + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → + ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1. /3 width=8 by cpm_cpms/ qed-. fact cnv_cpms_strip_lpr_sub (a) (h): ∀G0,L0,T0. - (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → - ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1. + (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) → + ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1. /3 width=8 by cpm_cpms/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma index 20b5259d0..17a22da58 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv.ma @@ -20,7 +20,7 @@ include "basic_2/dynamic/cnv.ma". inductive lsubv (a) (h) (G): relation lenv ≝ | lsubv_atom: lsubv a h G (⋆) (⋆) | lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I}) -| lsubv_beta: ∀L1,L2,W,V. ⦃G, L1⦄ ⊢ ⓝW.V ![a,h] → +| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![a,h] → lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW) . @@ -45,7 +45,7 @@ lemma lsubv_inv_atom_sn (a) (h) (G): ∀L2. G ⊢ ⋆ ⫃![a,h] L2 → L2 = ⋆. fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → ∀I,K1. L1 = K1.ⓘ{I} → ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I} - | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] & + | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW. #a #h #G #L1 #L2 * -L1 -L2 @@ -58,7 +58,7 @@ qed-. (* Basic_2A1: uses: lsubsv_inv_pair1 *) lemma lsubv_inv_bind_sn (a) (h) (G): ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 → ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I} - | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] & + | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW. /2 width=3 by lsubv_inv_bind_sn_aux/ qed-. @@ -78,7 +78,7 @@ lemma lsubv_inv_atom2 (a) (h) (G): ∀L1. G ⊢ L1 ⫃![a,h] ⋆ → L1 = ⋆. fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → ∀I,K2. L2 = K2.ⓘ{I} → ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I} - | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] & + | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V. #a #h #G #L1 #L2 * -L1 -L2 [ #J #K2 #H destruct @@ -90,7 +90,7 @@ qed-. (* Basic_2A1: uses: lsubsv_inv_pair2 *) lemma lsubv_inv_bind_dx (a) (h) (G): ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} → ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I} - | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] & + | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![a,h] & G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V. /2 width=3 by lsubv_inv_bind_dx_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv_cnv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv_cnv.ma index b1782a021..e5c87b05f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv_cnv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubv_cnv.ma @@ -20,8 +20,8 @@ include "basic_2/dynamic/lsubv_cpms.ma". (* Basic_2A1: uses: lsubsv_snv_trans *) lemma lsubv_cnv_trans (a) (h) (G): - ∀L2,T. ⦃G, L2⦄ ⊢ T ![a,h] → - ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G, L1⦄ ⊢ T ![a,h]. + ∀L2,T. ⦃G,L2⦄ ⊢ T ![a,h] → + ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G,L1⦄ ⊢ T ![a,h]. #a #h #G #L2 #T #H elim H -G -L2 -T // [ #I #G #K2 #V #HV #IH #L1 #H elim (lsubv_inv_bind_dx … H) -H * /3 width=1 by cnv_zero/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma index 0d058a673..ebc6fd96d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_preserve.ma @@ -125,7 +125,7 @@ elim (cpms_inv_lref_sn … H2) -H2 * qed-. lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L): - ∀X2, i. ⦃G,L⦄ ⊢ #i :[a,h] X2 → + ∀X2,i. ⦃G,L⦄ ⊢ #i :[a,h] X2 → ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h] | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]. #a #h #G #L #X2 #i #H @@ -241,7 +241,7 @@ elim (cpms_inv_cast1 … H2) -H2 [ * || * ] qed-. (* Basic_1: uses: ty3_gen_lift *) -(* Note: "⦃G,L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G,L⦄ ⊢ X2 ➡*[h] U2" *) +(* Note: "⦃G, L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G, L⦄ ⊢ X2 ➡*[h] U2" *) lemma nta_inv_lifts_sn (a) (h) (G): ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 → ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 → diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas.ma index 3c65cbd7c..99d2b7e12 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas.ma @@ -20,7 +20,7 @@ include "basic_2/dynamic/cnv.ma". (* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************) definition ntas (a) (h) (n) (G) (L): relation term ≝ λT,U. - ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[n,h] U0. + ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[n,h] U0. interpretation "iterated native type assignment (term)" 'Colon a h n G L T U = (ntas a h n G L T U). diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_etc.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_etc.ma index 0d531af18..f6ffde847 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_etc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_etc.ma @@ -19,13 +19,13 @@ include "basic_2/hod/ntas.ma". (* Advanced properties on native type assignment for terms ******************) -lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U → - ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U. +lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U → + ∀V. ⦃h,L⦄ ⊢ V : W → ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U. #h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/ qed. -axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U → - ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z. +axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U → + ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z. (* REQUIRES SUBJECT CONVERSION #h #L #T #W #Y #H @(ntas_ind_dx … H) -T [ #U #HYU @@ -35,9 +35,9 @@ axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T (* Advanced inversion lemmas on native type assignment for terms ************) -fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X → - ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & - L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X → + ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T. #h #L #Z #U #H elim H -L -Z -U [ #L #k #X #Y #H destruct | #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct @@ -54,13 +54,13 @@ fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X qed. (* Basic_1: was only: ty3_gen_appl *) -lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U → - ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & - L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U → + ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T. /2 width=3/ qed-. -axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U → - ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W & +axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U → + ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W & L ⊢ ⓐZ.ⓛY.W ⬌* U. (* REQUIRES SUBJECT REDUCTION #h #L #Z #Y #X #U #H diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma index c9ae6d1eb..7374d2df4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_dynamic/ntas_nta.ma @@ -25,8 +25,8 @@ definition ntas: sh → lenv → relation term ≝ (* Basic eliminators ********************************************************) axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 → - (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) → - ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1. + (∀T1,T. ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → R T → R T1) → + ∀T1. ⦃h,L⦄ ⊢ T1 :* T2 → R T1. (* #h #L #T2 #R #HT2 #IHT2 #T1 #HT12 @(star_ind_dx … HT2 IHT2 … HT12) // @@ -35,10 +35,10 @@ qed-. (* Basic properties *********************************************************) lemma ntas_strap1: ∀h,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2. + ⦃h,L⦄ ⊢ T1 :* T → ⦃h,L⦄ ⊢ T : T2 → ⦃h,L⦄ ⊢ T1 :* T2. /2 width=3/ qed. lemma ntas_strap2: ∀h,L,T1,T,T2. - ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2. + ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → ⦃h,L⦄ ⊢ T1 :* T2. /2 width=3/ qed. *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/prediteval_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/prediteval_5.ma index db0a4aa6d..b87e86381 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/prediteval_5.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/prediteval_5.ma @@ -9,6 +9,9 @@ (* \ / *) (* \ / This file is distributed under the terms of the *) (* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⥲* [ break term 46 h ] 𝐍 ⦃ break term 46 T2 ⦄ )" diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpme.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpme.ma index 78cf43d23..931c8168b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpme.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpme.ma @@ -20,7 +20,7 @@ include "basic_2/rt_computation/cpms.ma". (* Basic_2A1: uses: cpre *) definition cpme (h) (n) (G) (L): relation2 term term ≝ - λT1,T2. ∧∧ ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T2⦄. + λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T2⦄. interpretation "evaluation for t-bound context-sensitive parallel rt-transition (term)" 'PRedEval h n G L T1 T2 = (cpme h n G L T1 T2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma index 454ee8312..d1817d6ff 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma @@ -35,15 +35,15 @@ interpretation lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …): Q 0 T2 → - (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) → - ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1. + (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) → + ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T1. #h #G #L #T2 #Q @ltc_ind_sn_refl // qed-. lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …): Q 0 T1 → - (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) → - ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2. + (∀n1,n2,T,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → Q n1 T → ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → Q (n1+n2) T2) → + ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T2. #h #G #L #T1 #Q @ltc_ind_dx_refl // qed-. @@ -52,36 +52,36 @@ qed-. (* Basic_1: includes: pr1_pr0 *) (* Basic_1: uses: pr3_pr2 *) (* Basic_2A1: includes: cpr_cprs *) -lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2. +lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2. /2 width=1 by ltc_rc/ qed. -lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → - ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2. +lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → + ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2. /2 width=3 by ltc_sn/ qed-. -lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → - ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2. +lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → + ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2. /2 width=3 by ltc_dx/ qed-. (* Basic_2A1: uses: cprs_bind_dx *) lemma cpms_bind_dx (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2. #n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed. lemma cpms_appl_dx (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2. #n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/ qed. lemma cpms_zeta (n) (h) (G) (L): ∀T1,T. ⬆*[1] T ≘ T1 → - ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2. + ∀V,T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2. #n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2 /3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/ qed. @@ -89,22 +89,22 @@ qed. (* Basic_2A1: uses: cprs_zeta *) lemma cpms_zeta_dx (n) (h) (G) (L): ∀T2,T. ⬆*[1] T2 ≘ T → - ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2. + ∀V,T1. ⦃G,L.ⓓV⦄ ⊢ T1 ➡*[n,h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2. #n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/ qed. (* Basic_2A1: uses: cprs_eps *) lemma cpms_eps (n) (h) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → - ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → + ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ➡*[n,h] T2. #n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/ qed. lemma cpms_ee (n) (h) (G) (L): - ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 → - ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2. + ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 → + ∀T. ⦃G,L⦄ ⊢ ⓝU1.T ➡*[↑n,h] U2. #n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n [ /3 width=1 by cpm_cpms, cpm_ee/ | #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1 @@ -114,21 +114,21 @@ qed. (* Basic_2A1: uses: cprs_beta_dx *) lemma cpms_beta_dx (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 → - ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 → + ∀T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2. #n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2 /4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/ qed. (* Basic_2A1: uses: cprs_theta_dx *) lemma cpms_theta_dx (n) (h) (G) (L): - ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V → + ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 → - ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 → - ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2. + ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 → + ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2. #n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2 /4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/ qed. @@ -150,17 +150,17 @@ qed. (* Basic inversion lemmas ***************************************************) -lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s). +lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s). #n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 // #n1 #n2 #X #X2 #_ #IH #HX2 destruct elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct // qed-. lemma cpms_inv_cast1 (h) (n) (G) (L): - ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 → - ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2 - | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2 - | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m. + ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 → + ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2 + | ⦃G,L⦄ ⊢ T1 ➡*[n,h] X2 + | ∃∃m. ⦃G,L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m. #h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2 [ /3 width=5 by or3_intro0, ex3_2_intro/ | #n1 #n2 #X #X2 #_ * [ * || * ] diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma index 41d134080..f4cc579aa 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_aaa.ma @@ -26,7 +26,7 @@ lemma cpms_aaa_conf (n) (h): ∀G,L. Conf3 … (aaa G L) (cpms h G L n). /3 width=5 by cpms_fwd_cpxs, cpxs_aaa_conf/ qed-. lemma cpms_total_aaa (h) (G) (L) (n) (A): - ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U. + ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U. #h #G #L #n elim n -n [ /2 width=3 by ex_intro/ | #n #IH #A #T1 #HT1 (plus_n_O … n) -HT12 @@ -34,9 +34,9 @@ theorem cpms_bind (n) (h) (G) (L): qed. theorem cpms_appl (n) (h) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → - ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 → + ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2. #n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2 [ /2 width=1 by cpms_appl_dx/ | #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12 @@ -46,10 +46,10 @@ qed. (* Basic_2A1: includes: cprs_beta_rc *) theorem cpms_beta_rc (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2. #n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2 [ /2 width=1 by cpms_beta_dx/ | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12 @@ -59,10 +59,10 @@ qed. (* Basic_2A1: includes: cprs_beta *) theorem cpms_beta (n) (h) (G) (L): - ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 → - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2. + ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 → + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2. #n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2 [ /2 width=1 by cpms_beta_rc/ | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12 @@ -72,10 +72,10 @@ qed. (* Basic_2A1: includes: cprs_theta_rc *) theorem cpms_theta_rc (n) (h) (G) (L): - ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 → - ∀W1,T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2. + ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 → + ∀W1,T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2. #n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2 [ /2 width=3 by cpms_theta_dx/ | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12 @@ -85,10 +85,10 @@ qed. (* Basic_2A1: includes: cprs_theta *) theorem cpms_theta (n) (h) (G) (L): - ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 → - ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 → - ∀V1. ⦃G, L⦄ ⊢ V1 ➡*[h] V → - ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2. + ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 → + ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀V1. ⦃G,L⦄ ⊢ V1 ➡*[h] V → + ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2. #n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1 [ /2 width=3 by cpms_theta_rc/ | #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12 @@ -98,30 +98,30 @@ qed. (* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *) theorem cpms_trans (h) (G) (L): - ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → - ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2. + ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → + ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2. /2 width=3 by ltc_trans/ qed-. (* Basic_2A1: uses: scpds_cprs_trans *) theorem cpms_cprs_trans (n) (h) (G) (L): - ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T → - ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2. + ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T → + ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2. #n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n) /2 width=3 by cpms_trans/ qed-. (* Advanced inversion lemmas ************************************************) lemma cpms_inv_appl_sn (n) (h) (G) (L): - ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] X2 → + ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] X2 → ∨∨ ∃∃V2,T2. - ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 & + ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓐV2.T2 | ∃∃n1,n2,p,W,T. - ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2, h] X2 & + ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2,h] X2 & n1 + n2 = n | ∃∃n1,n2,p,V0,V2,V,T. - ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 & - ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2, h] X2 & + ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 & + ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2,h] X2 & n1 + n2 = n. #n #h #G #L #V1 #T1 #U2 #H @(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/ @@ -145,8 +145,8 @@ lemma cpms_inv_appl_sn (n) (h) (G) (L): ] qed-. -lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T & ⦃G, L⦄ ⊢ T ➡*[n2, h] T2. +lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2 → + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2. #h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/ #n1 #IH #n2 #T1 #T2 [h] ⦃G2, L2, T⦄ → - ∀T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ → + ∀T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbg_fpbs_trans, cpms_fwd_fpbs/ qed-. lemma cpms_fpbg_trans (h) (n): - ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ➡*[n,h] T → - ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ➡*[n,h] T → + ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fpbg_trans, cpms_fwd_fpbs/ qed-. lemma fqup_cpms_fwd_fpbg (h): - ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T⦄ → - ∀n,T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. -/3 width=5 by cpms_fwd_fpbs, fqup_fpbg,fpbg_fpbs_trans/ qed-. + ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T⦄ → + ∀n,T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. +/3 width=5 by cpms_fwd_fpbs, fqup_fpbg, fpbg_fpbs_trans/ qed-. lemma cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg (h) (G) (L) (T1): ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) → diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_fpbs.ma index 47aca5419..0fb54e8a6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_fpbs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_fpbs.ma @@ -21,5 +21,5 @@ include "basic_2/rt_computation/cpms_cpxs.ma". (* Basic_2A1: uses: cprs_fpbs *) lemma cpms_fwd_fpbs (n) (h): - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄. /3 width=2 by cpms_fwd_cpxs, cpxs_fpbs/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_lpr.ma index a636655eb..dc6c0f065 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_lpr.ma @@ -20,8 +20,8 @@ include "basic_2/rt_computation/cpms_cpms.ma". (* Properties with parallel rt-transition for full local environments *******) lemma lpr_cpm_trans (n) (h) (G): - ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 → - ∀L1. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2. + ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 → + ∀L1. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2. #n #h #G #L2 #T1 #T2 #H @(cpm_ind … H) -n -G -L2 -T1 -T2 [ /2 width=3 by/ | /3 width=2 by cpm_cpms/ @@ -46,8 +46,8 @@ lemma lpr_cpm_trans (n) (h) (G): qed-. lemma lpr_cpms_trans (n) (h) (G): - ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2. + ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → + ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2. #n #h #G #L1 #L2 #HL12 #T1 #T2 #H @(cpms_ind_sn … H) -n -T1 /3 width=3 by lpr_cpm_trans, cpms_trans/ qed-. @@ -56,14 +56,14 @@ qed-. (* Basic_2A1: includes cpr_bind2 *) lemma cpm_bind2 (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2. /4 width=5 by lpr_cpm_trans, cpms_bind_dx, lpr_pair/ qed. (* Basic_2A1: includes cprs_bind2_dx *) lemma cpms_bind2_dx (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2. /4 width=5 by lpr_cpms_trans, cpms_bind_dx, lpr_pair/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_rdeq.ma index f2470dbe3..f6c8198ac 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms_rdeq.ma @@ -20,11 +20,11 @@ include "basic_2/rt_computation/cpms_cpxs.ma". (* Properties with sort-irrelevant equivalence for local environments *******) lemma cpms_rdeq_conf_sn (h) (n) (G) (L1) (L2): - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 → + ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 → L1 ≛[T1] L2 → L1 ≛[T2] L2. /3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_sn/ qed-. lemma cpms_rdeq_conf_dx (h) (n) (G) (L1) (L2): - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n,h] T2 → + ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 → L1 ≛[T1] L2 → L1 ≛[T2] L2. /3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_dx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_cpre.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_cpre.ma index 202e1ccad..5ddc528f4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_cpre.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_cpre.ma @@ -32,7 +32,7 @@ qed-. (* Basic_1: was: nf2_pr3_confluence *) theorem cpre_mono (h) (G) (L) (T): - ∀T1. ⦃G, L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2. + ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2. #h #G #L #T0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2 elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20 >(cprs_inv_cnr_sn … HT10 HT1) -T1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_csx.ma index 1b1e44804..3d1a34c8b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpre_csx.ma @@ -23,7 +23,7 @@ include "basic_2/rt_computation/cpre.ma". (* Basic_1: was just: nf2_sn3 *) lemma cpre_total_csx (h) (G) (L): - ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄. + ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄. #h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #_ #IHT1 elim (cnr_dec_tdeq h G L T1) [ /3 width=3 by ex_intro, conj/ ] * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma index ff8271ebd..7a9df49a5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma @@ -22,8 +22,8 @@ include "basic_2/rt_computation/cpms.ma". (* Basic_2A1: was: cprs_ind_dx *) lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …): Q T2 → - (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) → - ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T1. + (∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ⦃G,L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) → + ∀T1. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T1. #h #G #L #T2 #Q #IH1 #IH2 #T1 @(insert_eq_0 … 0) #n #H @(cpms_ind_sn … H) -n -T1 // @@ -35,8 +35,8 @@ qed-. (* Basic_2A1: was: cprs_ind *) lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …): Q T1 → - (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) → - ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T2. + (∀T,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T → ⦃G,L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) → + ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T2. #h #G #L #T1 #Q #IH1 #IH2 #T2 @(insert_eq_0 … 0) #n #H @(cpms_ind_dx … H) -n -T2 // @@ -50,28 +50,28 @@ qed-. (* Basic_1: was: pr3_step *) (* Basic_2A1: was: cprs_strap2 *) lemma cprs_step_sn (h) (G) (L): - ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2. + ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2. /2 width=3 by cpms_step_sn/ qed-. (* Basic_2A1: was: cprs_strap1 *) lemma cprs_step_dx (h) (G) (L): - ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2. + ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2. /2 width=3 by cpms_step_dx/ qed-. (* Basic_1: was only: pr3_thin_dx *) lemma cprs_flat_dx (h) (I) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → - ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → + ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2. #h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1 /3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/ qed. lemma cprs_flat_sn (h) (I) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → - ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 → + ⦃G,L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2. #h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1 /3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/ qed. @@ -79,13 +79,13 @@ qed. (* Basic inversion lemmas ***************************************************) (* Basic_1: was: pr3_gen_sort *) -lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s. +lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s. /2 width=4 by cpms_inv_sort1/ qed-. (* Basic_1: was: pr3_gen_cast *) -lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 → - ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2 - | ⦃G, L⦄ ⊢ T1 ➡*[h] X2. +lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 → + ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2 + | ⦃G,L⦄ ⊢ T1 ➡*[h] X2. #h #G #L #W1 #T1 #X2 #H elim (cpms_inv_cast1 … H) -H [ /2 width=1 by or_introl/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cnr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cnr.ma index 2849fa0b7..8cbb2ee6f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cnr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cnr.ma @@ -22,7 +22,7 @@ include "basic_2/rt_computation/cprs.ma". (* Basic_1: was: nf2_pr3_unfold *) (* Basic_2A1: was: cprs_inv_cnr1 *) lemma cprs_inv_cnr_sn (h) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ → T1 = T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ → T1 = T2. #h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T1 // #T1 #T0 #HT10 #_ #IH #HT1 lapply (HT1 … HT10) -HT10 #H destruct /2 width=1 by/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cprs.ma index 990051c3a..069323fb4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_cprs.ma @@ -38,9 +38,9 @@ qed-. (* Basic_1: was: pr3_flat *) theorem cprs_flat (h) (G) (L): - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2. + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 → + ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2. #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2 [ /2 width=3 by cprs_flat_dx/ | /3 width=3 by cpr_pair_sn, cprs_step_dx/ @@ -52,15 +52,15 @@ qed. (* Basic_1: was pr3_gen_appl *) (* Basic_2A1: was: cprs_inv_appl1 *) lemma cprs_inv_appl_sn (h) (G) (L): - ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & - ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & + ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & + ⦃G,L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓐV2. T2 - | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T & - ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2 - | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 & - ⦃G, L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T & - ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2. + | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T & + ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2 + | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 & + ⦃G,L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T & + ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2. #h #G #L #V1 #T1 #X2 #H elim (cpms_inv_appl_sn … H) -H * [ /3 width=5 by or3_intro0, ex3_2_intro/ | #n1 #n2 #p #V2 #T2 #HT12 #HTX2 #H diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_drops.ma index 19829738f..bd1afeec3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_drops.ma @@ -20,9 +20,9 @@ include "basic_2/rt_computation/cpms_drops.ma". (* Basic_1: was: pr3_gen_lref *) (* Basic_2A1: was: cprs_inv_lref1 *) -lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h] T2 → +lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G,L⦄ ⊢ #i ➡*[h] T2 → ∨∨ T2 = #i - | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡*[h] T1 & + | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G,K⦄ ⊢ V1 ➡*[h] T1 & ⬆*[↑i] T1 ≘ T2. #h #G #L #T2 #i #H elim (cpms_inv_lref1_drops … H) -H * [ /2 width=1 by or_introl/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_lpr.ma index 1ea7cc963..8a578cf5b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs_lpr.ma @@ -31,8 +31,8 @@ lemma lpr_cprs_trans (h) (G): s_rs_transitive … (λL. cpm h G L 0) (λ_. lpr h qed-. lemma cprs_lpr_conf_dx (h) (G): - ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. + ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. #h #G #L0 #T0 #T1 #H @(cprs_ind_dx … H) -T1 /2 width=3 by ex2_intro/ #T #T1 #_ #HT1 #IHT0 #L1 #HL01 @@ -43,9 +43,9 @@ elim (cprs_strip … HT2 … HT3) -T qed-. lemma cprs_lpr_conf_sn (h) (G): - ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → - ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. + ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → + ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → + ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. #h #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1 #HT0 /3 width=3 by lpr_cpms_trans, ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue.ma index 95c2b56ff..a22002aed 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue.ma @@ -19,7 +19,7 @@ include "basic_2/rt_computation/cpms.ma". (* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************) definition cpue (h) (G) (L): relation2 term term ≝ - λT1,T2. ∃∃n. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄. + λT1,T2. ∃∃n. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄. interpretation "evaluation for t-unbound context-sensitive parallel rt-transition (term)" 'PRedITEval h G L T1 T2 = (cpue h G L T1 T2). diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue_csx.ma index 9c1fb3707..2ba2ddb02 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpue_csx.ma @@ -22,7 +22,7 @@ include "basic_2/rt_computation/cpue.ma". (* Properties with strong normalization for unbound rt-transition for terms *) lemma cpue_total_csx (h) (G) (L): - ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G, L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄. + ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄. #h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #_ #IHT1 elim (cnu_dec_tdeq h G L T1) [ /3 width=4 by ex2_intro, ex_intro/ ] * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma index ce01889c1..e0048f7d7 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma @@ -27,71 +27,71 @@ interpretation "unbound context-sensitive parallel rt-computation (term)" (* Basic eliminators ********************************************************) lemma cpxs_ind: ∀h,G,L,T1. ∀Q:predicate term. Q T1 → - (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) → - ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T2. + (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) → + ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T2. #h #L #G #T1 #Q #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) // qed-. lemma cpxs_ind_dx: ∀h,G,L,T2. ∀Q:predicate term. Q T2 → - (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T → ⦃G, L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) → - ∀T1. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T1. + (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T → ⦃G,L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) → + ∀T1. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T1. #h #G #L #T2 #Q #HT2 #IHT2 #T1 #HT12 @(TC_star_ind_dx … HT2 IHT2 … HT12) // qed-. (* Basic properties *********************************************************) -lemma cpxs_refl: ∀h,G,L,T. ⦃G, L⦄ ⊢ T ⬈*[h] T. +lemma cpxs_refl: ∀h,G,L,T. ⦃G,L⦄ ⊢ T ⬈*[h] T. /2 width=1 by inj/ qed. -lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2. +lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2. /2 width=1 by inj/ qed. -lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2. +lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2. normalize /2 width=3 by step/ qed-. -lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2. +lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2. normalize /2 width=3 by TC_strap/ qed-. (* Basic_2A1: was just: cpxs_sort *) -lemma cpxs_sort: ∀h,G,L,s,n. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s). +lemma cpxs_sort: ∀h,G,L,s,n. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s). #h #G #L #s #n elim n -n /2 width=1 by cpx_cpxs/ #n >iter_S /2 width=3 by cpxs_strap1/ qed. -lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. +lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀I,T1,T2. ⦃G,L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/ qed. -lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. +lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. #h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/ qed. -lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. +lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/ qed. -lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T. +lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T. #h #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=3 by cpxs_strap1, cpx_pair_sn/ qed. lemma cpxs_zeta (h) (G) (L) (V): ∀T1,T. ⬆*[1] T ≘ T1 → - ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2. + ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2. #h #G #L #V #T1 #T #HT1 #T2 #H @(cpxs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_zeta/ qed. @@ -99,34 +99,34 @@ qed. (* Basic_2A1: was: cpxs_zeta *) lemma cpxs_zeta_dx (h) (G) (L) (V): ∀T2,T. ⬆*[1] T2 ≘ T → - ∀T1. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2. + ∀T1. ⦃G,L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2. #h #G #L #V #T2 #T #HT2 #T1 #H @(cpxs_ind_dx … H) -T1 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/ qed. -lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ⬈*[h] T2. +lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ⬈*[h] T2. #h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/ qed. (* Basic_2A1: was: cpxs_ct *) -lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ⬈*[h] V2. +lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ∀T. ⦃G,L⦄ ⊢ ⓝV1.T ⬈*[h] V2. #h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ee/ qed. lemma cpxs_beta_dx: ∀h,p,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ qed. lemma cpxs_theta_dx: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → - ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → + ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 /4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ qed. @@ -134,7 +134,7 @@ qed. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: wa just: cpxs_inv_sort1 *) -lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] X2 → +lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] X2 → ∃n. X2 = ⋆((next h)^n s). #h #G #L #X2 #s #H @(cpxs_ind … H) -X2 /2 width=2 by ex_intro/ #X #X2 #_ #HX2 * #n #H destruct @@ -142,10 +142,10 @@ elim (cpx_inv_sort1 … HX2) -HX2 #H destruct /2 width=2 by ex_intro/ @(ex_intro … (↑n)) >iter_S // qed-. -lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 → - ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2 - | ⦃G, L⦄ ⊢ T1 ⬈*[h] U2 - | ⦃G, L⦄ ⊢ W1 ⬈*[h] U2. +lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G,L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 → + ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2 + | ⦃G,L⦄ ⊢ T1 ⬈*[h] U2 + | ⦃G,L⦄ ⊢ W1 ⬈*[h] U2. #h #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/ #U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ * #W #T #HW1 #HT1 #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cnx.ma index 2135a0cee..6dd802001 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cnx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cnx.ma @@ -19,7 +19,7 @@ include "basic_2/rt_computation/cpxs.ma". (* Inversion lemmas with normal terms ***************************************) -lemma cpxs_inv_cnx1: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → +lemma cpxs_inv_cnx1: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → T1 ≛ T2. #h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1 /5 width=9 by cnx_tdeq_trans, tdeq_trans/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma index 4740fc092..60ddf6394 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_cpxs.ma @@ -22,58 +22,58 @@ include "basic_2/rt_computation/cpxs.ma". theorem cpxs_trans: ∀h,G,L. Transitive … (cpxs h G L). normalize /2 width=3 by trans_TC/ qed-. -theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 → - ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. +theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 → + ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. #h #p #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /3 width=5 by cpxs_trans, cpxs_bind_dx/ qed. -theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. +theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2. #h #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /3 width=5 by cpxs_trans, cpxs_flat_dx/ qed. theorem cpxs_beta_rc: ∀h,p,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2 /4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/ qed. theorem cpxs_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. + ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2. #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2 /4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/ qed. theorem cpxs_theta_rc: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → - ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → + ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2 /3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/ qed. theorem cpxs_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2. - ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → - ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V → - ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. + ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → + ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V → + ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2. #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/ qed. (* Advanced inversion lemmas ************************************************) -lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 & +lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓐV2.T2 - | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2 - | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 & - ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2. + | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2 + | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 & + ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2. #h #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ] #U #U2 #_ #HU2 * * [ #V0 #T0 #HV10 #HT10 #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_drops.ma index c42f94f83..09ba0f251 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_drops.ma @@ -20,8 +20,8 @@ include "basic_2/rt_computation/cpxs.ma". (* Advanced properties ******************************************************) -lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 → - ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2. +lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 → + ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2. #h #I #G #K #V1 #V2 #H @(cpxs_ind … H) -V2 [ /3 width=3 by cpx_cpxs, cpx_delta/ | #V #V2 #_ #HV2 #IH #W2 #HVW2 @@ -30,8 +30,8 @@ lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 → ] qed. -lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G, K⦄ ⊢ #i ⬈*[h] T → - ∀U. ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U. +lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G,K⦄ ⊢ #i ⬈*[h] T → + ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U. #h #I #G #K #T #i #H @(cpxs_ind … H) -T [ /3 width=3 by cpx_cpxs, cpx_lref/ | #T0 #T #_ #HT2 #IH #U #HTU @@ -42,8 +42,8 @@ qed. (* Basic_2A1: was: cpxs_delta *) lemma cpxs_delta_drops: ∀h,I,G,L,K,V1,V2,i. - ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 → - ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h] W2. + ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 → + ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈*[h] W2. #h #I #G #L #K #V1 #V2 #i #HLK #H @(cpxs_ind … H) -V2 [ /3 width=7 by cpx_cpxs, cpx_delta_drops/ | #V #V2 #_ #HV2 #IH #W2 #HVW2 @@ -54,9 +54,9 @@ qed. (* Advanced inversion lemmas ************************************************) -lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈*[h] T2 → +lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈*[h] T2 → T2 = #0 ∨ - ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 & + ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓑ{I}V1. #h #G #L #T2 #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/ #T #T2 #_ #HT2 * @@ -69,9 +69,9 @@ lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈*[h] T2 → ] qed-. -lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈*[h] T2 → +lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈*[h] T2 → T2 = #(↑i) ∨ - ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. + ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. #h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/ #T #T2 #_ #HT2 * [ #H destruct @@ -84,9 +84,9 @@ lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈*[h] T2 → qed-. (* Basic_2A1: was: cpxs_inv_lref1 *) -lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h] T2 → +lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈*[h] T2 → T2 = #i ∨ - ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h] T1 & + ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G,K⦄ ⊢ V1 ⬈*[h] T1 & ⬆*[↑i] T1 ≘ T2. #h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/ #T #T2 #_ #HT2 * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fdeq.ma index ed477d71c..1b12ae605 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fdeq.ma @@ -19,9 +19,9 @@ include "basic_2/rt_computation/cpxs_rdeq.ma". (* Properties with sort-irrelevant equivalence for closures *****************) -lemma fdeq_cpxs_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T⦄ → - ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 → - ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛ ⦃G2, L2, T2⦄. +lemma fdeq_cpxs_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ → + ∀T2. ⦃G2,L2⦄ ⊢ T ⬈*[h] T2 → + ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2 elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct elim (rdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fqus.ma index bb0ce14ab..5fc7879fe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_fqus.ma @@ -21,33 +21,33 @@ include "basic_2/rt_computation/cpxs_cpxs.ma". (* Properties on supclosure *************************************************) -lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → - ∀T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → + ∀T1. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/ #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/ qed-. -lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → - ∀T1. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → + ∀T1. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/ #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fquq_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/ qed-. -lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → - ∀T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄. +lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → + ∀T1. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/ #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/ qed-. -lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → - ∀T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄. +lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → + ∀T1. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/ #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/ @@ -55,9 +55,9 @@ qed-. (* Note: a proof based on fqu_cpx_trans_tdneq might exist *) (* Basic_2A1: uses: fqu_cpxs_trans_neq *) -lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵) #U2 #HVU2 @(ex3_intro … U2) @@ -88,9 +88,9 @@ lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] qed-. (* Basic_2A1: uses: fquq_cpxs_trans_neq *) -lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12 [ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqu_fquq, ex3_intro/ @@ -99,9 +99,9 @@ lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b qed-. (* Basic_2A1: uses: fqup_cpxs_trans_neq *) -lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄. +lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqu_fqup, ex3_intro/ @@ -112,9 +112,9 @@ lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] qed-. (* Basic_2A1: uses: fqus_cpxs_trans_neq *) -lemma fqus_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄. +lemma fqus_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12 [ #H12 elim (fqup_cpxs_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqup_fqus, ex3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lpx.ma index f5cefda3a..dbefb4db5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_lpx.ma @@ -44,22 +44,22 @@ qed-. (* Advanced properties ******************************************************) -lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. +lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. /4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed. -lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. +lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. /4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed. (* Properties with plus-iterated structural successor for closures **********) (* Basic_2A1: uses: lpx_fqup_trans *) -lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2. +lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 → + ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⊐+[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 [ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/ @@ -73,9 +73,9 @@ qed-. (* Properties with star-iterated structural successor for closures **********) (* Basic_2A1: uses: lpx_fqus_trans *) -lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2. +lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 → + ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⊐*[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H [ #H12 elim (lpx_fqup_trans … H12 … HKL1) -L1 /3 width=5 by fqup_fqus, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_rdeq.ma index 58c6d29ff..1d3dfa50d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_rdeq.ma @@ -20,9 +20,9 @@ include "basic_2/rt_computation/cpxs_tdeq.ma". (* Properties with sort-irrelevant equivalence for local environments *******) (* Basic_2A1: was just: lleq_cpxs_trans *) -lemma rdeq_cpxs_trans: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 → +lemma rdeq_cpxs_trans: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 → ∀L2. L2 ≛[T0] L0 → - ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1. + ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1. #h #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/ #T0 #T #HT0 #_ #IH #L2 #HL2 elim (rdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2 @@ -32,18 +32,18 @@ elim (tdeq_cpxs_trans … H2 … H3) -T #U0 #H2 #H3 qed-. (* Basic_2A1: was just: cpxs_lleq_conf *) -lemma cpxs_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 → +lemma cpxs_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 → ∀L2. L0 ≛[T0] L2 → - ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1. + ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1. /3 width=3 by rdeq_cpxs_trans, rdeq_sym/ qed-. (* Basic_2A1: was just: cpxs_lleq_conf_dx *) -lemma cpxs_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 → +lemma cpxs_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈*[h] T2 → ∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2. #h #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/ qed-. (* Basic_2A1: was just: lleq_conf_sn *) -lemma cpxs_rdeq_conf_sn: ∀h,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 → +lemma cpxs_rdeq_conf_sn: ∀h,G,L1,T1,T2. ⦃G,L1⦄ ⊢ T1 ⬈*[h] T2 → ∀L2. L1 ≛[T1] L2 → L1 ≛[T2] L2. /4 width=6 by cpxs_rdeq_conf_dx, rdeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.ma index c313dc7ab..b1d2c9464 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_tdeq.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/cpxs.ma". (* Properties with sort-irrelevant equivalence for terms ********************) -lemma tdeq_cpxs_trans: ∀h,U1,T1. U1 ≛ T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛ T2. +lemma tdeq_cpxs_trans: ∀h,U1,T1. U1 ≛ T1 → ∀G,L,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + ∃∃U2. ⦃G,L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛ T2. #h #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/ #T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1 /3 width=3 by ex2_intro, cpxs_strap1/ @@ -28,8 +28,8 @@ qed-. (* Note: this requires tdeq to be symmetric *) (* Nasic_2A1: uses: cpxs_neq_inv_step_sn *) -lemma cpxs_tdneq_fwd_step_sn: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → - ∃∃T,T0. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 ≛ T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T0 & T0 ≛ T2. +lemma cpxs_tdneq_fwd_step_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → + ∃∃T,T0. ⦃G,L⦄ ⊢ T1 ⬈[h] T & T1 ≛ T → ⊥ & ⦃G,L⦄ ⊢ T ⬈*[h] T0 & T0 ≛ T2. #h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1 [ #H elim H -H // | #T1 #T0 #HT10 #HT02 #IH #Hn12 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma index 58ee19613..5174890e0 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq.ma @@ -21,7 +21,7 @@ include "basic_2/rt_computation/lpxs_cpxs.ma". (* Forward lemmas with head equivalence for terms ***************************) -lemma cpxs_fwd_sort: ∀h,G,L,X2,s1. ⦃G, L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2. +lemma cpxs_fwd_sort: ∀h,G,L,X2,s1. ⦃G,L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2. #h #G #L #X2 #s1 #H elim (cpxs_inv_sort1 … H) -H #s2 #H destruct // qed-. @@ -30,8 +30,8 @@ qed-. (* Basic_2A1: was: cpxs_fwd_delta *) lemma cpxs_fwd_delta_drops: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 → ∀V2. ⬆*[↑i] V1 ≘ V2 → - ∀X2. ⦃G, L⦄ ⊢ #i ⬈*[h] X2 → - ∨∨ #i ⩳ X2 | ⦃G, L⦄ ⊢ V2 ⬈*[h] X2. + ∀X2. ⦃G,L⦄ ⊢ #i ⬈*[h] X2 → + ∨∨ #i ⩳ X2 | ⦃G,L⦄ ⊢ V2 ⬈*[h] X2. #h #I #G #L #K #V1 #i #HLK #V2 #HV12 #X2 #H elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/ * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0 @@ -40,8 +40,8 @@ lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct qed-. (* Basic_1: was just: pr3_iso_beta *) -lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 → - ∨∨ ⓐV.ⓛ{p}W.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] X2. +lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 → + ∨∨ ⓐV.ⓛ{p}W.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] X2. #h #p #G #L #V #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H * [ #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/ | #b #W0 #T0 #HT0 #HU @@ -53,9 +53,9 @@ lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,X2. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] ] qed-. -lemma cpxs_fwd_theta: ∀h,p,G,L,V1,V,T,X2. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 → +lemma cpxs_fwd_theta: ∀h,p,G,L,V1,V,T,X2. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 → ∀V2. ⬆*[1] V1 ≘ V2 → - ∨∨ ⓐV1.ⓓ{p}V.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] X2. + ∨∨ ⓐV1.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] X2. #h #p #G #L #V1 #V #T #X2 #H #V2 #HV12 elim (cpxs_inv_appl1 … H) -H * [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/ @@ -84,13 +84,13 @@ elim (cpxs_inv_appl1 … H) -H * ] qed-. -lemma cpxs_fwd_cast: ∀h,G,L,W,T,X2. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] X2 → - ∨∨ ⓝW. T ⩳ X2 | ⦃G, L⦄ ⊢ T ⬈*[h] X2 | ⦃G, L⦄ ⊢ W ⬈*[h] X2. +lemma cpxs_fwd_cast: ∀h,G,L,W,T,X2. ⦃G,L⦄ ⊢ ⓝW.T ⬈*[h] X2 → + ∨∨ ⓝW. T ⩳ X2 | ⦃G,L⦄ ⊢ T ⬈*[h] X2 | ⦃G,L⦄ ⊢ W ⬈*[h] X2. #h #G #L #W #T #X2 #H elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ * #W0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/ qed-. -lemma cpxs_fwd_cnx: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → - ∀X2. ⦃G, L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2. +lemma cpxs_fwd_cnx: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → + ∀X2. ⦃G,L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2. /3 width=5 by cpxs_inv_cnx1, tdeq_theq/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq_vector.ma index ec7f35fee..40c508756 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_theq_vector.ma @@ -20,7 +20,7 @@ include "basic_2/rt_computation/cpxs_theq.ma". (* Vector form of forward lemmas with head equivalence for terms ************) -lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2. +lemma cpxs_fwd_sort_vector: ∀h,G,L,s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2. #h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/ #V #Vs #IHVs #X2 #H elim (cpxs_inv_appl1 … H) -H * @@ -37,8 +37,8 @@ qed-. (* Basic_2A1: was: cpxs_fwd_delta_vector *) lemma cpxs_fwd_delta_drops_vector: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 → ∀V2. ⬆*[↑i] V1 ≘ V2 → - ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 → - ∨∨ ⒶVs.#i ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2. + ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 → + ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2. #h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/ #V #Vs #IHVs #X2 #H -K -V1 @@ -62,8 +62,8 @@ elim (cpxs_inv_appl1 … H) -H * qed-. (* Basic_1: was just: pr3_iso_appls_beta *) -lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 → - ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2. +lemma cpxs_fwd_beta_vector: ∀h,p,G,L,Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 → + ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2. #h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/ #V0 #Vs #IHVs #V #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H * @@ -87,8 +87,8 @@ qed-. (* Basic_1: was just: pr3_iso_appls_abbr *) lemma cpxs_fwd_theta_vector: ∀h,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b → - ∀p,V,T,X2. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 → - ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2. + ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 → + ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2. #h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/ #V1b #V2b #V1a #V2a #HV12a #HV12b #p generalize in match HV12a; -HV12a @@ -135,10 +135,10 @@ elim (cpxs_inv_appl1 … H) -H * qed-. (* Basic_1: was just: pr3_iso_appls_cast *) -lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 → +lemma cpxs_fwd_cast_vector: ∀h,G,L,Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 → ∨∨ ⒶVs. ⓝW. T ⩳ X2 - | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2 - | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] X2. + | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 + | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2. #h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/ #V #Vs #IHVs #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H * @@ -166,8 +166,8 @@ elim (cpxs_inv_appl1 … H) -H * qed-. (* Basic_1: was just: nf2_iso_appls_lref *) -lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → - ∀Vs,X2. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2. +lemma cpxs_fwd_cnx_vector: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → + ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2. #h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *) #V #Vs #IHVs #X2 #H elim (cpxs_inv_appl1 … H) -H * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx.ma index 384d014cb..7fe343b8a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx.ma @@ -28,11 +28,11 @@ interpretation (* Basic eliminators ********************************************************) lemma csx_ind: ∀h,G,L. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → + (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T. + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T. #h #G #L #Q #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/ qed-. @@ -41,14 +41,14 @@ qed-. (* Basic_1: was just: sn3_pr2_intro *) lemma csx_intro: ∀h,G,L,T1. - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄. + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄. /4 width=1 by SN_intro/ qed. (* Basic forward lemmas *****************************************************) -fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → - ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. +fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → + ∀I,V,T. U = ②{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. #h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct @csx_intro #V2 #HLV2 #HV2 @(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2 @@ -56,23 +56,23 @@ fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → qed-. (* Basic_1: was just: sn3_gen_head *) -lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. +lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. /2 width=5 by csx_fwd_pair_sn_aux/ qed-. -fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → - ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → + ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct @csx_intro #T2 #HLT2 #HT2 -@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2 +@(IH (ⓑ{p, I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2 #H elim (tdeq_inv_pair … H) -H /2 width=1 by/ qed-. (* Basic_1: was just: sn3_gen_bind *) -lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /2 width=4 by csx_fwd_bind_dx_aux/ qed-. -fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → - ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → + ∀I,V,T. U = ⓕ{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct @csx_intro #T2 #HLT2 #HT2 @(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2 @@ -80,15 +80,15 @@ fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → qed-. (* Basic_1: was just: sn3_gen_flat *) -lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /2 width=5 by csx_fwd_flat_dx_aux/ qed-. -lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-. -lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-. (* Basic_1: removed theorems 14: diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_aaa.ma index c44bf1d53..dfbcf7e48 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_aaa.ma @@ -21,7 +21,7 @@ include "basic_2/rt_computation/csx_gcr.ma". (* Main properties with atomic arity assignment *****************************) -theorem aaa_csx: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +theorem aaa_csx: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #T #A #H @(gcr_aaa … (csx_gcp h) (csx_gcr h) … H) qed. @@ -29,32 +29,32 @@ qed. (* Advanced eliminators *****************************************************) fact aaa_ind_csx_aux: ∀h,G,L,A. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T. + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T. #h #G #L #A #Q #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/ qed-. lemma aaa_ind_csx: ∀h,G,L,A. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T. + ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T. /5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-. fact aaa_ind_csx_cpxs_aux: ∀h,G,L,A. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T. + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T. #h #G #L #A #Q #IH #T #H @(csx_ind_cpxs … H) -T /4 width=5 by cpxs_aaa_conf/ qed-. (* Basic_2A1: was: aaa_ind_csx_alt *) lemma aaa_ind_csx_cpxs: ∀h,G,L,A. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T. + ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T. /5 width=9 by aaa_ind_csx_cpxs_aux, aaa_csx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx.ma index eaaaa9aa2..3cd1edff9 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx.ma @@ -20,10 +20,10 @@ include "basic_2/rt_computation/csx.ma". (* Properties with normal terms for unbound parallel rt-transition **********) (* Basic_1: was just: sn3_nf2 *) -lemma cnx_csx: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma cnx_csx: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /2 width=1 by NF_to_SN/ qed. (* Advanced properties ******************************************************) -lemma csx_sort: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃⋆s⦄. +lemma csx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃⋆s⦄. /3 width=4 by cnx_csx, cnx_sort/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx_vector.ma index d0072aa83..91a7f2925 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cnx_vector.ma @@ -23,8 +23,8 @@ include "basic_2/rt_computation/csx_vector.ma". (* Properties with normal terms for unbound parallel rt-transition **********) (* Basic_1: was just: sn3_appls_lref *) -lemma csx_applv_cnx: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → - ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄. +lemma csx_applv_cnx: ∀h,G,L,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → + ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄. #h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ normalize in ⊢ (????%); /2 width=1 by cnx_csx/ | #V #Vs #IHV #H @@ -38,5 +38,5 @@ qed. (* Advanced properties ******************************************************) -lemma csx_applv_sort: ∀h,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.⋆s⦄. +lemma csx_applv_sort: ∀h,G,L,s,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.⋆s⦄. /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma index 586c2847f..aa206225c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma @@ -22,13 +22,13 @@ include "basic_2/rt_computation/csx_csx.ma". (* Basic_1: was just: sn3_intro *) lemma csx_intro_cpxs: ∀h,G,L,T1. - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄. + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄. /4 width=1 by cpx_cpxs, csx_intro/ qed-. (* Basic_1: was just: sn3_pr3_trans *) -lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/ qed-. @@ -36,11 +36,11 @@ qed-. (* Eliminators with unbound context-sensitive rt-computation for terms ******) lemma csx_ind_cpxs_tdeq: ∀h,G,L. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2. + ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀T0. ⦃G,L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2. #h #G #L #Q #IH #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02 @IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2 @@ -60,10 +60,10 @@ qed-. (* Basic_2A1: was: csx_ind_alt *) lemma csx_ind_cpxs: ∀h,G,L. ∀Q:predicate term. - (∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 + (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1 ) → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T. + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T. #h #G #L #Q #IH #T #HT @(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *) qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma index 3702ba245..456f38417 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma @@ -19,22 +19,22 @@ include "basic_2/rt_computation/csx_drops.ma". (* Advanced properties ******************************************************) -lemma csx_tdeq_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀T2. T1 ≛ T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_tdeq_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2 @csx_intro #T1 #HT21 #HnT21 elim (tdeq_cpx_trans … HT2 … HT21) -HT21 /4 width=5 by tdeq_repl/ qed-. -lemma csx_cpx_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_cpx_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12 elim (tdeq_dec T1 T2) /3 width=4 by csx_tdeq_trans/ qed-. (* Basic_1: was just: sn3_cast *) -lemma csx_cast: ∀h,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄. +lemma csx_cast: ∀h,G,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ → + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄. #h #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT @csx_intro @@ -52,7 +52,7 @@ qed. (* Basic_1: was just: sn3_abbr *) (* Basic_2A1: was: csx_lref_bind *) lemma csx_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → - ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄. + ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄. #h #I #G #L #K #V #i #HLK #HV @csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H [ #H destruct elim Hi // @@ -67,16 +67,16 @@ qed. (* Basic_1: was: sn3_gen_def *) (* Basic_2A1: was: csx_inv_lref_bind *) lemma csx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. #h #I #G #L #K #V #i #HLK #Hi elim (lifts_total V (𝐔❴↑i❵)) /4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/ qed-. -lemma csx_inv_lref: ∀h,G,L,i. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → - ∨∨ ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ +lemma csx_inv_lref: ∀h,G,L,i. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → + ∨∨ ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ | ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I} - | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. + | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄. #h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/ * * /4 width=9 by csx_inv_lref_pair, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma index 4899dc66d..889eddcfe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma @@ -23,8 +23,8 @@ include "basic_2/rt_computation/csx_vector.ma". (* Advanced properties ************************************* ****************) (* Basic_1: was just: sn3_appls_beta *) -lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄. +lemma csx_applv_beta: ∀h,p,G,L,Vs,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄. #h #p #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/ #V0 #Vs #IHV #V #W #T #H1T lapply (csx_fwd_pair_sn … H1T) #HV0 @@ -39,7 +39,7 @@ qed. lemma csx_applv_delta: ∀h,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 → ∀V2. ⬆*[↑i] V1 ≘ V2 → - ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄. + ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄. #h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs [ /4 width=11 by csx_inv_lifts, csx_lref_pair, drops_isuni_fwd_drop2/ | #V #Vs #IHV #H1T @@ -56,8 +56,8 @@ qed. (* Basic_1: was just: sn3_appls_abbr *) lemma csx_applv_theta: ∀h,p,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b → - ∀V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄. #h #p #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/ #V1b #V2b #V1 #V2 #HV12 #H generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1 @@ -74,8 +74,8 @@ elim (cpxs_fwd_theta_vector … (V2⨮V2b) … H1) -H1 /2 width=1 by liftsv_cons qed. (* Basic_1: was just: sn3_appls_cast *) -lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ → - ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄. +lemma csx_applv_cast: ∀h,G,L,Vs,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ → + ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄. #h #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/ #V #Vs #IHV #U #H1U #T #H1T lapply (csx_fwd_pair_sn … H1U) #HV diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fdeq.ma index 2ffd57655..fcb2f9af8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fdeq.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/csx_rdeq.ma". (* Properties with sort-irrelevant equivalence for closures *****************) -lemma csx_fdeq_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fdeq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2 /3 width=3 by csx_rdeq_conf, csx_tdeq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fpbq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fpbq.ma index a08e70926..6027a70f5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fpbq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fpbq.ma @@ -22,8 +22,8 @@ include "basic_2/rt_computation/csx_lpx.ma". (* Properties with parallel rst-transition for closures *********************) (* Basic_2A1: was: csx_fpb_conf *) -lemma csx_fpbq_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fpbq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * /2 width=6 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_fdeq_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fqus.ma index ee76d6762..03ae7cc37 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_fqus.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/csx_lsubr.ma". (* Properties with extended supclosure **************************************) -lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ /3 width=5 by csx_inv_lref_pair, drops_refl/ | /2 width=3 by csx_fwd_pair_sn/ @@ -31,20 +31,20 @@ lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, ] qed-. -lemma csx_fquq_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fquq_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=6 by csx_fqu_conf/ * #HG #HL #HT destruct // qed-. -lemma csx_fqup_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fqup_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 /3 width=6 by csx_fqu_conf/ qed-. -lemma csx_fqus_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma csx_fqus_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -H /3 width=6 by csx_fquq_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpx.ma index 77cf480fa..ed06dcdea 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpx.ma @@ -19,16 +19,16 @@ include "basic_2/rt_computation/csx_cpxs.ma". (* Properties with unbound parallel rt-transition on all entries ************) -lemma csx_lpx_conf: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → - ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_lpx_conf: ∀h,G,L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → + ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L1 #T #H @(csx_ind_cpxs … H) -T /4 width=3 by csx_intro, lpx_cpx_trans/ qed-. (* Advanced properties ******************************************************) -lemma csx_abst: ∀h,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ → - ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄. +lemma csx_abst: ∀h,p,G,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ → + ∀T. ⦃G,L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄. #h #p #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT @@ -42,8 +42,8 @@ elim (tdneq_inv_pair … H2) -H2 ] qed. -lemma csx_abbr: ∀h,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → - ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄. +lemma csx_abbr: ∀h,p,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → + ∀T. ⦃G,L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄. #h #p #G #L #V #HV @(csx_ind … HV) -V #V #_ #IHV #T #HT @(csx_ind_cpxs … HT) -T #T #HT #IHT @@ -60,8 +60,8 @@ elim (cpx_inv_abbr1 … H1) -H1 * ] qed. -fact csx_appl_theta_aux: ∀h,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 → - ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄. +fact csx_appl_theta_aux: ∀h,p,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 → + ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄. #h #p #G #L #X #H @(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct lapply (csx_fwd_pair_sn … HVT) #HV @@ -94,6 +94,6 @@ elim (cpx_inv_appl1 … HL) -HL * ] qed-. -lemma csx_appl_theta: ∀h,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ → - ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄. +lemma csx_appl_theta: ∀h,p,G,L,V,V2,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ → + ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄. /2 width=5 by csx_appl_theta_aux/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpxs.ma index 14f7a9d3a..84086bcb6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lpxs.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/lpxs_lpx.ma". (* Properties with unbound parallel rt-computation on all entries ***********) -lemma csx_lpxs_conf: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_lpxs_conf: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2 /3 by lpxs_step_dx, csx_lpx_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lsubr.ma index b96942572..07226df19 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lsubr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_lsubr.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/csx_csx.ma". (* Advanced properties ******************************************************) -fact csx_appl_beta_aux: ∀h,p,G,L,U1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U1⦄ → - ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄. +fact csx_appl_beta_aux: ∀h,p,G,L,U1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U1⦄ → + ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄. #h #p #G #L #X #H @(csx_ind … H) -X #X #HT1 #IHT1 #V #W #T1 #H1 destruct @csx_intro #X #H1 #H2 @@ -42,23 +42,23 @@ elim (cpx_inv_appl1 … H1) -H1 * qed-. (* Basic_1: was just: sn3_beta *) -lemma csx_appl_beta: ∀h,p,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T⦄. +lemma csx_appl_beta: ∀h,p,G,L,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T⦄. /2 width=3 by csx_appl_beta_aux/ qed. (* Advanced forward lemmas **************************************************) -fact csx_fwd_bind_dx_unit_aux: ∀h,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → - ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +fact csx_fwd_bind_dx_unit_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → + ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #U #H elim H -H #U0 #_ #IH #p #I #J #V #T #H destruct @csx_intro #T2 #HLT2 #HT2 -@(IH (ⓑ{p,I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2 +@(IH (ⓑ{p, I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2 #H elim (tdeq_inv_pair … H) -H /2 width=1 by/ qed-. -lemma csx_fwd_bind_dx_unit: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → - ∀J. ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_bind_dx_unit: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → + ∀J. ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /2 width=6 by csx_fwd_bind_dx_unit_aux/ qed-. -lemma csx_fwd_bind_unit: ∀h,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → - ∀J. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_bind_unit: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → + ∀J. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /3 width=4 by csx_fwd_pair_sn, csx_fwd_bind_dx_unit, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_rdeq.ma index a254d9751..dcd0462e1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_rdeq.ma @@ -20,8 +20,8 @@ include "basic_2/rt_computation/csx_csx.ma". (* Properties with sort-irrelevant equivalence for local environments *******) (* Basic_2A1: uses: csx_lleq_conf *) -lemma csx_rdeq_conf: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → - ∀L2. L1 ≛[T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_rdeq_conf: ∀h,G,L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → + ∀L2. L1 ≛[T] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L1 #T #H @(csx_ind … H) -T #T1 #_ #IH #L2 #HL12 @csx_intro #T2 #HT12 #HnT12 @@ -31,5 +31,5 @@ qed-. (* Basic_2A1: uses: csx_lleq_conf *) lemma csx_rdeq_trans: ∀h,L1,L2,T. L1 ≛[T] L2 → - ∀G. ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. + ∀G. ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. /3 width=3 by csx_rdeq_conf, rdeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple.ma index 95cb186f5..50a460732 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple.ma @@ -19,9 +19,9 @@ include "basic_2/rt_computation/csx_csx.ma". (* Properties with simple terms *********************************************) -lemma csx_appl_simple: ∀h,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) → - 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄. +lemma csx_appl_simple: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) → + 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄. #h #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1 @csx_intro #X #H1 #H2 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple_theq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple_theq.ma index 478f6fa05..bb3b10b01 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple_theq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_simple_theq.ma @@ -24,9 +24,9 @@ include "basic_2/rt_computation/csx_csx.ma". (* Basic_1: was just: sn3_appl_appl *) (* Basic_2A1: was: csx_appl_simple_tsts *) -lemma csx_appl_simple_theq: ∀h,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) → - 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄. +lemma csx_appl_simple_theq: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) → + 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄. #h #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_vector.ma index 0e11278b9..f4b85eb9d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_vector.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_vector.ma @@ -26,14 +26,14 @@ interpretation (* Basic inversion lemmas ***************************************************) -lemma csxv_inv_cons: ∀h,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⨮Ts⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Ts⦄. +lemma csxv_inv_cons: ∀h,G,L,T,Ts. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⨮Ts⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Ts⦄. normalize // qed-. (* Basic forward lemmas *****************************************************) -lemma csx_fwd_applv: ∀h,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → - ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma csx_fwd_applv: ∀h,G,L,T,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → + ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/ #V #Vs #IHVs #HVs lapply (csx_fwd_pair_sn … HVs) #HV diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg.ma index ba56fd0c5..2c2f375f2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg.ma @@ -20,45 +20,45 @@ include "basic_2/rt_computation/fpbs.ma". definition fpbg: ∀h. tri_relation genv lenv term ≝ λh,G1,L1,T1,G2,L2,T2. - ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄. + ∃∃G,L,T. ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ & ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄. interpretation "proper parallel rst-computation (closure)" 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2). (* Basic properties *********************************************************) -lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /2 width=5 by ex2_3_intro/ qed. lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 * /3 width=9 by fpbs_strap1, ex2_3_intro/ qed-. lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐ ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 /4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/ qed-. (* Note: this is used in the closure proof *) -lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/ qed-. (* Basic_2A1: uses: fpbg_fleq_trans *) -lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h] ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-. (* Properties with t-bound rt-transition for terms **************************) lemma cpm_tdneq_cpm_fpbg (h) (G) (L): - ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) → - ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2,h] T2 → ⦃G, L, T1⦄ >[h] ⦃G, L, T2⦄. + ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) → + ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄. /4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_cpxs.ma index d5d24893f..6fef34261 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_cpxs.ma @@ -21,13 +21,13 @@ include "basic_2/rt_computation/fpbg_fpbs.ma". (* Properties with unbound context-sensitive parallel rt-computation ********) (* Basic_2A1: was: cpxs_fpbg *) -lemma cpxs_tdneq_fpbg (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - (T1 ≛ T2 → ⊥) → ⦃G, L, T1⦄ >[h] ⦃G, L, T2⦄. +lemma cpxs_tdneq_fpbg (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + (T1 ≛ T2 → ⊥) → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄. #h #G #L #T1 #T2 #H #H0 elim (cpxs_tdneq_fwd_step_sn … H … H0) -H -H0 /4 width=5 by cpxs_tdeq_fpbs, fpb_cpx, ex2_3_intro/ qed. -lemma cpxs_fpbg_trans (h): ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → - ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma cpxs_fpbg_trans (h): ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → + ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fpbg_trans, cpxs_fpbs/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fpbs.ma index 57f4a81af..1eb2d110b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fpbs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fpbs.ma @@ -22,7 +22,7 @@ include "basic_2/rt_computation/fpbg.ma". (* Advanced forward lemmas **************************************************) lemma fpbg_fwd_fpbs: ∀h,G1,G2,L1,L2,T1,T2. - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 * /3 width=5 by fpbs_strap2, fpb_fpbq/ qed-. @@ -30,8 +30,8 @@ qed-. (* Advanced properties with sort-irrelevant equivalence on closures *********) (* Basic_2A1: uses: fleq_fpbg_trans *) -lemma fdeq_fpbg_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fdeq_fpbg_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1 elim (fdeq_fpb_trans … H1 … H0) -G -L -T /4 width=9 by fpbs_strap2, fpbq_fdeq, ex2_3_intro/ @@ -40,15 +40,15 @@ qed-. (* Properties with parallel proper rst-reduction on closures ****************) lemma fpb_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ≻[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-. (* Properties with parallel rst-reduction on closures ***********************) lemma fpbq_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpbq_inv_fpb … H1) -H1 /2 width=5 by fdeq_fpbg_trans, fpb_fpbg_trans/ @@ -56,8 +56,8 @@ qed-. (* Properties with parallel rst-compuutation on closures ********************) -lemma fpbs_fpbg_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ >[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpbs_fpbg_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/ qed-. @@ -71,9 +71,9 @@ lemma fqup_fpbg_trans (h): (* Advanced inversion lemmas of parallel rst-computation on closures ********) (* Basic_2A1: was: fpbs_fpbg *) -lemma fpbs_inv_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∨∨ ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ - | ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpbs_inv_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ + | ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 [ /2 width=1 by or_introl/ | #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 @@ -89,9 +89,9 @@ qed-. (* Advanced properties of parallel rst-computation on closures **************) -lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h] ⦃F2, K2, T2⦄ → - ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h] ⦃G2, L2, U2⦄ → - ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h] ⦃G2, L2, U2⦄. +lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≥[h] ⦃F2,K2,T2⦄ → + ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ → + ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄. #h #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H [ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2 /3 width=5 by fdeq_fpbs, ex2_3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fqup.ma index c72dcc821..9e40865af 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fqup.ma @@ -19,14 +19,14 @@ include "basic_2/rt_computation/fpbg.ma". (* Advanced properties with sort-irrelevant equivalence for terms ***********) -lemma fpbg_tdeq_div: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T⦄ → - ∀T2. T2 ≛ T → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fpbg_tdeq_div: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ → + ∀T2. T2 ≛ T → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. /4 width=5 by fpbg_fdeq_trans, tdeq_fdeq, tdeq_sym/ qed-. (* Properties with plus-iterated structural successor for closures **********) (* Note: this is used in the closure proof *) -lemma fqup_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄. +lemma fqup_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H /3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_lpxs.ma index 4abcd30de..0921a5e12 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_lpxs.ma @@ -20,8 +20,8 @@ include "basic_2/rt_computation/fpbg.ma". (* Properties with unbound rt-computation on full local environments ********) (* Basic_2A1: uses: lpxs_fpbg *) -lemma lpxs_rdneq_fpbg: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - (L1 ≛[T] L2 → ⊥) → ⦃G, L1, T⦄ >[h] ⦃G, L2, T⦄. +lemma lpxs_rdneq_fpbg: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + (L1 ≛[T] L2 → ⊥) → ⦃G,L1,T⦄ >[h] ⦃G,L2,T⦄. #h #G #L1 #L2 #T #H #H0 elim (lpxs_rdneq_inv_step_sn … H … H0) -H -H0 /4 width=7 by fpb_lpx, lpxs_fdeq_fpbs, fdeq_intro_sn, ex2_3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs.ma index ab2994823..9d6f5c280 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs.ma @@ -27,13 +27,13 @@ interpretation "parallel rst-computation (closure)" (* Basic eliminators ********************************************************) lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2. + (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2. /3 width=8 by tri_TC_star_ind/ qed-. lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G1 L1 T1. + (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G1 L1 T1. /3 width=8 by tri_TC_star_ind_dx/ qed-. (* Basic properties *********************************************************) @@ -41,34 +41,34 @@ lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 → lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h). /2 width=1 by tri_inj/ qed. -lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /2 width=1 by tri_inj/ qed. -lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ≽[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /2 width=5 by tri_step/ qed-. -lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /2 width=5 by tri_TC_strap/ qed-. (* Basic_2A1: uses: lleq_fpbs fleq_fpbs *) -lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=1 by fpbq_fpbs, fpbq_fdeq/ qed. (* Basic_2A1: uses: fpbs_lleq_trans *) -lemma fpbs_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=9 by fpbs_strap1, fpbq_fdeq/ qed-. (* Basic_2A1: uses: lleq_fpbs_trans *) -lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_strap2, fpbq_fdeq/ qed-. lemma tdeq_rdeq_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 → - ∀G,L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃G, L1, T1⦄ ≥[h] ⦃G, L2, T2⦄. + ∀G,L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃G,L1,T1⦄ ≥[h] ⦃G,L2,T2⦄. /4 width=5 by fdeq_fpbs, fpbs_strap1, fpbq_lpx, fdeq_intro_dx/ qed. (* Basic_2A1: removed theorems 3: diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_aaa.ma index 0ecacf3e4..9d27bacb8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_aaa.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/fpbs.ma". (* Properties with atomic arity assignment for terms ************************) -lemma fpbs_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma fpbs_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/ #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A /2 width=8 by fpbq_aaa_conf/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpx.ma index b716e5d43..5dd4e5378 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpx.ma @@ -21,9 +21,9 @@ include "basic_2/rt_computation/fpbs_lpxs.ma". (* Properties with unbound context-sensitive parallel rt-transition *********) (* Basic_2A1: uses: fpbs_cpx_trans_neq *) -lemma fpbs_cpx_tdneq_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h] ⦃G2, L2, U2⦄. +lemma fpbs_cpx_tdneq_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_star … H) -H #G0 #L0 #L3 #T0 #T3 #HT10 #H10 #HL03 #H32 elim (fdeq_cpx_trans … H32 … HTU2) -HTU2 #T4 #HT34 #H42 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpxs.ma index 09a7919b1..f995181c3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_cpxs.ma @@ -19,40 +19,40 @@ include "basic_2/rt_computation/fpbs_fqup.ma". (* Properties with unbound context-sensitive parallel rt-computation ********) -lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄. +lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄. #h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=5 by fpbq_cpx, fpbs_strap1/ qed. -lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T2⦄. +lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T2⦄. #h #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2 /3 width=5 by fpbs_strap1, fpbq_cpx/ qed-. -lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1, L1, T⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀T1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀T1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1 /3 width=5 by fpbs_strap2, fpbq_cpx/ qed-. -lemma cpxs_tdeq_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → +lemma cpxs_tdeq_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → ∀T0. T ≛ T0 → - ∀G2,L2,T2. ⦃G1, L1, T0⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. + ∀G2,L2,T2. ⦃G1,L1,T0⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=3 by cpxs_fpbs_trans, tdeq_fpbs_trans/ qed-. -lemma cpxs_tdeq_fpbs: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T → - ∀T2. T ≛ T2 → ⦃G, L, T1⦄ ≥[h] ⦃G, L, T2⦄. +lemma cpxs_tdeq_fpbs: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T → + ∀T2. T ≛ T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄. /4 width=3 by cpxs_fpbs_trans, fdeq_fpbs, tdeq_fdeq/ qed. (* Properties with star-iterated structural successor for closures **********) -lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → - ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → + ∀G2,L2,T2. ⦃G1,L1,T⦄ ⊐* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed. (* Properties with plus-iterated structural successor for closures **********) -lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → - ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → + ∀G2,L2,T2. ⦃G1,L1,T⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_csx.ma index 71f12aeff..583f5e31f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_csx.ma @@ -20,8 +20,8 @@ include "basic_2/rt_computation/fpbs.ma". (* Properties with sn for unbound parallel rt-transition for terms **********) (* Basic_2A1: was: csx_fpbs_conf *) -lemma fpbs_csx_conf: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. +lemma fpbs_csx_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄. #h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=5 by csx_fpbq_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fpb.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fpb.ma index 2c6cb6a8b..428644da8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fpb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fpb.ma @@ -19,6 +19,6 @@ include "basic_2/rt_computation/fpbs.ma". (* Properties with proper parallel rst-reduction on closures ****************) -lemma fpb_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpb_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=1 by fpbq_fpbs, fpb_fpbq/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqup.ma index 8c588f3b3..ccb50f624 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqup.ma @@ -21,25 +21,25 @@ include "basic_2/rt_computation/fpbs_fqus.ma". (* Advanced properties ******************************************************) lemma tdeq_fpbs_trans: ∀h,T1,T. T1 ≛ T → - ∀G1,G2,L1,L2,T2. ⦃G1, L1, T⦄ ≥[h] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. + ∀G1,G2,L1,L2,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fdeq_fpbs_trans, tdeq_fdeq/ qed-. -lemma fpbs_tdeq_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T⦄ → - ∀T2. T ≛ T2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_tdeq_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T⦄ → + ∀T2. T ≛ T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fdeq_trans, tdeq_fdeq/ qed-. (* Properties with plus-iterated structural successor for closures **********) -lemma fqup_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fqup_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 /4 width=5 by fqu_fquq, fpbq_fquq, tri_step/ qed. -lemma fpbs_fqup_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_fqup_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ ⊐+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-. -lemma fqup_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fqup_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐+ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqus.ma index 57de42633..c869f3d9f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_fqus.ma @@ -19,20 +19,20 @@ include "basic_2/rt_computation/fpbs.ma". (* Properties with star-iterated structural successor for closures **********) -lemma fqus_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fqus_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /3 width=5 by fpbq_fquq, tri_step/ qed. -lemma fpbs_fqus_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h] ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_fqus_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ ⊐* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /3 width=5 by fpbs_strap1, fpbq_fquq/ qed-. -lemma fqus_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fqus_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐* ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqus_ind_dx … H) -G1 -L1 -T1 /3 width=5 by fpbs_strap2, fpbq_fquq/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_lpxs.ma index c5ad6426a..07df0c624 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs_lpxs.ma @@ -23,49 +23,49 @@ include "basic_2/rt_computation/fpbs_cpxs.ma". (* Properties with unbound rt-computation on full local environments *******) -lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → ⦃G, L1, T⦄ ≥[h] ⦃G, L2, T⦄. +lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → ⦃G,L1,T⦄ ≥[h] ⦃G,L2,T⦄. #h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2 /3 width=5 by fpbq_lpx, fpbs_strap1/ qed. -lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L, T2⦄ → - ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ → + ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2 /3 width=5 by fpbs_strap1, fpbq_lpx/ qed-. -lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∀L1. ⦃G1, L1⦄ ⊢ ⬈*[h] L → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1,L,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∀L1. ⦃G1,L1⦄ ⊢ ⬈*[h] L → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1 /3 width=5 by fpbs_strap2, fpbq_lpx/ qed-. (* Basic_2A1: uses: lpxs_lleq_fpbs *) -lemma lpxs_fdeq_fpbs: ∀h,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] L → - ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma lpxs_fdeq_fpbs: ∀h,G1,L1,L,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] L → + ∀G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=3 by lpxs_fpbs_trans, fdeq_fpbs/ qed. -lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L, T2⦄ → - ∀L2. ⦃G2, L⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ → + ∀L2. ⦃G2,L⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-. (* Properties with star-iterated structural successor for closures **********) -lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ → - ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L,T2⦄ → + ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed. (* Properties with unbound context-sensitive parallel rt-computation ********) -lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → - ∀G2,L,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → - ∀L2.⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄. +lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → + ∀G2,L,T2. ⦃G1,L1,T⦄ ⊐* ⦃G2,L,T2⦄ → + ∀L2.⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄. /3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed. -lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1, L1, T1⦄ ≥ [h] ⦃G3, L3, T3⦄ → - ∀T4. ⦃G3, L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 → - ∀G2,L4,T2. ⦃G3, L3, T5⦄ ⊐+ ⦃G2, L4, T2⦄ → - ∀L2. ⦃G2, L4⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥ [h] ⦃G2, L2, T2⦄. +lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1,L1,T1⦄ ≥ [h] ⦃G3,L3,T3⦄ → + ∀T4. ⦃G3,L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 → + ∀G2,L4,T2. ⦃G3,L3,T5⦄ ⊐+ ⦃G2,L4,T2⦄ → + ∀L2. ⦃G2,L4⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥ [h] ⦃G2,L2,T2⦄. #h #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42 @(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *) @(fpbs_fqup_trans … H34) -G2 -L4 -T2 @@ -75,18 +75,18 @@ qed-. (* Advanced properties ******************************************************) (* Basic_2A1: uses: fpbs_intro_alt *) -lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → - ∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ → - ∀L0. ⦃G, L⦄ ⊢ ⬈*[h] L0 → - ∀G2,L2,T2. ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ . +lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → + ∀G,L,T0. ⦃G1,L1,T⦄ ⊐* ⦃G,L,T0⦄ → + ∀L0. ⦃G,L⦄ ⊢ ⬈*[h] L0 → + ∀G2,L2,T2. ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ . /3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_fdeq/ qed. (* Advanced inversion lemmas *************************************************) (* Basic_2A1: uses: fpbs_inv_alt *) -lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → - ∃∃G,L,L0,T,T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ - & ⦃G, L⦄ ⊢ ⬈*[h] L0 & ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄. +lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → + ∃∃G,L,L0,T,T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,L1,T⦄ ⊐* ⦃G,L,T0⦄ + & ⦃G,L⦄ ⊢ ⬈*[h] L0 & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1 [ /2 width=9 by ex4_5_intro/ | #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb.ma index 936fabeca..e5a0108c3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb.ma @@ -19,7 +19,7 @@ include "basic_2/rt_transition/fpb.ma". inductive fsb (h): relation3 genv lenv term ≝ | fsb_intro: ∀G1,L1,T1. ( - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → fsb h G2 L2 T2 + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → fsb h G2 L2 T2 ) → fsb h G1 L1 T1 . @@ -32,11 +32,11 @@ interpretation (* Note: eliminator with shorter ground hypothesis *) (* Note: to be named fsb_ind when fsb becomes a definition like csx, lfsx ***) lemma fsb_ind_alt: ∀h. ∀Q: relation3 …. ( - ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → ( - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2 + ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → ( + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2 ) → Q G1 L1 T1 ) → - ∀G,L,T. ≥[h] 𝐒⦃G, L, T⦄ → Q G L T. + ∀G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → Q G L T. #h #Q #IH #G #L #T #H elim H -G -L -T /4 width=1 by fsb_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_aaa.ma index 93e9c0ec4..7ea1776a0 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_aaa.ma @@ -22,17 +22,17 @@ include "basic_2/rt_computation/fsb_csx.ma". (* Main properties with atomic arity assignment for terms *******************) (* Note: this is the "big tree" theorem *) -theorem aaa_fsb: ∀h,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ≥[h] 𝐒⦃G, L, T⦄. +theorem aaa_fsb: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ≥[h] 𝐒⦃G,L,T⦄. /3 width=2 by aaa_csx, csx_fsb/ qed. (* Advanced eliminators with atomic arity assignment for terms **************) fact aaa_ind_fpb_aux: ∀h. ∀Q:relation3 …. - (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T. + ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T. #h #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH // #G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1 @@ -40,19 +40,19 @@ fact aaa_ind_fpb_aux: ∀h. ∀Q:relation3 …. qed-. lemma aaa_ind_fpb: ∀h. ∀Q:relation3 …. - (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T. + ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T. /4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-. fact aaa_ind_fpbg_aux: ∀h. ∀Q:relation3 …. - (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T. + ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T. #h #Q #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH // #G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1 @@ -60,9 +60,9 @@ fact aaa_ind_fpbg_aux: ∀h. ∀Q:relation3 …. qed-. lemma aaa_ind_fpbg: ∀h. ∀Q:relation3 …. - (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T. + ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T. /4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma index 2d33b783e..83d407f75 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma @@ -21,14 +21,14 @@ include "basic_2/rt_computation/fsb_fpbg.ma". (* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****) -lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. +lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄. #h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/ qed-. (* Propreties with context-sensitive stringly rt-normalizing terms **********) -lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄. +lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄. #h #G1 #L1 #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2 #G0 #L0 #T0 #IHu #H10 @@ -56,23 +56,23 @@ generalize in match IHu; -IHu generalize in match H10; -H10 ] qed. -lemma csx_fsb: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G, L, T⦄. +lemma csx_fsb: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G,L,T⦄. /2 width=5 by csx_fsb_fpbs/ qed. (* Advanced eliminators *****************************************************) lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term. - (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T. + ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T. /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-. lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term. - (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T. + ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T. /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fdeq.ma index 84703ccb6..a24b46ffe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fdeq.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/fsb.ma". (* Properties with sort-irrelevant equivalence for closures *****************) -lemma fsb_fdeq_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄. +lemma fsb_fdeq_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄. #h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1 #G1 #L1 #T1 #_ #IH #G2 #L2 #T2 #H12 @fsb_intro #G #L #T #H2 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fpbg.ma index 806180ead..0ebebad3a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fpbg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_fpbg.ma @@ -19,8 +19,8 @@ include "basic_2/rt_computation/fsb_fdeq.ma". (* Properties with parallel rst-computation for closures ********************) -lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄. +lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄. #h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1 #G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12 elim (fpbs_inv_fpbg … H12) -H12 @@ -32,19 +32,19 @@ qed-. (* Properties with proper parallel rst-computation for closures *************) lemma fsb_intro_fpbg: ∀h,G1,L1,T1. ( - ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄ - ) → ≥[h] 𝐒⦃G1, L1, T1⦄. + ∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄ + ) → ≥[h] 𝐒⦃G1,L1,T1⦄. /4 width=1 by fsb_intro, fpb_fpbg/ qed. (* Eliminators with proper parallel rst-computation for closures ************) lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term. - (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2. + ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2. #h #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1 #G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12 @IH1 -IH1 @@ -56,11 +56,11 @@ lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term. qed-. lemma fsb_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term. - (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → - (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → + (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → - ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ → Q G1 L1 T1. + ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → Q G1 L1 T1. #h #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H /3 width=1 by/ qed-. @@ -68,7 +68,7 @@ qed-. (* Inversion lemmas with proper parallel rst-computation for closures *******) lemma fsb_fpbg_refl_false (h) (G) (L) (T): - ≥[h] 𝐒⦃G, L, T⦄ → ⦃G, L, T⦄ >[h] ⦃G, L, T⦄ → ⊥. + ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥. #h #G #L #T #H @(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H /2 width=5 by/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs.ma index 880278d22..7a3de7dca 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs.ma @@ -28,38 +28,38 @@ interpretation (* Basic properties *********************************************************) (* Basic_2A1: uses: lprs_pair_refl *) -lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → - ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}. +lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → + ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}. /2 width=1 by lex_bind_refl_dx/ qed. -lemma lprs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → - ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h] V2 → - ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2. +lemma lprs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → + ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ➡*[h] V2 → + ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2. /2 width=1 by lex_pair/ qed. -lemma lprs_refl (h) (G): ∀L. ⦃G, L⦄ ⊢ ➡*[h] L. +lemma lprs_refl (h) (G): ∀L. ⦃G,L⦄ ⊢ ➡*[h] L. /2 width=1 by lex_refl/ qed. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: uses: lprs_inv_atom1 *) -lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆. +lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆. /2 width=2 by lex_inv_atom_sn/ qed-. (* Basic_2A1: was: lprs_inv_pair1 *) lemma lprs_inv_pair_sn (h) (G): - ∀I,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 → - ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2. + ∀I,K1,L2,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 → + ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2. /2 width=1 by lex_inv_pair_sn/ qed-. (* Basic_2A1: uses: lprs_inv_atom2 *) -lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆. +lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆. /2 width=2 by lex_inv_atom_dx/ qed-. (* Basic_2A1: was: lprs_inv_pair2 *) lemma lprs_inv_pair_dx (h) (G): - ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 → - ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1. + ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 → + ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1. /2 width=1 by lex_inv_pair_dx/ qed-. (* Basic eliminators ********************************************************) @@ -68,12 +68,12 @@ lemma lprs_inv_pair_dx (h) (G): lemma lprs_ind (h) (G): ∀Q:relation lenv. Q (⋆) (⋆) → ( ∀I,K1,K2. - ⦃G, K1⦄ ⊢ ➡*[h] K2 → + ⦃G,K1⦄ ⊢ ➡*[h] K2 → Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I}) ) → ( ∀I,K1,K2,V1,V2. - ⦃G, K1⦄ ⊢ ➡*[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 → + ⦃G,K1⦄ ⊢ ➡*[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 → Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2) ) → - ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1 L2. + ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1 L2. /3 width=4 by lex_ind/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma index ad24e5876..591dbd1c1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma @@ -19,22 +19,22 @@ include "basic_2/rt_computation/lprs_lpr.ma". (* Properties with t-bound context-sensitive rt-computarion for terms *******) lemma lprs_cpms_trans (n) (h) (G): - ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 → - ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2. + ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 → + ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2. #n #h #G #L2 #T1 #T2 #HT12 #L1 #H @(lprs_ind_sn … H) -L1 /2 width=3 by lpr_cpms_trans/ qed-. lemma lprs_cpm_trans (n) (h) (G): - ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 → - ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2. + ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 → + ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2. /3 width=3 by lprs_cpms_trans, cpm_cpms/ qed-. (* Basic_2A1: includes cprs_bind2 *) lemma cpms_bind_dx (n) (h) (G) (L): - ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2. + ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2. /4 width=5 by lprs_cpms_trans, lprs_pair, cpms_bind/ qed. (* Inversion lemmas with t-bound context-sensitive rt-computarion for terms *) @@ -43,8 +43,8 @@ lemma cpms_bind_dx (n) (h) (G) (L): (* Basic_2A1: includes: cprs_inv_abst1 *) (* Basic_2A1: uses: scpds_inv_abst1 *) lemma cpms_inv_abst_sn (n) (h) (G) (L): - ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 & + ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n,h] X2 → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓛ{p}V2.T2. #n #h #G #L #p #V1 #T1 #X2 #H @(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/ @@ -63,8 +63,8 @@ qed-. (* Basic_2A1: includes: cprs_inv_abst *) lemma cpms_inv_abst_bi (n) (h) (p1) (p2) (G) (L): - ∀W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n, h] ⓛ{p2}W2.T2 → - ∧∧ p1 = p2 & ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2. + ∀W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n,h] ⓛ{p2}W2.T2 → + ∧∧ p1 = p2 & ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2. #n #h #p1 #p2 #G #L #W1 #W2 #T1 #T2 #H elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1 by and3_intro/ @@ -73,9 +73,9 @@ qed-. (* Basic_1: was pr3_gen_abbr *) (* Basic_2A1: includes: cprs_inv_abbr1 *) lemma cpms_inv_abbr_sn_dx (n) (h) (G) (L): - ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n, h] X2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T2 & X2 = ⓓ{p}V2.T2 - | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ. + ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n,h] X2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓓ{p}V2.T2 + | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ. #n #h #G #L #p #V1 #T1 #X2 #H @(cpms_ind_dx … H) -X2 -n /3 width=5 by ex3_2_intro, or_introl/ #n1 #n2 #X #X2 #_ * * @@ -95,8 +95,8 @@ qed-. (* Basic_2A1: uses: scpds_inv_abbr_abst *) lemma cpms_inv_abbr_abst (n) (h) (G) (L): - ∀p1,p2,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n, h] ⓛ{p2}W2.T2 → - ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ. + ∀p1,p2,V1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n,h] ⓛ{p2}W2.T2 → + ∃∃T. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ. #n #h #G #L #p1 #p2 #V1 #W2 #T1 #T2 #H elim (cpms_inv_abbr_sn_dx … H) -H * [ #V #T #_ #_ #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cprs.ma index 958fcab3b..1706ae98e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cprs.ma @@ -20,16 +20,16 @@ include "basic_2/rt_computation/lprs_cpms.ma". (* Advanced properties ******************************************************) (* Basic_2A1: was: lprs_pair2 *) -lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → - ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h] V2 → - ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2. +lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → + ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ➡*[h] V2 → + ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2. /3 width=3 by lprs_pair, lprs_cpms_trans/ qed. (* Properties on context-sensitive parallel r-computation for terms *********) -lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. +lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → + ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. #h #G #L0 #T0 #T1 #HT01 #L1 #H @(lprs_ind_dx … H) -L1 /2 width=3 by ex2_intro/ #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0 @@ -39,21 +39,21 @@ elim (cprs_conf … HT2 … HT3) -T /3 width=5 by cprs_trans, ex2_intro/ qed-. -lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. +lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → + ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 → + ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. /3 width=3 by lprs_cprs_conf_dx, cpm_cpms/ qed-. (* Note: this can be proved on its own using lprs_ind_sn *) -lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 → - ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. +lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → + ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 → + ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. #h #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 /3 width=3 by lprs_cpms_trans, ex2_intro/ qed-. -lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → - ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 → - ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T. +lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → + ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 → + ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T. /3 width=3 by lprs_cprs_conf_sn, cpm_cpms/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_ctc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_ctc.ma index ecef51fe5..742a0038a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_ctc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_ctc.ma @@ -20,11 +20,11 @@ include "basic_2/rt_computation/lprs.ma". (* Properties with contextual transitive closure ****************************) lemma lprs_CTC (h) (G): - ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G, L1⦄⊢ ➡*[h] L2. + ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G,L1⦄⊢ ➡*[h] L2. /3 width=3 by cprs_CTC, lex_co/ qed. (* Inversion lemmas with contextual transitive closure **********************) lemma lprs_inv_CTC (h) (G): - ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2. + ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2. /3 width=3 by cprs_inv_CTC, lex_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_length.ma index ae38e76a4..4327dcb7b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_length.ma @@ -19,5 +19,5 @@ include "basic_2/rt_computation/lprs.ma". (* Forward lemmas with length for local environments ************************) -lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|. +lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|. /2 width=2 by lex_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpr.ma index 120177208..ea0456434 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpr.ma @@ -20,29 +20,29 @@ include "basic_2/rt_computation/lprs_tc.ma". (* Basic_2A1: was: lprs_ind_dx *) lemma lprs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 → - (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) → - ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1. + (∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L → ⦃G,L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) → + ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1. /4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-. (* Basic_2A1: was: lprs_ind *) lemma lprs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 → - (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → Q L → Q L2) → - ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L2. + (∀L,L2. ⦃G,L1⦄ ⊢ ➡*[h] L → ⦃G,L⦄ ⊢ ➡[h] L2 → Q L → Q L2) → + ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L2. /4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-. (* Properties with unbound rt-transition for full local environments ********) -lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2. +lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2. /4 width=3 by lprs_CTC, lpr_cprs_trans, lex_CTC_inj/ qed. (* Basic_2A1: was: lprs_strap2 *) -lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → - ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2. +lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L → + ∀L2.⦃G,L⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2. /4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_sn/ qed-. (* Basic_2A1: was: lpxs_strap1 *) -lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L → - ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2. +lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡*[h] L → + ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2. /4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_dx/ qed-. lemma lprs_strip (h) (G): confluent2 … (lprs h G) (lpr h G). diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpxs.ma index fe2d59ee7..5709df1f3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_lpxs.ma @@ -22,5 +22,5 @@ include "basic_2/rt_computation/lprs.ma". (* Basic_2A1: was: lprs_lpxs *) (* Note: original proof uses lpr_fwd_lpx and monotonic_TC *) -lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2. +lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2. /3 width=3 by cpms_fwd_cpxs, lex_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_tc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_tc.ma index 122051b4d..3b031c6d5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_tc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_tc.ma @@ -21,11 +21,11 @@ include "basic_2/rt_computation/cprs_lpr.ma". (* Properties with transitive closure ***************************************) lemma lprs_TC (h) (G): - ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G, L1⦄⊢ ➡*[h] L2. + ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G,L1⦄⊢ ➡*[h] L2. /4 width=3 by lprs_CTC, lex_CTC, lpr_cprs_trans/ qed. (* Inversion lemmas with transitive closure *********************************) lemma lprs_inv_TC (h) (G): - ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2. + ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2. /3 width=3 by lprs_inv_CTC, lex_inv_CTC/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma index 322f43712..d532c62bf 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma @@ -28,13 +28,13 @@ interpretation (* Basic properties *********************************************************) (* Basic_2A1: uses: lpxs_pair_refl *) -lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}. +lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}. /2 width=1 by lex_bind_refl_dx/ qed. -lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ⬈*[h] V2 → - ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2. +lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ⬈*[h] V2 → + ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2. /2 width=1 by lex_pair/ qed. lemma lpxs_refl (h) (G): reflexive … (lpxs h G). @@ -43,25 +43,25 @@ lemma lpxs_refl (h) (G): reflexive … (lpxs h G). (* Basic inversion lemmas ***************************************************) (* Basic_2A1: was: lpxs_inv_atom1 *) -lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆. +lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆. /2 width=2 by lex_inv_atom_sn/ qed-. -lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 → - ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}. +lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 → + ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}. /2 width=1 by lex_inv_bind_sn/ qed-. (* Basic_2A1: was: lpxs_inv_pair1 *) -lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 → - ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2. +lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 → + ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2. /2 width=1 by lex_inv_pair_sn/ qed-. (* Basic_2A1: was: lpxs_inv_atom2 *) -lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆. +lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆. /2 width=2 by lex_inv_atom_dx/ qed-. (* Basic_2A1: was: lpxs_inv_pair2 *) -lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 → - ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1. +lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 → + ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1. /2 width=1 by lex_inv_pair_dx/ qed-. (* Basic eliminators ********************************************************) @@ -70,12 +70,12 @@ lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I} lemma lpxs_ind (h) (G): ∀Q:relation lenv. Q (⋆) (⋆) → ( ∀I,K1,K2. - ⦃G, K1⦄ ⊢ ⬈*[h] K2 → + ⦃G,K1⦄ ⊢ ⬈*[h] K2 → Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I}) ) → ( ∀I,K1,K2,V1,V2. - ⦃G, K1⦄ ⊢ ⬈*[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 → + ⦃G,K1⦄ ⊢ ⬈*[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 → Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2) ) → - ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2. + ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2. /3 width=4 by lex_ind/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_cpxs.ma index 71742516c..611aab47b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_cpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_cpxs.ma @@ -19,15 +19,15 @@ include "basic_2/rt_computation/lpxs_lpx.ma". (* Properties with context-sensitive extended rt-computation for terms ******) (* Basic_2A1: was: cpxs_bind2 *) -lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. +lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2. /4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed. (* Inversion lemmas with context-sensitive ext rt-computation for terms *****) -lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 & +lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓛ{p}V2.T2. #h #G #p #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/ #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct @@ -38,10 +38,10 @@ qed-. (* Basic_2A1: was: cpxs_inv_abbr1 *) lemma cpxs_inv_abbr1_dx (h) (p) (G) (L): - ∀V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & + ∀V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓓ{p}V2.T2 - | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ. + | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ. #h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/ #U0 #U2 #_ #HU02 * * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_fdeq.ma index 30bbb3e75..e21438d63 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_fdeq.ma @@ -19,9 +19,9 @@ include "basic_2/rt_computation/lpxs_rdeq.ma". (* Properties with sort-irrelevant equivalence on closures ******************) -lemma fdeq_lpxs_trans (h): ∀G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L0, T2⦄ → - ∀L2. ⦃G2, L0⦄ ⊢⬈*[h] L2 → - ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h] L & ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄. +lemma fdeq_lpxs_trans (h): ∀G1,G2,L1,L0,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L0,T2⦄ → + ∀L2. ⦃G2,L0⦄ ⊢⬈*[h] L2 → + ∃∃L. ⦃G1,L1⦄ ⊢⬈*[h] L & ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02 elim (fdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct elim (rdeq_lpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_length.ma index fd664b95c..8ab26578e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_length.ma @@ -19,5 +19,5 @@ include "basic_2/rt_computation/lpxs.ma". (* Forward lemmas with length for local environments ************************) -lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|. +lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|. /2 width=2 by lex_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_lpx.ma index 52a418617..c27507776 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_lpx.ma @@ -20,31 +20,31 @@ include "basic_2/rt_computation/lpxs.ma". (* Properties with unbound rt-transition for full local environments ********) -lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2. +lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2. /3 width=3 by lpx_cpxs_trans, lex_CTC_inj/ qed. (* Basic_2A1: was: lpxs_strap2 *) -lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L → - ∀L2. ⦃G, L⦄ ⊢ ⬈*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2. +lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L → + ∀L2. ⦃G,L⦄ ⊢ ⬈*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2. /3 width=3 by lpx_cpxs_trans, lex_CTC_step_sn/ qed-. (* Basic_2A1: was: lpxs_strap1 *) -lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈*[h] L → - ∀L2. ⦃G, L⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2. +lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈*[h] L → + ∀L2. ⦃G,L⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2. /3 width=3 by lpx_cpxs_trans, lex_CTC_step_dx/ qed-. (* Eliminators with unbound rt-transition for full local environments *******) (* Basic_2A1: was: lpxs_ind_dx *) lemma lpxs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 → - (∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L → ⦃G, L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) → - ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1. + (∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L → ⦃G,L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) → + ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1. /3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_sn/ qed-. (* Basic_2A1: was: lpxs_ind *) lemma lpxs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 → - (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L → ⦃G, L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) → - ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L2. + (∀L,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L → ⦃G,L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) → + ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L2. /3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_dx/ qed-. (* Properties with context-sensitive extended rt-transition for terms *******) @@ -65,7 +65,7 @@ qed-. (* Advanced properties ******************************************************) (* Basic_2A1: was: lpxs_pair2 *) -lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ⬈*[h] V2 → - ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2. +lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ⬈*[h] V2 → + ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2. /3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma index 2bc873339..dd0a02ce2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs_rdeq.ma @@ -21,9 +21,9 @@ include "basic_2/rt_computation/lpxs_lpx.ma". (* Basic_2A1: uses: lleq_lpxs_trans *) lemma rdeq_lpxs_trans (h) (G) (T:term): - ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 → + ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈*[h] K2 → ∀L1. L1 ≛[T] L2 → - ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[T] K2. + ∃∃K1. ⦃G,L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[T] K2. #h #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/ #L #L2 #HL2 #_ #IH #L1 #HT elim (rdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT @@ -33,9 +33,9 @@ qed-. (* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *) lemma lpxs_rdneq_inv_step_sn (h) (G) (T:term): - ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → - ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[T] L → ⊥ & - ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[T] L2. + ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → + ∃∃L,L0. ⦃G,L1⦄ ⊢ ⬈[h] L & L1 ≛[T] L → ⊥ & + ⦃G,L⦄ ⊢ ⬈*[h] L0 & L0 ≛[T] L2. #h #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1 [ #H elim H -H // | #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec L1 L T) #H diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma index d976e52d1..e2fb36648 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx.ma @@ -25,7 +25,7 @@ inductive lsubsx (h) (G): rtmap → relation lenv ≝ lsubsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I}) | lsubsx_unit: ∀f,I,K1,K2. lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ) -| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ → +| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → lsubsx h G f K1 K2 → lsubsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ) . @@ -35,7 +35,7 @@ interpretation (* Basic inversion lemmas ***************************************************) -fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → L1 = ⋆ → L2 = ⋆. #h #g #G #L1 #L2 * -g -L1 -L2 // [ #f #I #K1 #K2 #_ #H destruct @@ -44,12 +44,12 @@ fact lsubsx_inv_atom_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, g] L2 → L2 = ⋆. +lemma lsubsx_inv_atom_sn: ∀h,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h,g] L2 → L2 = ⋆. /2 width=7 by lsubsx_inv_atom_sn_aux/ qed-. -fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}. + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #_ #H destruct | #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct @@ -61,13 +61,13 @@ fact lsubsx_inv_push_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, ⫯f] L2 → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓘ{I}. +lemma lsubsx_inv_push_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h,⫯f] L2 → + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓘ{I}. /2 width=5 by lsubsx_inv_push_sn_aux/ qed-. -fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #_ #H destruct | #f #I #K1 #K2 #_ #g #J #L1 #H @@ -78,14 +78,14 @@ fact lsubsx_inv_unit_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → ] qed-. -lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, ↑f] L2 → - ∃∃K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. +lemma lsubsx_inv_unit_sn: ∀h,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h,↑f] L2 → + ∃∃K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. /2 width=6 by lsubsx_inv_unit_sn_aux/ qed-. -fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → +fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h,g] L2 → ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V → - ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. + ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. #h #g #G #L1 #L2 * -g -L1 -L2 [ #f #g #J #L1 #W #_ #H destruct | #f #I #K1 #K2 #_ #g #J #L1 #W #H @@ -97,17 +97,17 @@ fact lsubsx_inv_pair_sn_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, g] L2 → qed-. (* Basic_2A1: uses: lcosx_inv_pair *) -lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, ↑f] L2 → - ∃∃K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & L2 = K2.ⓧ. +lemma lsubsx_inv_pair_sn: ∀h,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,↑f] L2 → + ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & L2 = K2.ⓧ. /2 width=6 by lsubsx_inv_pair_sn_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, g] L2 → - ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V - | ∃∃f,K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ & - G ⊢ K1 ⊆ⓧ[h, f] K2 & g = ↑f & L2 = K2.ⓧ. +lemma lsubsx_inv_pair_sn_gen: ∀h,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h,g] L2 → + ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V + | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & + G ⊢ K1 ⊆ⓧ[h,f] K2 & g = ↑f & L2 = K2.ⓧ. #h #g #I #G #K1 #L2 #V #H elim (pn_split g) * #f #Hf destruct [ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/ @@ -117,8 +117,8 @@ qed-. (* Advanced forward lemmas **************************************************) -lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, g] L2 → - ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, ⫱g] K2 & L2 = K2.ⓘ{I2}. +lemma lsubsx_fwd_bind_sn: ∀h,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h,g] L2 → + ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h,⫱g] K2 & L2 = K2.ⓘ{I2}. #h #g #I1 #G #K1 #L2 elim (pn_split g) * #f #Hf destruct [ #H elim (lsubsx_inv_push_sn … H) -H @@ -132,7 +132,7 @@ qed-. (* Basic properties *********************************************************) -lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2). +lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2). #h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 // [ #f #I #L1 #L2 #_ #IH #x #H elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/ @@ -143,7 +143,7 @@ lemma lsubsx_eq_repl_back: ∀h,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[ ] qed-. -lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, f] L2). +lemma lsubsx_eq_repl_fwd: ∀h,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h,f] L2). #h #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_lsubsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_lsubsx.ma index 0a06062e6..82038855e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_lsubsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_lsubsx.ma @@ -18,8 +18,8 @@ include "basic_2/rt_computation/lsubsx.ma". (* Main properties **********************************************************) -theorem lsubsx_fix: ∀h,f,G,L1,L. G ⊢ L1 ⊆ⓧ[h, f] L → - ∀L2. G ⊢ L ⊆ⓧ[h, f] L2 → L = L2. +theorem lsubsx_fix: ∀h,f,G,L1,L. G ⊢ L1 ⊆ⓧ[h,f] L → + ∀L2. G ⊢ L ⊆ⓧ[h,f] L2 → L = L2. #h #f #G #L1 #L #H elim H -f -L1 -L [ #f #L2 #H >(lsubsx_inv_atom_sn … H) -L2 // diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_rdsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_rdsx.ma index 70ba811a9..7240fb3b6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_rdsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsubsx_rdsx.ma @@ -22,9 +22,9 @@ include "basic_2/rt_computation/lsubsx.ma". (* Basic_2A1: uses: lsx_cpx_trans_lcosx *) lemma rdsx_cpx_trans_lsubsx (h): - ∀G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ⬈[h] T2 → - ∀f,L. G ⊢ L0 ⊆ⓧ[h, f] L → - G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄. + ∀G,L0,T1,T2. ⦃G,L0⦄ ⊢ T1 ⬈[h] T2 → + ∀f,L. G ⊢ L0 ⊆ⓧ[h,f] L → + G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄. #h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2 // [ #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #L #HK0 #HL elim (lsubsx_inv_pair_sn_gen … HK0) -HK0 * @@ -63,13 +63,13 @@ qed-. (* Basic_2A1: uses: lsx_cpx_trans_O *) lemma rdsx_cpx_trans (h): - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → - G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → + G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄. /3 width=6 by rdsx_cpx_trans_lsubsx, lsubsx_refl/ qed-. lemma rdsx_cpxs_trans (h): - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → - G ⊢ ⬈*[h, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, T2] 𝐒⦃L⦄. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → + G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄. #h #G #L #T1 #T2 #H @(cpxs_ind_dx ???????? H) -T1 // /3 width=3 by rdsx_cpx_trans/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx.ma index 45dd9f908..46b287bb9 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx.ma @@ -30,11 +30,11 @@ interpretation (* Basic_2A1: uses: lsx_ind *) lemma rdsx_ind (h) (G) (T): ∀Q:predicate lenv. - (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → + (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → Q L1 ) → - ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → Q L. + ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L. #h #G #T #Q #H0 #L1 #H elim H -L1 /5 width=1 by SN_intro/ qed-. @@ -44,16 +44,16 @@ qed-. (* Basic_2A1: uses: lsx_intro *) lemma rdsx_intro (h) (G) (T): ∀L1. - (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) → - G ⊢ ⬈*[h, T] 𝐒⦃L1⦄. + (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) → + G ⊢ ⬈*[h,T] 𝐒⦃L1⦄. /5 width=1 by SN_intro/ qed. (* Basic forward lemmas *****************************************************) (* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *) lemma rdsx_fwd_pair_sn (h) (G): - ∀I,L,V,T. G ⊢ ⬈*[h, ②{I}V.T] 𝐒⦃L⦄ → - G ⊢ ⬈*[h, V] 𝐒⦃L⦄. + ∀I,L,V,T. G ⊢ ⬈*[h,②{I}V.T] 𝐒⦃L⦄ → + G ⊢ ⬈*[h,V] 𝐒⦃L⦄. #h #G #I #L #V #T #H @(rdsx_ind … H) -L #L1 #_ #IHL1 @rdsx_intro #L2 #HL12 #HnL12 @@ -62,8 +62,8 @@ qed-. (* Basic_2A1: uses: lsx_fwd_flat_dx *) lemma rdsx_fwd_flat_dx (h) (G): - ∀I,L,V,T. G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄ → - G ⊢ ⬈*[h, T] 𝐒⦃L⦄. + ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ → + G ⊢ ⬈*[h,T] 𝐒⦃L⦄. #h #G #I #L #V #T #H @(rdsx_ind … H) -L #L1 #_ #IHL1 @rdsx_intro #L2 #HL12 #HnL12 @@ -71,23 +71,23 @@ lemma rdsx_fwd_flat_dx (h) (G): qed-. fact rdsx_fwd_pair_aux (h) (G): - ∀L. G ⊢ ⬈*[h, #0] 𝐒⦃L⦄ → - ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, V] 𝐒⦃K⦄. + ∀L. G ⊢ ⬈*[h,#0] 𝐒⦃L⦄ → + ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄. #h #G #L #H @(rdsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct /5 width=5 by lpx_pair, rdsx_intro, rdeq_fwd_zero_pair/ qed-. lemma rdsx_fwd_pair (h) (G): - ∀I,K,V. G ⊢ ⬈*[h, #0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h, V] 𝐒⦃K⦄. + ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄. /2 width=4 by rdsx_fwd_pair_aux/ qed-. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: uses: lsx_inv_flat *) lemma rdsx_inv_flat (h) (G): - ∀I,L,V,T. G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄ → - ∧∧ G ⊢ ⬈*[h, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, T] 𝐒⦃L⦄. + ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ → + ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L⦄. /3 width=3 by rdsx_fwd_pair_sn, rdsx_fwd_flat_dx, conj/ qed-. (* Basic_2A1: removed theorems 9: diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_csx.ma index 436c2b224..f6c85a089 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_csx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_csx.ma @@ -22,9 +22,9 @@ include "basic_2/rt_computation/lsubsx_rdsx.ma". (* Basic_2A1: uses: lsx_lref_be_lpxs *) lemma rdsx_pair_lpxs (h) (G): - ∀K1,V. ⦃G, K1⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → - ∀K2. G ⊢ ⬈*[h, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h] K2 → - ∀I. G ⊢ ⬈*[h, #0] 𝐒⦃K2.ⓑ{I}V⦄. + ∀K1,V. ⦃G,K1⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → + ∀K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → ⦃G,K1⦄ ⊢ ⬈*[h] K2 → + ∀I. G ⊢ ⬈*[h,#0] 𝐒⦃K2.ⓑ{I}V⦄. #h #G #K1 #V #H @(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H @(rdsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I @@ -42,8 +42,8 @@ qed. (* Basic_2A1: uses: lsx_lref_be *) lemma rdsx_lref_pair_drops (h) (G): - ∀K,V. ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h, V] 𝐒⦃K⦄ → - ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄. + ∀K,V. ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄ → + ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄. #h #G #K #V #HV #HK #I #i elim i -i [ #L #H >(drops_fwd_isid … H) -H /2 width=3 by rdsx_pair_lpxs/ | #i #IH #L #H @@ -55,7 +55,7 @@ qed. (* Main properties **********************************************************) (* Basic_2A1: uses: csx_lsx *) -theorem csx_rdsx (h): ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → G ⊢ ⬈*[h, T] 𝐒⦃L⦄. +theorem csx_rdsx (h): ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → G ⊢ ⬈*[h,T] 𝐒⦃L⦄. #h #G #L #T @(fqup_wf_ind_eq (Ⓕ) … G L T) -G -L -T #Z #Y #X #IH #G #L * * // [ #i #HG #HL #HT #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_drops.ma index a27c2a71d..3c1c6ac7e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_drops.ma @@ -23,7 +23,7 @@ include "basic_2/rt_computation/rdsx_fqup.ma". (* Note: this uses length *) (* Basic_2A1: uses: lsx_lift_le lsx_lift_ge *) -lemma rdsx_lifts (h) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄). +lemma rdsx_lifts (h) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄). #h #G #K #T #H @(rdsx_ind … H) -K #K1 #_ #IH #b #f #L1 #HLK1 #Hf #U #HTU @rdsx_intro #L2 #HL12 #HnL12 elim (lpx_drops_conf … HLK1 … HL12) @@ -33,7 +33,7 @@ qed-. (* Inversion lemmas on relocation *******************************************) (* Basic_2A1: uses: lsx_inv_lift_le lsx_inv_lift_be lsx_inv_lift_ge *) -lemma rdsx_inv_lifts (h) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄). +lemma rdsx_inv_lifts (h) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄). #h #G #L #U #H @(rdsx_ind … H) -L #L1 #_ #IH #b #f #K1 #HLK1 #Hf #T #HTU @rdsx_intro #K2 #HK12 #HnK12 elim (drops_lpx_trans … HLK1 … HK12) -HK12 @@ -43,13 +43,13 @@ qed-. (* Advanced properties ******************************************************) (* Basic_2A1: uses: lsx_lref_free *) -lemma rdsx_lref_atom (h) (G): ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄. +lemma rdsx_lref_atom (h) (G): ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄. #h #G #L1 #i #HL1 @(rdsx_lifts … (#0) … HL1) -HL1 // qed. (* Basic_2A1: uses: lsx_lref_skip *) -lemma rdsx_lref_unit (h) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h, #i] 𝐒⦃L⦄. +lemma rdsx_lref_unit (h) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄. #h #G #I #L1 #K1 #i #HL1 @(rdsx_lifts … (#0) … HL1) -HL1 // qed. @@ -58,8 +58,8 @@ qed. (* Basic_2A1: uses: lsx_fwd_lref_be *) lemma rdsx_fwd_lref_pair (h) (G): - ∀L,i. G ⊢ ⬈*[h, #i] 𝐒⦃L⦄ → - ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, V] 𝐒⦃K⦄. + ∀L,i. G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ → + ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄. #h #G #L #i #HL #I #K #V #HLK lapply (rdsx_inv_lifts … HL … HLK … (#0) ?) -L /2 width=2 by rdsx_fwd_pair/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_fqup.ma index 7906406eb..38293606c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_fqup.ma @@ -20,7 +20,7 @@ include "basic_2/rt_computation/rdsx.ma". (* Advanced properties ******************************************************) (* Basic_2A1: uses: lsx_atom *) -lemma lfsx_atom (h) (G) (T): G ⊢ ⬈*[h, T] 𝐒⦃⋆⦄. +lemma lfsx_atom (h) (G) (T): G ⊢ ⬈*[h,T] 𝐒⦃⋆⦄. #h #G #T @rdsx_intro #Y #H #HnT lapply (lpx_inv_atom_sn … H) -H #H destruct @@ -33,8 +33,8 @@ qed. (* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *) (* Note: the old proof without the exclusion binder requires lreq *) lemma rdsx_fwd_bind_dx (h) (G): - ∀p,I,L,V,T. G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄ → - G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄. + ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ → + G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄. #h #G #p #I #L #V #T #H @(rdsx_ind … H) -L #L1 #_ #IH @rdsx_intro #Y #H #HT @@ -46,6 +46,6 @@ qed-. (* Basic_2A1: uses: lsx_inv_bind *) lemma rdsx_inv_bind (h) (G): - ∀p,I,L,V,T. G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄ → - ∧∧ G ⊢ ⬈*[h, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄. + ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ → + ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄. /3 width=4 by rdsx_fwd_pair_sn, rdsx_fwd_bind_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_length.ma index 7c3c5703b..75177be45 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_length.ma @@ -21,18 +21,18 @@ include "basic_2/rt_computation/rdsx.ma". (* Advanced properties ******************************************************) (* Basic_2A1: uses: lsx_sort *) -lemma rdsx_sort (h) (G): ∀L,s. G ⊢ ⬈*[h, ⋆s] 𝐒⦃L⦄. +lemma rdsx_sort (h) (G): ∀L,s. G ⊢ ⬈*[h,⋆s] 𝐒⦃L⦄. #h #G #L1 #s @rdsx_intro #L2 #H #Hs elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_sort_length/ qed. (* Basic_2A1: uses: lsx_gref *) -lemma rdsx_gref (h) (G): ∀L,l. G ⊢ ⬈*[h, §l] 𝐒⦃L⦄. +lemma rdsx_gref (h) (G): ∀L,l. G ⊢ ⬈*[h,§l] 𝐒⦃L⦄. #h #G #L1 #s @rdsx_intro #L2 #H #Hs elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_gref_length/ qed. -lemma rdsx_unit (h) (G): ∀I,L. G ⊢ ⬈*[h, #0] 𝐒⦃L.ⓤ{I}⦄. +lemma rdsx_unit (h) (G): ∀I,L. G ⊢ ⬈*[h,#0] 𝐒⦃L.ⓤ{I}⦄. #h #G #I #L1 @rdsx_intro #Y #HY #HnY elim HnY -HnY elim (lpx_inv_unit_sn … HY) -HY #L2 #HL12 #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma index 0368929ea..77867e07a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_lpxs.ma @@ -22,14 +22,14 @@ include "basic_2/rt_computation/rdsx_rdsx.ma". (* Basic_2A1: uses: lsx_intro_alt *) lemma rdsx_intro_lpxs (h) (G): - ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄) → - G ⊢ ⬈*[h, T] 𝐒⦃L1⦄. + ∀L1,T. (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) → + G ⊢ ⬈*[h,T] 𝐒⦃L1⦄. /4 width=1 by lpx_lpxs, rdsx_intro/ qed-. (* Basic_2A1: uses: lsx_lpxs_trans *) lemma rdsx_lpxs_trans (h) (G): - ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄. + ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄. #h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2 /2 width=3 by rdsx_lpx_trans/ qed-. @@ -38,12 +38,12 @@ qed-. lemma rdsx_ind_lpxs_rdeq (h) (G): ∀T. ∀Q:predicate lenv. - (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → + (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → Q L1 ) → - ∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2. + ∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + ∀L0. ⦃G,L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2. #h #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1 #L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02 @IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2 @@ -64,11 +64,11 @@ qed-. (* Basic_2A1: uses: lsx_ind_alt *) lemma rdsx_ind_lpxs (h) (G): ∀T. ∀Q:predicate lenv. - (∀L1. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → + (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) → Q L1 ) → - ∀L. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → Q L. + ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L. #h #G #T #Q #IH #L #HL @(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *) qed-. @@ -76,10 +76,10 @@ qed-. (* Advanced properties ******************************************************) fact rdsx_bind_lpxs_aux (h) (G): - ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ → - ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ → - ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄. + ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ → + ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ → + ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄. #h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1 #L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y #Y #HY #IHY #L2 #H #HL12 destruct @@ -97,16 +97,16 @@ qed-. (* Basic_2A1: uses: lsx_bind *) lemma rdsx_bind (h) (G): - ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ → - ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓑ{I}V⦄ → - G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄. + ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ → + ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓑ{I}V⦄ → + G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄. /2 width=3 by rdsx_bind_lpxs_aux/ qed. (* Basic_2A1: uses: lsx_flat_lpxs *) lemma rdsx_flat_lpxs (h) (G): - ∀I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ → - ∀L2,T. G ⊢ ⬈*[h, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L2⦄. + ∀I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ → + ∀L2,T. G ⊢ ⬈*[h,T] 𝐒⦃L2⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄. #h #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1 #L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2 #L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs @@ -123,15 +123,15 @@ qed-. (* Basic_2A1: uses: lsx_flat *) lemma rdsx_flat (h) (G): - ∀I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ → - ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, ⓕ{I}V.T] 𝐒⦃L⦄. + ∀I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ → + ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄. /2 width=3 by rdsx_flat_lpxs/ qed. fact rdsx_bind_lpxs_void_aux (h) (G): - ∀p,I,L1,V. G ⊢ ⬈*[h, V] 𝐒⦃L1⦄ → - ∀Y,T. G ⊢ ⬈*[h, T] 𝐒⦃Y⦄ → - ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 → - G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L2⦄. + ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ → + ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ → + ∀L2. Y = L2.ⓧ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 → + G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄. #h #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1 #L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y #Y #HY #IHY #L2 #H #HL12 destruct @@ -148,7 +148,7 @@ elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ] qed-. lemma rdsx_bind_void (h) (G): - ∀p,I,L,V. G ⊢ ⬈*[h, V] 𝐒⦃L⦄ → - ∀T. G ⊢ ⬈*[h, T] 𝐒⦃L.ⓧ⦄ → - G ⊢ ⬈*[h, ⓑ{p,I}V.T] 𝐒⦃L⦄. + ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ → + ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄ → + G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄. /2 width=3 by rdsx_bind_lpxs_void_aux/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_rdsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_rdsx.ma index 8066dad4b..861fe1f45 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_rdsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/rdsx_rdsx.ma @@ -21,8 +21,8 @@ include "basic_2/rt_computation/rdsx.ma". (* Basic_2A1: uses: lsx_lleq_trans *) lemma rdsx_rdeq_trans (h) (G): - ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - ∀L2. L1 ≛[T] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄. + ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + ∀L2. L1 ≛[T] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄. #h #G #L1 #T #H @(rdsx_ind … H) -L1 #L1 #_ #IHL1 #L2 #HL12 @rdsx_intro #L #HL2 #HnL2 elim (rdeq_lpx_trans … HL2 … HL12) -HL2 @@ -31,8 +31,8 @@ qed-. (* Basic_2A1: uses: lsx_lpx_trans *) lemma rdsx_lpx_trans (h) (G): - ∀L1,T. G ⊢ ⬈*[h, T] 𝐒⦃L1⦄ → - ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h, T] 𝐒⦃L2⦄. + ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ → + ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄. #h #G #L1 #T #H @(rdsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 elim (rdeq_dec L1 L2 T) /3 width=4 by rdsx_rdeq_trans/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc.ma index 156374e3f..df250a17b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc.ma @@ -18,7 +18,7 @@ include "basic_2/rt_transition/cpm.ma". (* CONTEXT-SENSITIVE PARALLEL R-CONVERSION FOR TERMS ************************) definition cpc: sh → relation4 genv lenv term term ≝ - λh,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G, L⦄ ⊢ T2 ➡[h] T1. + λh,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G,L⦄ ⊢ T2 ➡[h] T1. interpretation "context-sensitive parallel r-conversion (term)" @@ -35,7 +35,7 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h] T & ⦃G, L⦄ ⊢ T2 ➡[h] T. +lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 → + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[h] T & ⦃G,L⦄ ⊢ T2 ➡[h] T. #h #G #L #T1 #T2 * /2 width=3 by ex2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc_cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc_cpc.ma index b5b9ac86c..d3151122a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc_cpc.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/cpc_cpc.ma @@ -18,6 +18,6 @@ include "basic_2/rt_conversion/cpc.ma". (* Main properties **********************************************************) -theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G, L⦄ ⊢ T0 ⬌[h] T1 → ⦃G, L⦄ ⊢ T0 ⬌[h] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ⬌[h] T & ⦃G, L⦄ ⊢ T2 ⬌[h] T. +theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G,L⦄ ⊢ T0 ⬌[h] T1 → ⦃G,L⦄ ⊢ T0 ⬌[h] T2 → + ∃∃T. ⦃G,L⦄ ⊢ T1 ⬌[h] T & ⦃G,L⦄ ⊢ T2 ⬌[h] T. /3 width=3 by cpc_sym, ex2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/lpce.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/lpce.ma index b74313dae..8da6c1af4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/lpce.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_conversion/lpce.ma @@ -62,7 +62,7 @@ lemma lpce_inv_bind_dx (h) (G): lemma lpce_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 → - ∃∃K2. ⦃G, K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}. + ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}. /2 width=1 by lex_inv_unit_sn/ qed-. lemma lpce_inv_pair_sn (h) (G): diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs.ma index 59848b068..4daecd3eb 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs.ma @@ -29,16 +29,16 @@ interpretation "context-sensitive parallel r-equivalence (term)" (* Basic_2A1: was: cpcs_ind_dx *) lemma cpcs_ind_sn (h) (G) (L) (T2): ∀Q:predicate term. Q T2 → - (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) → - ∀T1. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T1. + (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T → ⦃G,L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) → + ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T1. normalize /3 width=6 by TC_star_ind_dx/ qed-. (* Basic_2A1: was: cpcs_ind *) lemma cpcs_ind_dx (h) (G) (L) (T1): ∀Q:predicate term. Q T1 → - (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) → - ∀T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T2. + (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → ⦃G,L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) → + ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T2. normalize /3 width=6 by TC_star_ind/ qed-. @@ -54,50 +54,50 @@ lemma cpcs_sym (h) (G) (L): symmetric … (cpcs h G L). /2 width=1 by cpc_sym/ qed-. -lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /2 width=1 by inj/ qed. (* Basic_2A1: was: cpcs_strap2 *) -lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. normalize /2 width=3 by TC_strap/ qed-. (* Basic_2A1: was: cpcs_strap1 *) -lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. normalize /2 width=3 by step/ qed-. (* Basic_1: was: pc3_pr2_r *) -lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=1 by cpc_cpcs, or_introl/ qed. (* Basic_1: was: pc3_pr2_x *) -lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=1 by cpc_cpcs, or_intror/ qed. (* Basic_1: was: pc3_pr2_u *) (* Basic_2A1: was: cpcs_cpr_strap2 *) -lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpcs_step_sn, or_introl/ qed-. (* Basic_2A1: was: cpcs_cpr_strap1 *) -lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpcs_step_dx, or_introl/ qed-. -lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpcs_step_dx, or_intror/ qed-. -lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpr_cpcs_dx, cpcs_step_dx, or_intror/ qed-. (* Basic_1: was: pc3_pr2_u2 *) -lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡[h] T1 → - ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡[h] T1 → + ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpcs_step_sn, or_intror/ qed-. (* Basic_1: removed theorems 9: diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_aaa.ma index 13d12e5ad..2c1be91f4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_aaa.ma @@ -20,8 +20,8 @@ include "basic_2/rt_equivalence/cpcs_cprs.ma". (* Main inversion lemmas with atomic arity assignment on terms **************) (* Note: lemma 1500 *) -theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → - ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 → +theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → + ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 → A1 = A2. #h #G #L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma index a29055eee..85cd8c732 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cpcs.ma @@ -31,21 +31,21 @@ theorem cpcs_canc_dx (h) (G) (L): right_cancellable … (cpcs h G L). (* Advanced properties ******************************************************) -lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2. +lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2. /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. -lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 → - ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2. +lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 → + ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2. /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed. (* Advanced properties with r-transition for full local environments ********) (* Basic_1: was: pc3_wcpr0 *) -lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2. +lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → + ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2. #h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma index 5292a2665..bd40f01c3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma @@ -20,8 +20,8 @@ include "basic_2/rt_equivalence/cpcs.ma". (* Inversion lemmas with context sensitive r-computation on terms ***********) -lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[h] T & ⦃G, L⦄ ⊢ T2 ➡*[h] T. +lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[h] T & ⦃G,L⦄ ⊢ T2 ➡*[h] T. #h #G #L #T1 #T2 #H @(cpcs_ind_dx … H) -T2 [ /3 width=3 by ex2_intro/ | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0 @@ -35,7 +35,7 @@ qed-. (* Basic_1: was: pc3_gen_sort *) (* Basic_2A1: was: cpcs_inv_sort *) -lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2. +lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G,L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2. #h #G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H #T #H1 >(cprs_inv_sort1 … H1) -T #H2 lapply (cprs_inv_sort1 … H2) -L #H destruct // @@ -43,8 +43,8 @@ qed-. (* Basic_2A1: was: cpcs_inv_abst1 *) lemma cpcs_inv_abst_sn (h) (G) (L): - ∀p,W1,T1,X. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X → - ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2. + ∀p,W1,T1,X. ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X → + ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2. #h #G #L #p #W1 #T1 #T #H elim (cpcs_inv_cprs … H) -H #X #H1 #H2 elim (cpms_inv_abst_sn … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct @@ -53,13 +53,13 @@ qed-. (* Basic_2A1: was: cpcs_inv_abst2 *) lemma cpcs_inv_abst_dx (h) (G) (L): - ∀p,W1,T1,X. ⦃G, L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 → - ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2. + ∀p,W1,T1,X. ⦃G,L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 → + ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2. /3 width=1 by cpcs_inv_abst_sn, cpcs_sym/ qed-. (* Basic_1: was: pc3_gen_sort_abst *) lemma cpcs_inv_sort_abst (h) (G) (L): - ∀p,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥. + ∀p,W,T,s. ⦃G,L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥. #h #G #L #p #W #T #s #H elim (cpcs_inv_cprs … H) -H #X #H1 >(cprs_inv_sort1 … H1) -X #H2 @@ -69,97 +69,97 @@ qed-. (* Properties with context sensitive r-computation on terms *****************) (* Basic_1: was: pc3_pr3_r *) -lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /3 width=3 by cpcs_cpr_step_dx, cpcs_step_dx, cpc_cpcs/ qed. (* Basic_1: was: pc3_pr3_x *) -lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T2 /3 width=3 by cpcs_cpr_div, cpcs_step_sn, cpcs_cprs_dx/ qed. (* Basic_2A1: was: cpcs_cprs_strap1 *) -lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_step_dx/ qed-. (* Basic_2A1: was: cpcs_cprs_strap2 *) -lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_sn … H) -T1 /2 width=3 by cpcs_cpr_step_sn/ qed-. -lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2 /2 width=3 by cpcs_cpr_div/ qed-. (* Basic_1: was: pc3_pr3_conf *) -lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡*[h] T1 → - ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡*[h] T1 → + ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_conf/ qed-. (* Basic_1: was: pc3_pr3_t *) (* Basic_1: note: pc3_pr3_t should be renamed *) -lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2 /2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/ qed. -lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=5 by cpm_cpms, cprs_div/ qed-. -lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. /3 width=3 by cpm_cpms, cprs_div/ qed-. -lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 → - ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 → + ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2 /2 width=3 by cpr_cprs_div/ qed-. -lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 → - ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T2 ⬌*[h] T1. +lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 → + ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h] T1. #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2 /2 width=3 by cprs_cpr_div/ qed-. -lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 → - ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2. +lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 → + ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2. #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2 /2 width=3 by cprs_div/ qed-. (* Basic_1: was only: pc3_thin_dx *) -lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 → - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2. +lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 → + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → + ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2. #h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HV12) -HV12 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ qed. -lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V2 ➡[h] V1 → - ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → - ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2. +lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V2 ➡[h] V1 → + ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → + ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2. /3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed. -lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2. +lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2. #h #G #L #I #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cpms_bind/ qed. -lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 → - ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T. +lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 → + ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T. #h #G #L #I #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cpms_bind/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_lprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_lprs.ma index 3564a480a..6da444ce1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_lprs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_lprs.ma @@ -19,20 +19,20 @@ include "basic_2/rt_equivalence/cpcs_cprs.ma". (* Properties with parallel r-computation for full local environments *******) -lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2. +lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → + ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2. #h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H /4 width=5 by cprs_div, lpr_cpms_trans/ qed-. -lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → - ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2. +lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → + ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2. #h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H /4 width=5 by cprs_div, lprs_cpms_trans/ qed-. -lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2. +lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → + ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2. #h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1 /2 width=3 by cprs_div/ qed-. @@ -40,23 +40,23 @@ qed-. (* Basic_1: was: pc3_wcpr0_t *) (* Basic_1: note: pc3_wcpr0_t should be renamed *) (* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *) -lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2. +lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → + ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2. #h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1 /2 width=3 by cprs_div/ qed-. (* Basic_1: was only: pc3_pr0_pr2_t *) (* Basic_1: note: pc3_pr0_pr2_t should be renamed *) -lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → - ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2. +lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → + ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2. /3 width=5 by lpr_cprs_conf, cpm_cpms/ qed-. (* Advanced inversion lemmas ************************************************) (* Note: there must be a proof suitable for lfpr *) -lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 → - ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2. +lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 → + ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2. #h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H #T #H1 #H2 elim (cpms_inv_abst_sn … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct @@ -66,8 +66,8 @@ lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/ qed-. -lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 → - ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2. +lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 → + ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2. #h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12 #HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes.ma index c864c81a9..de444a0d7 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes.ma @@ -20,7 +20,7 @@ include "basic_2/rt_computation/cpms.ma". (* Basic_2A1: uses: scpes *) definition cpes (h) (n1) (n2): relation4 genv lenv term term ≝ λG,L,T1,T2. - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G, L⦄ ⊢ T2 ➡*[n2,h] T. + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T. interpretation "t-bound context-sensitive parallel rt-equivalence (term)" 'PConvStar h n1 n2 G L T1 T2 = (cpes h n1 n2 G L T1 T2). @@ -29,8 +29,8 @@ interpretation "t-bound context-sensitive parallel rt-equivalence (term)" (* Basic_2A1: uses: scpds_div *) lemma cpms_div (h) (n1) (n2): - ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1,h] T → - ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[n2,h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2. + ∀G,L,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → + ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2. /2 width=3 by ex2_intro/ qed. lemma cpes_refl (h): ∀G,L. reflexive … (cpes h 0 0 G L). @@ -38,6 +38,6 @@ lemma cpes_refl (h): ∀G,L. reflexive … (cpes h 0 0 G L). (* Basic_2A1: uses: scpes_sym *) lemma cpes_sym (h) (n1) (n2): - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → ⦃G, L⦄ ⊢ T2 ⬌*[h,n2,n1] T1. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h,n2,n1] T1. #h #n1 #n2 #G #L #T1 #T2 * /2 width=3 by cpms_div/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes_aaa.ma index 2aa50131f..06eacc674 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpes_aaa.ma @@ -21,7 +21,7 @@ include "basic_2/rt_equivalence/cpes.ma". (* Basic_2A1: uses: scpes_refl *) lemma cpes_refl_aaa (h) (n): - ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ T ⬌*[h,n,n] T. + ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ T ⬌*[h,n,n] T. #h #n #G #L #T #A #HA elim (cpms_total_aaa h … n … HA) #U #HTU /2 width=3 by cpms_div/ @@ -31,8 +31,8 @@ qed. (* Basic_2A1: uses: scpes_aaa_mono *) theorem cpes_aaa_mono (h) (n1) (n2): - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → - ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 → A1 = A2. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → + ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 → A1 = A2. #h #n1 #n2 #G #L #T1 #T2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2 lapply (cpms_aaa_conf … HA1 … HT1) -T1 #HA1 lapply (cpms_aaa_conf … HA2 … HT2) -T2 #HA2 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr.ma index aeb33e6dd..aa739d678 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr.ma @@ -26,7 +26,7 @@ interpretation (* Basic inversion lemmas ***************************************************) lemma cnr_inv_abst (h) (p) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}V.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L.ⓛV⦄ ⊢ ➡[h] 𝐍⦃T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}V.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓛV⦄ ⊢ ➡[h] 𝐍⦃T⦄. #h #p #G #L #V1 #T1 #HVT1 @conj [ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct // | #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct // @@ -35,7 +35,7 @@ qed-. (* Basic_2A1: was: cnr_inv_abbr *) lemma cnr_inv_abbr_neg (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄. #h #G #L #V1 #T1 #HVT1 @conj [ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct // | #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct // @@ -43,7 +43,7 @@ lemma cnr_inv_abbr_neg (h) (G) (L): qed-. (* Basic_2A1: was: cnr_inv_eps *) -lemma cnr_inv_cast (h) (G) (L): ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓝV.T⦄ → ⊥. +lemma cnr_inv_cast (h) (G) (L): ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓝV.T⦄ → ⊥. #h #G #L #V #T #H lapply (H T ?) -H /2 width=4 by cpm_eps, discr_tpair_xy_y/ qed-. @@ -51,26 +51,26 @@ qed-. (* Basic properties *********************************************************) (* Basic_1: was: nf2_sort *) -lemma cnr_sort (h) (G) (L): ∀s. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃⋆s⦄. +lemma cnr_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃⋆s⦄. #h #G #L #s #X #H >(cpr_inv_sort1 … H) // qed. -lemma cnr_gref (h) (G) (L): ∀l. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃§l⦄. +lemma cnr_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃§l⦄. #h #G #L #l #X #H >(cpr_inv_gref1 … H) // qed. (* Basic_1: was: nf2_abst *) lemma cnr_abst (h) (p) (G) (L): - ∀W,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}W.T⦄. + ∀W,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}W.T⦄. #h #p #G #L #W #T #HW #HT #X #H elim (cpm_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct <(HW … HW0) -W0 <(HT … HT0) -T0 // qed. lemma cnr_abbr_neg (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G, L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄. #h #G #L #V #T #HV #HT #X #H elim (cpm_inv_abbr1 … H) -H * [ #V0 #T0 #HV0 #HT0 #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_drops.ma index 6bc4661ca..8d9a8ce0e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_drops.ma @@ -21,7 +21,7 @@ include "basic_2/rt_transition/cnr.ma". (* Basic_1: was only: nf2_csort_lref *) lemma cnr_lref_atom (h) (b) (G) (L): - ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. + ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. #h #b #G #L #i #Hi #X #H elim (cpr_inv_lref1_drops … H) -H // * #K #V1 #V2 #HLK lapply (drops_gen b … HLK) -HLK #HLK @@ -30,7 +30,7 @@ qed. (* Basic_1: was: nf2_lref_abst *) lemma cnr_lref_abst (h) (G) (L): - ∀K,V,i. ⬇*[i] L ≘ K.ⓛV → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. + ∀K,V,i. ⬇*[i] L ≘ K.ⓛV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. #h #G #L #K #V #i #HLK #X #H elim (cpr_inv_lref1_drops … H) -H // * #K0 #V1 #V2 #HLK0 #_ #_ @@ -38,7 +38,7 @@ lapply (drops_mono … HLK … HLK0) -L #H destruct qed. lemma cnr_lref_unit (h) (I) (G) (L): - ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. + ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄. #h #I #G #L #K #i #HLK #X #H elim (cpr_inv_lref1_drops … H) -H // * #K0 #V1 #V2 #HLK0 #_ #_ @@ -59,7 +59,7 @@ qed-. (* Basic_2A1: was: cnr_inv_delta *) lemma cnr_inv_lref_abbr (h) (G) (L): - ∀K,V,i. ⬇*[i] L ≘ K.ⓓV → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃#i⦄ → ⊥. + ∀K,V,i. ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄ → ⊥. #h #G #L #K #V #i #HLK #H elim (lifts_total V 𝐔❴↑i❵) #W #HVW lapply (H W ?) -H [ /3 width=6 by cpm_delta_drops/ ] -HLK #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_simple.ma index 5dcbada49..20039dafe 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_simple.ma @@ -20,7 +20,7 @@ include "basic_2/rt_transition/cnr.ma". (* Inversion lemmas with simple terms ***************************************) lemma cnr_inv_appl (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄. #h #G #L #V1 #T1 #HVT1 @and3_intro [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct // | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct // @@ -36,7 +36,7 @@ qed-. (* Basic_1: was only: nf2_appl_lref *) lemma cnr_appl_simple (h) (G) (L): - ∀V,T. ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄. + ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄. #h #G #L #V #T #HV #HT #HS #X #H elim (cpm_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct <(HV … HV0) -V0 <(HT … HT0) -T0 // diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.ma index 80e4d8ff0..00d42e11d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.ma @@ -24,8 +24,8 @@ include "basic_2/rt_transition/cnr_drops.ma". (* Basic_1: was: nf2_dec *) (* Basic_2A1: uses: cnr_dec *) lemma cnr_dec_tdeq (h) (G) (L): - ∀T1. ∨∨ ⦃G, L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ - | ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥). + ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ + | ∃∃T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥). #h #G #L #T1 @(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * * [ #s #HG #HL #HT destruct -IH diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_drops.ma index 6b6aa0962..97e13d22a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_drops.ma @@ -21,7 +21,7 @@ include "basic_2/rt_transition/cnu.ma". (* Advanced properties ******************************************************) lemma cnu_atom_drops (h) (b) (G) (L): - ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄. + ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄. #h #b #G #L #i #Hi #n #X #H elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #K #V1 #V2 #HLK lapply (drops_gen b … HLK) -HLK #HLK @@ -29,7 +29,7 @@ lapply (drops_mono … Hi … HLK) -L #H destruct qed. lemma cnu_unit_drops (h) (I) (G) (L): - ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄. + ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄. #h #I #G #L #K #i #HLK #n #X #H elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #Y #V1 #V2 #HLY lapply (drops_mono … HLK … HLY) -L #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_tdeq.ma index 91676f500..f6eb1a805 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnu_tdeq.ma @@ -22,8 +22,8 @@ include "basic_2/rt_transition/cnu_cnr_simple.ma". (* Properties with context-free sort-irrelevant equivalence for terms *******) lemma cnu_dec_tdeq (h) (G) (L): - ∀T1. ∨∨ ⦃G, L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ - | ∃∃n,T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥). + ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ + | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥). #h #G #L #T1 @(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * * [ #s #HG #HL #HT destruct -IH diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx.ma index 6607013df..a79e8d74e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx.ma @@ -27,8 +27,8 @@ interpretation (* Basic inversion lemmas ***************************************************) -lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ → - ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ⬈[h] 𝐍⦃T⦄. +lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ → + ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓛV⦄ ⊢ ⬈[h] 𝐍⦃T⦄. #h #p #G #L #V1 #T1 #HVT1 @conj [ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 | #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 @@ -37,8 +37,8 @@ lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ qed-. (* Basic_2A1: was: cnx_inv_abbr *) -lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ → - ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ⬈[h] 𝐍⦃T⦄. +lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ → + ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓓV⦄ ⊢ ⬈[h] 𝐍⦃T⦄. #h #G #L #V1 #T1 #HVT1 @conj [ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 | #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 @@ -47,20 +47,20 @@ lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ qed-. (* Basic_2A1: was: cnx_inv_eps *) -lemma cnx_inv_cast: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓝV.T⦄ → ⊥. +lemma cnx_inv_cast: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓝV.T⦄ → ⊥. #h #G #L #V #T #H lapply (H T ?) -H /2 width=6 by cpx_eps, tdeq_inv_pair_xy_y/ qed-. (* Basic properties *********************************************************) -lemma cnx_sort: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃⋆s⦄. +lemma cnx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃⋆s⦄. #h #G #L #s #X #H elim (cpx_inv_sort1 … H) -H /2 width=1 by tdeq_sort/ qed. -lemma cnx_abst: ∀h,p,G,L,W,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → - ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}W.T⦄. +lemma cnx_abst: ∀h,p,G,L,W,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → + ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}W.T⦄. #h #p #G #L #W #T #HW #HT #X #H elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct @tdeq_pair [ @HW | @HT ] // (**) (* auto fails because δ-expansion gets in the way *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma index 08cf65476..abbc125a0 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma @@ -19,8 +19,8 @@ include "basic_2/rt_transition/cnx.ma". (* Advanced properties ******************************************************) -lemma cnx_tdeq_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → - ∀T2. T1 ≛ T2 → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T2⦄. +lemma cnx_tdeq_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → + ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T2⦄. #h #G #L #T1 #HT1 #T2 #HT12 #T #HT2 elim (tdeq_cpx_trans … HT12 … HT2) -HT2 #T0 #HT10 #HT0 lapply (HT1 … HT10) -HT1 -HT10 /2 width=5 by tdeq_repl/ (**) (* full auto fails *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_drops.ma index 3b396d94e..e532b364a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_drops.ma @@ -20,12 +20,12 @@ include "basic_2/rt_transition/cnx.ma". (* Properties with generic slicing ******************************************) -lemma cnx_lref_atom: ∀h,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄. +lemma cnx_lref_atom: ∀h,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄. #h #G #L #i #Hi #X #H elim (cpx_inv_lref1_drops … H) -H // * #I #K #V1 #V2 #HLK lapply (drops_mono … Hi … HLK) -L #H destruct qed. -lemma cnx_lref_unit: ∀h,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄. +lemma cnx_lref_unit: ∀h,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄. #h #I #G #L #K #i #HLK #X #H elim (cpx_inv_lref1_drops … H) -H // * #Z #Y #V1 #V2 #HLY lapply (drops_mono … HLK … HLY) -L #H destruct qed. @@ -40,7 +40,7 @@ qed-. (* Inversion lemmas with generic slicing ************************************) (* Basic_2A1: was: cnx_inv_delta *) -lemma cnx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄ → ⊥. +lemma cnx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄ → ⊥. #h #I #G #L #K #V #i #HLK #H elim (lifts_total V (𝐔❴↑i❵)) #W #HVW lapply (H W ?) -H /2 width=7 by cpx_delta_drops/ -HLK diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_simple.ma index e08892002..322930c7f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_simple.ma @@ -19,8 +19,8 @@ include "basic_2/rt_transition/cnx.ma". (* Inversion lemmas with simple terms ***************************************) -lemma cnx_inv_appl: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄ → - ∧∧ ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ & 𝐒⦃T⦄. +lemma cnx_inv_appl: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄ → + ∧∧ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ & 𝐒⦃T⦄. #h #G #L #V1 #T1 #HVT1 @and3_intro [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H elim (tdeq_inv_pair … H) -H // @@ -39,8 +39,8 @@ qed-. (* Properties with simple terms *********************************************) -lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → - ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄. +lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → + ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄. #h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct @tdeq_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma index 1ebd7f744..6d36b466c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma @@ -15,7 +15,7 @@ include "ground_2/steps/rtc_max.ma". include "ground_2/steps/rtc_plus.ma". include "basic_2/notation/relations/predty_7.ma". -include "static_2/syntax/item_sh.ma". +include "static_2/syntax/sort.ma". include "static_2/syntax/lenv.ma". include "static_2/syntax/genv.ma". include "static_2/relocation/lifts.ma". @@ -61,21 +61,21 @@ interpretation (* Basic properties *********************************************************) (* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *) -lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G, L⦄ ⊢ T ⬈[Rt, 𝟘𝟘, h] T. +lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G,L⦄ ⊢ T ⬈[Rt,𝟘𝟘,h] T. #Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/ * /2 width=1 by cpg_appl, cpg_cast/ qed. (* Basic inversion lemmas ***************************************************) -fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[Rt, c, h] T2 → ∀J. T1 = ⓪{J} → +fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪{J} → ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 - | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 & + | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I} & J = LRef (↑i). #Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2 [ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/ @@ -94,18 +94,18 @@ fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[Rt, c, h] T2 ] qed-. -lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[Rt, c, h] T2 → +lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[Rt,c,h] T2 → ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘 | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙 - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙 - | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 & + | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I} & J = LRef (↑i). /2 width=3 by cpg_inv_atom1_aux/ qed-. -lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[Rt, c, h] T2 → +lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[Rt,c,h] T2 → ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(next h s) ∧ c = 𝟘𝟙. #Rt #c #h #G #L #T2 #s #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ @@ -115,11 +115,11 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ ] qed-. -lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[Rt, c, h] T2 → +lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[Rt,c,h] T2 → ∨∨ T2 = #0 ∧ c = 𝟘𝟘 - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV - | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙. #Rt #c #h #G #L #T2 #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/ @@ -129,9 +129,9 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/ ] qed-. -lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[Rt, c, h] T2 → +lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[Rt,c,h] T2 → ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘 - | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. + | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. #Rt #c #h #G #L #T2 #i #H elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ [ #s #H destruct @@ -140,7 +140,7 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/ ] qed-. -lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[Rt, c, h] T2 → T2 = §l ∧ c = 𝟘𝟘. +lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘. #Rt #c #h #G #L #T2 #l #H elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/ [ #s #H destruct @@ -149,11 +149,11 @@ elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/ ] qed-. -fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 → +fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt, cT, h] T2 & + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & U2 = ⓑ{p,J}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 & + | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & p = true & J = Abbr & c = cT+𝟙𝟘. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #q #J #W #U1 #H destruct @@ -172,24 +172,24 @@ fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 ] qed-. -lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U2 = ⓑ{p,I}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 & + | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & p = true & I = Abbr & c = cT+𝟙𝟘. /2 width=3 by cpg_inv_bind1_aux/ qed-. -lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt, c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U2 = ⓓ{p}V2.T2 & c = ((↕*cV)∨cT) - | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 & + | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 & p = true & c = cT+𝟙𝟘. #Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * /3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/ qed-. -lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt, c, h] U2 → - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt,c,h] U2 → + ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U2 = ⓛ{p}V2.T2 & c = ((↕*cV)∨cT). #Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * [ /3 width=8 by ex4_4_intro/ @@ -197,13 +197,13 @@ lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt ] qed-. -fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 → +fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ∀V1,U1. U = ⓐV1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 & + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 - | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & + | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct @@ -222,21 +222,21 @@ fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 ] qed-. -lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ⬈[Rt, c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT) - | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & + | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘 - | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & + | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘. /2 width=3 by cpg_inv_appl1_aux/ qed-. -fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 → +fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 → ∀V1,U1. U = ⓝV1.U1 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 & + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) - | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘 - | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙. + | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2 [ #I #G #L #W #U1 #H destruct | #G #L #s #W #U1 #H destruct @@ -254,36 +254,36 @@ fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 ] qed-. -lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[Rt, c, h] U2 → - ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 → + ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 & Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT) - | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘 - | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙. + | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘 + | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙. /2 width=3 by cpg_inv_cast1_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt, c, h] T2 → +lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt,c,h] T2 → ∨∨ T2 = #0 ∧ c = 𝟘𝟘 - | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & I = Abbr & c = cV - | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙. #Rt #c #h #I #G #K #V1 #T2 #H elim (cpg_inv_zero1 … H) -H /2 width=1 by or3_intro0/ * #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/ qed-. -lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt, c, h] T2 → +lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt,c,h] T2 → ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘 - | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2. + | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2. #Rt #c #h #I #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/ * #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/ qed-. (* Basic forward lemmas *****************************************************) -lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt, c, h] T → ∀p. - ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] ⓑ{p,I}V2.T2 & +lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt,c,h] T → ∀p. + ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] ⓑ{p,I}V2.T2 & T = -ⓑ{I}V2.T2. #Rt #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H * [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma index 37ac3fa83..71d38a1ba 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma @@ -21,8 +21,8 @@ include "basic_2/rt_transition/cpg.ma". (* Advanced properties ******************************************************) -lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 → - ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2. +lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ⬈[Rt,c,h] V2 → + ⬆*[↑i] V2 ≘ T2 → ⦃G,L⦄ ⊢ #i ⬈[Rt,c,h] T2. #Rt #c #h #G #K #V #V2 #i elim i -i [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/ | #i #IH #L0 #T0 #H0 #HV2 #HVT2 @@ -31,8 +31,8 @@ lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, ] qed. -lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 → - ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2. +lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ⬈[Rt,c,h] V2 → + ⬆*[↑i] V2 ≘ T2 → ⦃G,L⦄ ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2. #Rt #c #h #G #K #V #V2 #i elim i -i [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/ | #i #IH #L0 #T0 #H0 #HV2 #HVT2 @@ -43,11 +43,11 @@ qed. (* Advanced inversion lemmas ************************************************) -lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 → +lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G,L⦄ ⊢ #i ⬈[Rt,c,h] T2 → ∨∨ T2 = #i ∧ c = 𝟘𝟘 - | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 & ⬆*[↑i] V2 ≘ T2 & c = cV - | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 & ⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙. #Rt #c #h #G #i elim i -i [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/ @@ -61,12 +61,12 @@ lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 ] qed-. -lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 → +lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ⬈[Rt,c,h] T2 → ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘 | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙 - | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV - | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 & + | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙. #Rt #c #h * #n #G #L #T2 #H [ elim (cpg_inv_sort1 … H) -H * diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma index 39228c247..7d4b85f06 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_simple.ma @@ -20,8 +20,8 @@ include "basic_2/rt_transition/cpg.ma". (* Properties with simple terms *********************************************) (* Note: the main property of simple terms *) -lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[Rt, c, h] U → 𝐒⦃T1⦄ → - ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[Rt, cT, h] T2 & +lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[Rt,c,h] U → 𝐒⦃T1⦄ → + ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[Rt,cT,h] T2 & U = ⓐV2.T2 & c = ((↕*cV)∨cT). #Rt #c #h #G #L #V1 #T1 #U #H #HT1 elim (cpg_inv_appl1 … H) -H * [ /2 width=8 by ex4_4_intro/ 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 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_aaa.ma index 8fa425fc5..b5a67ac4e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_aaa.ma @@ -25,7 +25,7 @@ lemma cpm_aaa_conf (n) (h): ∀G,L. Conf3 … (aaa G L) (cpm h G L n). (* Note: one of these U is the inferred type of T *) lemma aaa_cpm_SO (h) (G) (L) (A): - ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U. + ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U. #h #G #L #A #T #H elim H -G -L -T -A [ /3 width=2 by ex_intro/ | * #G #L #V #B #_ * #V0 #HV0 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_cpx.ma index 6f07b0647..8cbb8fc18 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_cpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_cpx.ma @@ -20,7 +20,7 @@ include "basic_2/rt_transition/cpm.ma". (* Forward lemmas with unbound context-sensitive rt-transition for terms ****) (* Basic_2A1: includes: cpr_cpx *) -lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2. +lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2. #n #h #G #L #T1 #T2 * #c #Hc #H elim H -L -T1 -T2 /2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_drops.ma index beddc2697..b6298c766 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_drops.ma @@ -50,27 +50,27 @@ qed-. (* Basic_1: includes: pr2_delta1 *) (* Basic_2A1: includes: cpr_delta *) lemma cpm_delta_drops: ∀n,h,G,L,K,V,V2,W2,i. - ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 → - ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2. + ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 → + ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[n,h] W2. #n #h #G #L #K #V #V2 #W2 #i #HLK * /3 width=8 by cpg_delta_drops, ex2_intro/ qed. lemma cpm_ell_drops: ∀n,h,G,L,K,V,V2,W2,i. - ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 → - ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[↑n, h] W2. + ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 → + ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[↑n,h] W2. #n #h #G #L #K #V #V2 #W2 #i #HLK * /3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/ qed. (* Advanced inversion lemmas ************************************************) -lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[n, h] T2 → +lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] T2 → ∨∨ T2 = ⓪{I} ∧ n = 0 | ∃∃s. T2 = ⋆(next h s) & I = Sort s & n = 1 - | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 & + | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i - | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 & + | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i & n = ↑m. #n #h #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H * [ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc @@ -85,11 +85,11 @@ lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[n, h] T2 ] qed-. -lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 → +lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[n,h] T2 → ∨∨ T2 = #i ∧ n = 0 - | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 & + | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 & ⬆*[↑i] V2 ≘ T2 - | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 & + | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 & ⬆*[↑i] V2 ≘ T2 & n = ↑m. #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H * [ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc @@ -104,9 +104,9 @@ qed-. (* Advanced forward lemmas **************************************************) -fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → +fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ∀n1,n2. n1+n2 = n → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2. + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2. #n #h #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n [ #I #G #L #n1 #n2 #H elim (plus_inv_O3 … H) -H #H1 #H2 destruct @@ -165,6 +165,6 @@ fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ] qed-. -lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n1+n2, h] T2 → - ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2. +lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n1+n2,h] T2 → + ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2. /2 width=3 by cpm_fwd_plus_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lsubr.ma index 29f43808c..7b4cde75a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lsubr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_lsubr.ma @@ -24,7 +24,7 @@ lemma lsubr_cpm_trans (n) (h) (G): lsub_trans … (λL. cpm h G L n) lsubr. #n #h #G #L1 #T1 #T2 * /3 width=5 by lsubr_cpg_trans, ex2_intro/ qed-. -lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ➡[n, h] T2 → - ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2. +lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ➡[n,h] T2 → + ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2. /4 width=4 by lsubr_cpm_trans, cpm_bind, lsubr_unit/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_simple.ma index 9a37802e6..b160f471c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm_simple.ma @@ -20,8 +20,8 @@ include "basic_2/rt_transition/cpm.ma". (* Properties with simple terms *********************************************) (* Basic_2A1: includes: cpr_inv_appl1_simple *) -lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] U → 𝐒⦃T1⦄ → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 & +lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] U → 𝐒⦃T1⦄ → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & U = ⓐV2.T2. #n #h #G #L #V1 #T1 #U * #c #Hc #H #HT1 elim (cpg_inv_appl1_simple … H HT1) -H -HT1 #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct elim (isrt_inv_max … Hc) -Hc diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma index fb1f479dd..9d984e384 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma @@ -22,24 +22,24 @@ include "basic_2/rt_transition/cpm.ma". (* Note: cpr_flat: does not hold in basic_1 *) (* Basic_1: includes: pr2_thin_dx *) lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2. - ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 → - ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2. + ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 → + ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2. #h * /2 width=1 by cpm_cast, cpm_appl/ qed. (* Basic_1: was: pr2_head_1 *) -lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → - ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T. +lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → + ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T. #h * /2 width=1 by cpm_bind, cpr_flat/ qed. (* Basic inversion properties ***********************************************) -lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 → +lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[h] T2 → ∨∨ T2 = ⓪{J} - | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 - | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & + | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I} & J = LRef (↑i). #h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H * [2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ] @@ -49,48 +49,48 @@ lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 → qed-. (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *) -lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s. +lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s. #h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H // qed-. -lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 → +lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[h] T2 → ∨∨ T2 = #0 - | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓓV1. #h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/ #n #K #V1 #V2 #_ #_ #_ #H destruct qed-. -lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[h] T2 → +lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[h] T2 → ∨∨ T2 = #(↑i) - | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. + | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. #h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/ qed-. -lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l. +lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[h] T2 → T2 = §l. #h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H // qed-. (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *) -lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 & +lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 & U2 = ⓝV2.T2 - | ⦃G, L⦄ ⊢ U1 ➡[h] U2. + | ⦃G,L⦄ ⊢ U1 ➡[h] U2. #h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H /2 width=1 by or_introl, or_intror/ * #n #_ #H destruct qed-. -lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 & +lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 & U2 = ⓕ{I}V2.T2 - | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast) - | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 & - ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 & + | (⦃G,L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast) + | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 & + ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl - | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 & - ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[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 ➡[h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl. #h * #G #L #V1 #U1 #U2 #H @@ -105,26 +105,26 @@ qed-. lemma cpr_ind (h): ∀Q:relation4 genv lenv term term. (∀I,G,L. Q G L (⓪{I}) (⓪{I})) → - (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 → + (∀G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 → ⬆*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2 - ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → Q G K (#i) T → + ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[h] T → Q G K (#i) T → ⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U) - ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 → + ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 → Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) - ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 → + ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[h] T2 → + ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[h] T2 → Q G L T T2 → Q G L (+ⓓV.T1) T2 - ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 → + ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 → Q G L (ⓝV.T1) T2 - ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 → + ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 → Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 → Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2) - ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 → + ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 → Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 → ⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2) ) → - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2. #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2 @(insert_eq_0 … 0) #n #H @(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ] diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops.ma index ba96c07fb..afd2d3fda 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops.ma @@ -19,9 +19,9 @@ include "basic_2/rt_transition/cpm_drops.ma". (* Advanced inversion lemmas ************************************************) (* Basic_2A1: includes: cpr_inv_atom1 *) -lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[h] T2 → +lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[h] T2 → ∨∨ T2 = ⓪{I} - | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 & + | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i. #h #I #G #L #T2 #H elim (cpm_inv_atom1_drops … H) -H * [ /2 width=1 by or_introl/ @@ -33,9 +33,9 @@ qed-. (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *) (* Basic_2A1: includes: cpr_inv_lref1 *) -lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 → +lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[h] T2 → ∨∨ T2 = #i - | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 & + | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 & ⬆*[↑i] V2 ≘ T2. #h #G #L #T2 #i #H elim (cpm_inv_lref1_drops … H) -H * [ /2 width=1 by or_introl/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops_basic.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops_basic.ma index 705031ffd..ab99dbfca 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops_basic.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr_drops_basic.ma @@ -22,7 +22,7 @@ include "basic_2/rt_transition/cpr.ma". lemma cpr_subst (h) (G) (L) (U1) (i): ∀K,V. ⬇*[i] L ≘ K.ⓓV → - ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ➡[h] U2 & ⬆[i,1] T2 ≘ U2. + ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ➡[h] U2 & ⬆[i,1] T2 ≘ U2. #h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1 #G0 #L0 #U0 #IH #G #L * * [ #s #HG #HL #HT #i #K #V #_ destruct -IH diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma index 3da67484a..e92d469a4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma @@ -18,7 +18,7 @@ include "basic_2/rt_transition/cpg.ma". (* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************) definition cpx (h): relation4 genv lenv term term ≝ - λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2. + λG,L,T1,T2. ∃c. ⦃G,L⦄ ⊢ T1 ⬈[eq_f,c,h] T2. interpretation "unbound context-sensitive parallel rt-transition (term)" @@ -27,64 +27,64 @@ interpretation (* Basic properties *********************************************************) (* Basic_2A1: was: cpx_st *) -lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s). +lemma cpx_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s). /2 width=2 by cpg_ess, ex_intro/ qed. -lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → - ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2. +lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → + ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2. #h * #G #K #V1 #V2 #W2 * /3 width=4 by cpg_delta, cpg_ell, ex_intro/ qed. -lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → - ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U. +lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → + ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U. #h #I #G #K #T #U #i * /3 width=4 by cpg_lref, ex_intro/ qed. lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 → - ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 → + ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2. #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 * /3 width=2 by cpg_bind, ex_intro/ qed. lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → - ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → + ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2. #h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 * /3 width=5 by cpg_appl, cpg_cast, ex_intro/ qed. lemma cpx_zeta (h) (G) (L): - ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 → - ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2. + ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 → + ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈[h] T2. #h #G #L #T1 #T #HT1 #T2 * /3 width=4 by cpg_zeta, ex_intro/ qed. -lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2. +lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ⬈[h] T2. #h #G #L #V #T1 #T2 * /3 width=2 by cpg_eps, ex_intro/ qed. (* Basic_2A1: was: cpx_ct *) -lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2. +lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ⬈[h] V2. #h #G #L #V1 #V2 #T * /3 width=2 by cpg_ee, ex_intro/ qed. lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2. - ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2. #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 * /3 width=2 by cpg_beta, ex_intro/ qed. lemma cpx_theta: ∀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 ⬈[h] T2 → - ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2. + ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → + ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 → + ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2. #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 * /3 width=4 by cpg_theta, ex_intro/ qed. @@ -95,8 +95,8 @@ lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L). (* Advanced properties ******************************************************) -lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T. +lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T. #h * /2 width=2 by cpx_flat, cpx_bind/ qed. @@ -108,115 +108,115 @@ qed. (* Basic inversion lemmas ***************************************************) -lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 → +lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[h] T2 → ∨∨ T2 = ⓪{J} | ∃∃s. T2 = ⋆(next h s) & J = Sort s - | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓑ{I}V1 & J = LRef 0 - | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & + | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I} & J = LRef (↑i). #h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H * /4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/ qed-. -lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 → +lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] T2 → ∨∨ T2 = ⋆s | T2 = ⋆(next h s). #h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H * /2 width=1 by or_introl, or_intror/ qed-. -lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 → +lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[h] T2 → ∨∨ T2 = #0 - | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 & + | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 & L = K.ⓑ{I}V1. #h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H * /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/ qed-. -lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[h] T2 → +lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[h] T2 → ∨∨ T2 = #(↑i) - | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. + | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}. #h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H * /4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/ qed-. -lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l. +lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[h] T2 → T2 = §l. #h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H // qed-. -lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 & +lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 & U2 = ⓑ{p,I}V2.T2 - | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & + | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 & p = true & I = Abbr. #h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H * /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/ qed-. -lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 & +lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 & U2 = ⓓ{p}V2.T2 - | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & p = true. + | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 & p = true. #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H * /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/ qed-. -lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 & +lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 & U2 = ⓛ{p}V2.T2. #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H /3 width=5 by ex3_2_intro, ex_intro/ qed-. -lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 & +lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[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 ⬈[h] T2 & + | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & + ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[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 ⬈[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 ⬈[h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2. #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H * /4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/ qed-. -lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 & +lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 & U2 = ⓝV2.T2 - | ⦃G, L⦄ ⊢ U1 ⬈[h] U2 - | ⦃G, L⦄ ⊢ V1 ⬈[h] U2. + | ⦃G,L⦄ ⊢ U1 ⬈[h] U2 + | ⦃G,L⦄ ⊢ V1 ⬈[h] U2. #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H * /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/ qed-. (* Advanced inversion lemmas ************************************************) -lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 → +lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 → ∨∨ T2 = #0 - | ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2. + | ∃∃V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2. #h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H * /4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/ qed-. -lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 → +lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 → ∨∨ T2 = #(↑i) - | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2. + | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2. #h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H * /4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/ qed-. -lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 → - ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 & +lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 → + ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 & U2 = ⓕ{I}V2.T2 - | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast) - | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast) - | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & - ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 & + | (⦃G,L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast) + | (⦃G,L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast) + | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & + ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl - | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 & - ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[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 ⬈[h] T2 & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl. #h * #G #L #V1 #U1 #U2 #H @@ -229,8 +229,8 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p. - ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 & +lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p. + ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 & T = -ⓑ{I}V2.T2. #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H /3 width=4 by ex2_2_intro, ex_intro/ @@ -241,28 +241,28 @@ qed-. lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term. (∀I,G,L. Q G L (⓪{I}) (⓪{I})) → (∀G,L,s. Q G L (⋆s) (⋆(next h s))) → - (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 → + (∀I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 → ⬆*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2 - ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T → + ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T → ⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U) - ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 → + ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 → Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2) - ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → + ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) - ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 → + ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 → Q G L (+ⓓV.T1) T2 - ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 → + ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 → Q G L (ⓝV.T1) T2 - ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 → + ) → (∀G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 → Q G L (ⓝV1.T) V2 - ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 → + ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 → Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 → Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2) - ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 → + ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 → Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 → ⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2) ) → - ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2. + ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2. #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2 * #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma index 4fcea0470..9f1350864 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops.ma @@ -21,8 +21,8 @@ include "basic_2/rt_transition/cpx.ma". (* Basic_2A1: was: cpx_delta *) lemma cpx_delta_drops: ∀h,I,G,L,K,V,V2,W2,i. - ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈[h] V2 → - ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈[h] W2. + ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⬈[h] V2 → + ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈[h] W2. #h * #G #L #K #V #V2 #W2 #i #HLK * /3 width=7 by cpg_ell_drops, cpg_delta_drops, ex_intro/ qed. @@ -30,19 +30,19 @@ qed. (* Advanced inversion lemmas ************************************************) (* Basic_2A1: was: cpx_inv_atom1 *) -lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] T2 → +lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ⬈[h] T2 → ∨∨ T2 = ⓪{I} | ∃∃s. T2 = ⋆(next h s) & I = Sort s - | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 & + | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 & ⬆*[↑i] V2 ≘ T2 & I = LRef i. #h #I #G #L #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H * /4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_intro, ex_intro/ qed-. (* Basic_2A1: was: cpx_inv_lref1 *) -lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈[h] T2 → +lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈[h] T2 → T2 = #i ∨ - ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 & + ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 & ⬆*[↑i] V2 ≘ T2. #h #G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H * /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops_basic.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops_basic.ma index 09b998bae..12d23e217 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops_basic.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_drops_basic.ma @@ -21,7 +21,7 @@ include "basic_2/rt_transition/cpx_drops.ma". lemma cpx_subst (h) (G) (L) (U1) (i): ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → - ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ⬈[h] U2 & ⬆[i,1] T2 ≘ U2. + ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ⬈[h] U2 & ⬆[i,1] T2 ≘ U2. #h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1 #G0 #L0 #U0 #IH #G #L * * [ #s #HG #HL #HT #i #I #K #V #_ destruct -IH diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fdeq.ma index 5a00dc678..f8224c247 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fdeq.ma @@ -20,9 +20,9 @@ include "basic_2/rt_transition/rpx_rdeq.ma". (* Properties with sort-irrelevant equivalence for closures *****************) -lemma fdeq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T⦄ → - ∀T2. ⦃G2, L2⦄ ⊢ T ⬈[h] T2 → - ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1, L1, T0⦄ ≛ ⦃G2, L2, T2⦄. +lemma fdeq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ → + ∀T2. ⦃G2,L2⦄ ⊢ T ⬈[h] T2 → + ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2 elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct elim (rdeq_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma index 0afb76e7f..ca625dd1b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.ma @@ -21,9 +21,9 @@ include "basic_2/rt_transition/cpx_lsubr.ma". (* Properties on supclosure *************************************************) -lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/ [ #I #G #L2 #V2 #X2 #HVX2 @@ -36,18 +36,18 @@ lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2 ] qed-. -lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/ | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/ ] qed-. -lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄. +lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2 /3 width=3 by fqu_fqup, ex2_intro/ @@ -57,18 +57,18 @@ lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, ] qed-. -lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄. +lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/ | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/ ] qed-. -lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵) #U2 #HVU2 @(ex3_intro … U2) @@ -98,9 +98,9 @@ lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃ ] qed-. -lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12 [ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqu_fquq, ex3_intro/ @@ -108,9 +108,9 @@ lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ] qed-. -lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄. +lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqu_fqup, ex3_intro/ @@ -120,9 +120,9 @@ lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ] qed-. -lemma fqus_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → - ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄. +lemma fqus_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) → + ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12 [ #H12 elim (fqup_cpx_trans_tdneq … H12 … HTU2 H) -T2 /3 width=4 by fqup_fqus, ex3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma index b11694680..7eb730a80 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_lsubr.ma @@ -23,8 +23,8 @@ lemma lsubr_cpx_trans (h) (G): lsub_trans … (cpx h G) lsubr. #h #G #L1 #T1 #T2 * /3 width=4 by lsubr_cpg_trans, ex_intro/ qed-. -lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 → - ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2. +lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 → + ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2. /4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_rdeq.ma index b90635d72..8e84ffab9 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_rdeq.ma @@ -24,6 +24,6 @@ lemma cpx_rdeq_conf_sn: ∀h,G. s_r_confluent1 … (cpx h G) rdeq. /3 width=6 by cpx_rex_conf/ qed-. (* Basic_2A1: was just: cpx_lleq_conf_dx *) -lemma cpx_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 → +lemma cpx_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 → ∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2. /4 width=5 by cpx_rdeq_conf_sn, rdeq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_req.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_req.ma index b0d2b9922..5f0c34f6c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_req.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_req.ma @@ -50,8 +50,8 @@ lemma req_cpx_trans: ∀h,G. req_transitive (cpx h G). qed-. (* (* Basic_2A1: was: cpx_lleq_conf *) -lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 → - ∀L1. L2 ≘[T1] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h] T2. +lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 → + ∀L1. L2 ≘[T1] L1 → ⦃G,L1⦄ ⊢ T1 ⬈[h] T2. /3 width=3 by req_cpx_trans, req_sym/ qed-. *) (* Basic_2A1: was: cpx_lleq_conf_sn *) @@ -59,7 +59,7 @@ lemma cpx_req_conf_sn: ∀h,G. s_r_confluent1 … (cpx h G) req. /2 width=5 by cpx_rex_conf/ qed-. (* (* Basic_2A1: was: cpx_lleq_conf_dx *) -lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 → +lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 → ∀L1. L1 ≘[T1] L2 → L1 ≘[T2] L2. /4 width=6 by cpx_req_conf_sn, req_sym/ qed-. *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma index a18e14b6c..1ce5b538d 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_simple.ma @@ -19,8 +19,8 @@ include "basic_2/rt_transition/cpx.ma". (* Inversion lemmas with simple terms ***************************************) -lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[h] T2 & +lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ → + ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[h] T2 & U = ⓐV2.T2. #h #G #L #V1 #T1 #U * #c #H #HT1 elim (cpg_inv_appl1_simple … H) -H /3 width=5 by ex3_2_intro, ex_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma index 4c262718f..5cfee0413 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb.ma @@ -20,9 +20,9 @@ include "basic_2/rt_transition/lpr_lpx.ma". (* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************) inductive fpb (h) (G1) (L1) (T1): relation3 genv lenv term ≝ -| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h G1 L1 T1 G2 L2 T2 -| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → fpb h G1 L1 T1 G1 L1 T2 -| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T1] L2 → ⊥) → fpb h G1 L1 T1 G1 L2 T1 +| fpb_fqu: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ → fpb h G1 L1 T1 G2 L2 T2 +| fpb_cpx: ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → fpb h G1 L1 T1 G1 L1 T2 +| fpb_lpx: ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T1] L2 → ⊥) → fpb h G1 L1 T1 G1 L2 T1 . interpretation @@ -32,10 +32,10 @@ interpretation (* Basic properties *********************************************************) (* Basic_2A1: includes: cpr_fpb *) -lemma cpm_fpb (n) (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≛ T2 → ⊥) → - ⦃G, L, T1⦄ ≻[h] ⦃G, L, T2⦄. +lemma cpm_fpb (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → (T1 ≛ T2 → ⊥) → + ⦃G,L,T1⦄ ≻[h] ⦃G,L,T2⦄. /3 width=2 by fpb_cpx, cpm_fwd_cpx/ qed. -lemma lpr_fpb (h) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → (L1 ≛[T] L2 → ⊥) → - ⦃G, L1, T⦄ ≻[h] ⦃G, L2, T⦄. +lemma lpr_fpb (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → (L1 ≛[T] L2 → ⊥) → + ⦃G,L1,T⦄ ≻[h] ⦃G,L2,T⦄. /3 width=1 by fpb_lpx, lpr_fwd_lpx/ qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_fdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_fdeq.ma index afa9995d0..a08244671 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_fdeq.ma @@ -21,9 +21,9 @@ include "basic_2/rt_transition/fpb_rdeq.ma". (* Properties with degree-based equivalence for closures ********************) (* Basic_2A1: uses: fleq_fpb_trans *) -lemma fdeq_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≛ ⦃F2, K2, T2⦄ → - ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h] ⦃G2, L2, U2⦄ → - ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≛ ⦃G2, L2, U2⦄. +lemma fdeq_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≛ ⦃F2,K2,T2⦄ → + ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ → + ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄. #h #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2 #K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12 elim (tdeq_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0 @@ -35,8 +35,8 @@ qed-. (* Inversion lemmas with degree-based equivalence for closures **************) (* Basic_2A1: uses: fpb_inv_fleq *) -lemma fpb_inv_fdeq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥. +lemma fpb_inv_fdeq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥. #h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 [ #G2 #L2 #T2 #H12 #H elim (fdeq_inv_gen_sn … H) -H /3 width=11 by rdeq_fwd_length, fqu_inv_tdeq/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.ma index 2546dd37e..b09005faf 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.ma @@ -22,8 +22,8 @@ include "basic_2/rt_transition/fpb.ma". (* Properties with sort-irrelevant equivalence for local environments *******) lemma tdeq_fpb_trans: ∀h,U2,U1. U2 ≛ U1 → - ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h] ⦃G2, L2, T1⦄ → - ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2. + ∀G1,G2,L1,L2,T1. ⦃G1,L1,U1⦄ ≻[h] ⦃G2,L2,T1⦄ → + ∃∃L,T2. ⦃G1,L1,U2⦄ ≻[h] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2. #h #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1 [ #G2 #L2 #T1 #H elim (tdeq_fqu_trans … H … HU21) -H @@ -37,8 +37,8 @@ qed-. (* Basic_2A1: was just: lleq_fpb_trans *) lemma rdeq_fpb_trans: ∀h,F,K1,K2,T. K1 ≛[T] K2 → - ∀G,L2,U. ⦃F, K2, T⦄ ≻[h] ⦃G, L2, U⦄ → - ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h] ⦃G, L1, U0⦄ & U0 ≛ U & L1 ≛[U] L2. + ∀G,L2,U. ⦃F,K2,T⦄ ≻[h] ⦃G,L2,U⦄ → + ∃∃L1,U0. ⦃F,K1,T⦄ ≻[h] ⦃G,L1,U0⦄ & U0 ≛ U & L1 ≛[U] L2. #h #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U [ #G #L2 #U #H2 elim (rdeq_fqu_trans … H2 … HT) -K2 /3 width=5 by fpb_fqu, ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq.ma index 9c29aca68..c50a6b2b4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq.ma @@ -21,10 +21,10 @@ include "basic_2/rt_transition/lpr_lpx.ma". (* Basic_2A1: includes: fleq_fpbq fpbq_lleq *) inductive fpbq (h) (G1) (L1) (T1): relation3 genv lenv term ≝ -| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h G1 L1 T1 G2 L2 T2 -| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h G1 L1 T1 G1 L1 T2 -| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h G1 L1 T1 G1 L2 T1 -| fpbq_fdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → fpbq h G1 L1 T1 G2 L2 T2 +| fpbq_fquq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2 +| fpbq_cpx : ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h G1 L1 T1 G1 L1 T2 +| fpbq_lpx : ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → fpbq h G1 L1 T1 G1 L2 T1 +| fpbq_fdeq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2 . interpretation @@ -37,10 +37,10 @@ lemma fpbq_refl (h): tri_reflexive … (fpbq h). /2 width=1 by fpbq_cpx/ qed. (* Basic_2A1: includes: cpr_fpbq *) -lemma cpm_fpbq (n) (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h] ⦃G, L, T2⦄. +lemma cpm_fpbq (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L,T1⦄ ≽[h] ⦃G,L,T2⦄. /3 width=2 by fpbq_cpx, cpm_fwd_cpx/ qed. -lemma lpr_fpbq (h) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1, T⦄ ≽[h] ⦃G, L2, T⦄. +lemma lpr_fpbq (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1,T⦄ ≽[h] ⦃G,L2,T⦄. /3 width=1 by fpbq_lpx, lpr_fwd_lpx/ qed. (* Basic_2A1: removed theorems 2: diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_aaa.ma index b12379dca..a29292aae 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_aaa.ma @@ -21,8 +21,8 @@ include "basic_2/rt_transition/fpbq.ma". (* Properties with atomic arity assignment for terms ************************) -lemma fpbq_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma fpbq_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_fdeq_conf, aaa_fquq_conf, ex_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_fpb.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_fpb.ma index 0f83111ac..21143003b 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_fpb.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpbq_fpb.ma @@ -19,23 +19,23 @@ include "basic_2/rt_transition/fpbq.ma". (* Properties with proper parallel rst-transition for closures **************) -lemma fpb_fpbq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄. +lemma fpb_fpbq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/ qed. (* Basic_2A1: fpb_fpbq_alt *) -lemma fpb_fpbq_ffdneq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄ → - ∧∧ ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥). +lemma fpb_fpbq_ffdneq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → + ∧∧ ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ & (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥). /3 width=10 by fpb_fpbq, fpb_inv_fdeq, conj/ qed-. (* Inversrion lemmas with proper parallel rst-transition for closures *******) (* Basic_2A1: uses: fpbq_ind_alt *) -lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → - ∨∨ ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ - | ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄. +lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → + ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ + | ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 [ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ] /3 width=1 by fpb_fqu, fdeq_intro_sn, or_intror, or_introl/ @@ -48,8 +48,8 @@ lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T qed-. (* Basic_2A1: fpbq_inv_fpb_alt *) -lemma fpbq_ffdneq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h] ⦃G2, L2, T2⦄ → - (⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h] ⦃G2, L2, T2⦄. +lemma fpbq_ffdneq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → + (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥) → ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄. #h #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr.ma index f69cb5e67..f47d6640e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr.ma @@ -27,8 +27,8 @@ interpretation (* Basic properties *********************************************************) -lemma lpr_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 → - ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ➡[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}. +lemma lpr_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 → + ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ➡[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}. /2 width=1 by lex_bind/ qed. (* Note: lemma 250 *) @@ -37,60 +37,60 @@ lemma lpr_refl (h) (G): reflexive … (lpr h G). (* Advanced properties ******************************************************) -lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 → - ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}. +lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 → + ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}. /2 width=1 by lex_bind_refl_dx/ qed. -lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h] V2 → - ∀I. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2. +lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡[h] V2 → + ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2. /2 width=1 by lex_pair/ qed. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: was: lpr_inv_atom1 *) (* Basic_1: includes: wcpr0_gen_sort *) -lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆. +lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆. /2 width=2 by lex_inv_atom_sn/ qed-. -lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 → - ∃∃I2,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 & +lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 → + ∃∃I2,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 & L2 = K2.ⓘ{I2}. /2 width=1 by lex_inv_bind_sn/ qed-. (* Basic_2A1: was: lpr_inv_atom2 *) -lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆. +lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆. /2 width=2 by lex_inv_atom_dx/ qed-. -lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓘ{I2} → - ∃∃I1,K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 & +lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓘ{I2} → + ∃∃I1,K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 & L1 = K1.ⓘ{I1}. /2 width=1 by lex_inv_bind_dx/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ➡[h] L2 → - ∃∃K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}. +lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ➡[h] L2 → + ∃∃K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}. /2 width=1 by lex_inv_unit_sn/ qed-. (* Basic_2A1: was: lpr_inv_pair1 *) (* Basic_1: includes: wcpr0_gen_head *) -lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 → - ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 & +lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 → + ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 & L2 = K2.ⓑ{I}V2. /2 width=1 by lex_inv_pair_sn/ qed-. -lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓤ{I} → - ∃∃K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}. +lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓤ{I} → + ∃∃K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}. /2 width=1 by lex_inv_unit_dx/ qed-. (* Basic_2A1: was: lpr_inv_pair2 *) -lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 → - ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 & +lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 → + ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 & L1 = K1.ⓑ{I}V1. /2 width=1 by lex_inv_pair_dx/ qed-. -lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 → - ∧∧ ⦃G, L1⦄ ⊢ ➡[h] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2. +lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 → + ∧∧ ⦃G,L1⦄ ⊢ ➡[h] L2 & ⦃G,L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2. /2 width=1 by lex_inv_pair/ qed-. (* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_fquq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_fquq.ma index e1256efc5..031de39f8 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_fquq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_fquq.ma @@ -22,9 +22,9 @@ include "basic_2/rt_transition/lpr.ma". (* Properties with extended structural successor for closures ***************) -lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 → - ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 → + ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/ | /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/ @@ -37,9 +37,9 @@ lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ] qed-. -lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 → - ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄. +lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 → + ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/ | /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/ @@ -52,9 +52,9 @@ lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ] qed-. -lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄. +lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 → + ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ /3 width=5 by lpr_bind_refl_dx, fqu_lref_O, ex3_2_intro/ | /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/ @@ -71,9 +71,9 @@ qed-. (* Note: does not hold in Basic_2A1 because it requires cpm *) (* Note: L1 = K0.ⓛV0 and T1 = #0 require n = 1 *) -lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 → - ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1. +lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 → + ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ * #G #K #V #K1 #H elim (lpr_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct @@ -91,36 +91,36 @@ qed-. (* Properties with extended optional structural successor for closures ******) -lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 → - ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 → + ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H [ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ ] qed-. -lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 → - ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄. +lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 → + ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⊐⸮[b] ⦃G2,L2,U2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H [ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ ] qed-. -lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄. +lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 → + ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H [ #H12 elim (fqu_lpr_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ ] qed-. -lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 → - ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1. +lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 → + ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H [ #H12 elim (lpr_fqu_trans … H12 … HKL1) -L1 /3 width=7 by fqu_fquq, ex4_3_intro/ | * #H1 #H2 #H3 destruct /2 width=7 by ex4_3_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_length.ma index 51f2da5f2..db3c8e334 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_length.ma @@ -17,5 +17,5 @@ include "basic_2/rt_transition/lpr.ma". (* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************) -lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|. +lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|. /2 width=2 by lex_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma index 7dedda628..312265d54 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma @@ -21,24 +21,24 @@ include "basic_2/rt_transition/lpr_drops.ma". (* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************) definition IH_cpr_conf_lpr (h): relation3 genv lenv term ≝ λG,L,T. - ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → - ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → - ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0. + ∀T1. ⦃G,L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → + ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → + ∃∃T0. ⦃G,L1⦄ ⊢ T1 ➡[h] T0 & ⦃G,L2⦄ ⊢ T2 ➡[h] T0. (* Main properties with context-sensitive parallel reduction for terms ******) fact cpr_conf_lpr_atom_atom (h): - ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T. + ∀I,G,L1,L2. ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡[h] T. /2 width=3 by cpr_refl, ex2_intro/ qed-. fact cpr_conf_lpr_atom_delta (h): ∀G0,L0,i. ( - ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,#i⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 → - ∀V2. ⦃G0, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ #i ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T. + ∀V2. ⦃G0,K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ #i ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T. #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1 elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct @@ -54,14 +54,14 @@ qed-. (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *) fact cpr_conf_lpr_delta_delta (h): ∀G0,L0,i. ( - ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,#i⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 → - ∀V1. ⦃G0, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 → + ∀V1. ⦃G0,K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 → ∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX → - ∀V2. ⦃G0, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T. + ∀V2. ⦃G0,KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T. #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 lapply (drops_mono … H … HLK0) -H #H destruct @@ -79,12 +79,12 @@ qed-. fact cpr_conf_lpr_bind_bind (h): ∀p,I,G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T. #h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HV01 … HV02 … HL01 … HL02) // @@ -94,12 +94,12 @@ qed-. fact cpr_conf_lpr_bind_zeta (h): ∀G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → - ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → + ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T. #h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02 elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01 [| /3 width=1 by drops_refl, drops_drop/ ] #T #HT1 #HT2 @@ -109,12 +109,12 @@ qed-. fact cpr_conf_lpr_zeta_zeta (h): ∀G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0, L0⦄ ⊢ T1 ➡[h] X1 → - ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ X1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T. + ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0,L0⦄ ⊢ T1 ➡[h] X1 → + ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ X1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T. #h #G0 #L0 #V0 #T0 #IH #T1 #HT10 #X1 #HTX1 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02 lapply (lifts_inj … HT20 … HT10) -HT20 #H destruct @@ -124,12 +124,12 @@ qed-. fact cpr_conf_lpr_flat_flat (h): ∀I,G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓕ{I}V0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T. #h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HV01 … HV02 … HL01 … HL02) // @@ -139,11 +139,11 @@ qed-. fact cpr_conf_lpr_flat_eps (h): ∀G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1,T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T. + ∀V1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T. #h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 @@ -152,11 +152,11 @@ qed-. fact cpr_conf_lpr_eps_eps (h): ∀G0,L0,V0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T. + ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T. #h #G0 #L0 #V0 #T0 #IH #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 @@ -165,12 +165,12 @@ qed-. fact cpr_conf_lpr_flat_beta (h): ∀p,G0,L0,V0,W0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct @@ -187,13 +187,13 @@ qed-. *) fact cpr_conf_lpr_flat_theta (h): ∀p,G0,L0,V0,W0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 → - ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 → + ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 @@ -212,12 +212,12 @@ qed-. fact cpr_conf_lpr_beta_beta (h): ∀p,G0,L0,V0,W0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T. #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 @@ -231,14 +231,14 @@ qed-. (* Basic_1: was: pr0_upsilon_upsilon *) fact cpr_conf_lpr_theta_theta (h): ∀p,G0,L0,V0,W0,T0. ( - ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T + ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T ) → - ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 → - ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 → - ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 → - ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → - ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 → - ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. + ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 → + ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 → + ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 → + ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 → + ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 → + ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T. #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2 @@ -308,15 +308,15 @@ qed-. (* Properties with context-sensitive parallel reduction for terms ***********) -lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T. +lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → + ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L1⦄ ⊢ T1 ➡[h] T. #h #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01 /2 width=3 by ex2_intro/ qed-. -lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → - ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T. +lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → + ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L0⦄ ⊢ T1 ➡[h] T. #h #G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01 /2 width=3 by ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpx.ma index 484ae046f..6cc94e31f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpx.ma @@ -21,5 +21,5 @@ include "basic_2/rt_transition/lpr.ma". (* Forward lemmas with unbound parallel rt-transition for ref local envs ****) (* Basic_2A1: was: lpr_lpx *) -lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h] L2. +lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h] L2. /3 width=3 by cpm_fwd_cpx, lex_co/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma index ae782872c..14d34b21a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx.ma @@ -27,8 +27,8 @@ interpretation (* Basic properties *********************************************************) -lemma lpx_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → - ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}. +lemma lpx_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 → + ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}. /2 width=1 by lex_bind/ qed. lemma lpx_refl (h) (G): reflexive … (lpx h G). @@ -36,56 +36,56 @@ lemma lpx_refl (h) (G): reflexive … (lpx h G). (* Advanced properties ******************************************************) -lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → - ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}. +lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 → + ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}. /2 width=1 by lex_bind_refl_dx/ qed. -lemma lpx_pair (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 → - ∀I.⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2. +lemma lpx_pair (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 → + ∀I.⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2. /2 width=1 by lex_pair/ qed. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: was: lpx_inv_atom1 *) -lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆. +lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆. /2 width=2 by lex_inv_atom_sn/ qed-. -lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 → - ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 & +lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 → + ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 & L2 = K2.ⓘ{I2}. /2 width=1 by lex_inv_bind_sn/ qed-. (* Basic_2A1: was: lpx_inv_atom2 *) -lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆. +lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G,L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆. /2 width=2 by lex_inv_atom_dx/ qed-. -lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} → - ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 & +lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} → + ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 & L1 = K1.ⓘ{I1}. /2 width=1 by lex_inv_bind_dx/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 → - ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}. +lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 → + ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}. /2 width=1 by lex_inv_unit_sn/ qed-. (* Basic_2A1: was: lpx_inv_pair1 *) -lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 → - ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 & +lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 → + ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 & L2 = K2.ⓑ{I}V2. /2 width=1 by lex_inv_pair_sn/ qed-. -lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓤ{I} → - ∃∃K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}. +lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓤ{I} → + ∃∃K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}. /2 width=1 by lex_inv_unit_dx/ qed-. (* Basic_2A1: was: lpx_inv_pair2 *) -lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 → - ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 & +lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 → + ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 & L1 = K1.ⓑ{I}V1. /2 width=1 by lex_inv_pair_dx/ qed-. -lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 → - ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2. +lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 → + ∧∧ ⦃G,L1⦄ ⊢ ⬈[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2. /2 width=1 by lex_inv_pair/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma index 85ff65ecc..6e2319f19 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_aaa.ma @@ -22,9 +22,9 @@ include "basic_2/rt_transition/lpx_drops.ma". (* Note: lemma 500 *) (* Basic_2A1: was: cpx_lpx_aaa_conf *) -lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → - ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 → - ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A. +lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → + ∀T2. ⦃G,L1⦄ ⊢ T1 ⬈[h] T2 → + ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A. #h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A [ #G #L1 #s #X #H elim (cpx_inv_sort1 … H) -H #H destruct // diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma index 11ae99e7d..e6393d4ae 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fquq.ma @@ -19,9 +19,9 @@ include "basic_2/rt_transition/lpx.ma". (* Properties with extended structural successor for closures ***************) -lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2. +lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 → + ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #K #V #K1 #H elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct @@ -37,9 +37,9 @@ lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2 ] qed-. -lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄. +lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 → + ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐[b] ⦃G2,K2,T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/ | /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/ @@ -56,18 +56,18 @@ qed-. (* Properties with extended optional structural successor for closures ******) -lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 → - ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2. +lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 → + ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H [ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ ] qed-. -lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 → - ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄. +lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 → + ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⊐⸮[b] ⦃G2,K2,T2⦄. #h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H [ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/ | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fsle.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fsle.ma index f40b6b627..ed3144625 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fsle.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_fsle.ma @@ -19,10 +19,10 @@ include "basic_2/rt_transition/rpx_lpx.ma". (* Forward lemmas with free variables inclusion for restricted closures *****) (* Basic_2A1: uses: lpx_cpx_frees_trans *) -lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 → - ∀L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄. +lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 → + ∀L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄. /3 width=4 by rpx_cpx_conf_fsge, lpx_rpx/ qed-. (* Basic_2A1: uses: lpx_frees_trans *) -lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄. +lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T0⦄ ⊆ ⦃L0,T0⦄. /2 width=4 by lpx_cpx_conf_fsge/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_length.ma index 29b3e5e46..1c416b934 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_length.ma @@ -19,5 +19,5 @@ include "basic_2/rt_transition/lpx.ma". (* Forward lemmas with length for local environments ************************) -lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|. +lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|. /2 width=2 by lex_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_rdeq.ma index 26fd8c129..8055506a6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_rdeq.ma @@ -21,9 +21,9 @@ include "basic_2/rt_transition/rpx_lpx.ma". (* Properties with sort-irrelevant equivalence for local environments *******) (* Basic_2A1: uses: lleq_lpx_trans *) -lemma rdeq_lpx_trans (h) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 → +lemma rdeq_lpx_trans (h) (G): ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈[h] K2 → ∀L1. ∀T:term. L1 ≛[T] L2 → - ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[T] K2. + ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h] K1 & K1 ≛[T] K2. #h #G #L2 #K2 #HLK2 #L1 #T #HL12 lapply (lpx_rpx … T HLK2) -HLK2 #HLK2 elim (rdeq_rpx_trans … HLK2 … HL12) -L2 #K #H #HK2 diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma index f45abc6d7..70351ad4f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma @@ -27,111 +27,111 @@ interpretation (* Basic properties ***********************************************************) -lemma rpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆. +lemma rpx_atom: ∀h,I,G. ⦃G,⋆⦄ ⊢ ⬈[h,⓪{I}] ⋆. /2 width=1 by rex_atom/ qed. lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s. - ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}. + ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2}. /2 width=1 by rex_sort/ qed. lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2. - ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2. + ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 → ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2. /2 width=1 by rex_pair/ qed. lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i. - ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}. + ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2}. /2 width=1 by rex_lref/ qed. lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l. - ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}. + ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2}. /2 width=1 by rex_gref/ qed. lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T. - ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} → - ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 → - ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}. + ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I1} → + ∀I2. ⦃G,L1⦄ ⊢ I ⬈[h] I2 → + ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I2}. /2 width=2 by rex_bind_repl_dx/ qed-. (* Basic inversion lemmas ***************************************************) -lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆. +lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G,⋆⦄ ⊢ ⬈[h,T] Y2 → Y2 = ⋆. /2 width=3 by rex_inv_atom_sn/ qed-. -lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆. +lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G,Y1⦄ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆. /2 width=3 by rex_inv_atom_dx/ qed-. -lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 → +lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & + | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. /2 width=1 by rex_inv_sort/ qed-. -lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 → +lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & + | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. /2 width=1 by rex_inv_lref/ qed-. -lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 → +lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & + | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. /2 width=1 by rex_inv_gref/ qed-. -lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → - ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V. +lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → + ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V. /2 width=2 by rex_inv_bind/ qed-. -lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → - ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2. +lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 → + ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1⦄ ⊢ ⬈[h,T] L2. /2 width=2 by rex_inv_flat/ qed-. (* Advanced inversion lemmas ************************************************) -lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 → - ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}. +lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] Y2 → + ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y2 = L2.ⓘ{I2}. /2 width=2 by rex_inv_sort_bind_sn/ qed-. -lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} → - ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}. +lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2} → + ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ{I1}. /2 width=2 by rex_inv_sort_bind_dx/ qed-. -lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 → - ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & +lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] Y2 → + ∃∃L2,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & Y2 = L2.ⓑ{I}V2. /2 width=1 by rex_inv_zero_pair_sn/ qed-. -lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 → - ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & +lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2 → + ∃∃L1,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & Y1 = L1.ⓑ{I}V1. /2 width=1 by rex_inv_zero_pair_dx/ qed-. -lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 → - ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}. +lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] Y2 → + ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y2 = L2.ⓘ{I2}. /2 width=2 by rex_inv_lref_bind_sn/ qed-. -lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} → - ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}. +lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2} → + ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ{I1}. /2 width=2 by rex_inv_lref_bind_dx/ qed-. -lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 → - ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}. +lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] Y2 → + ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y2 = L2.ⓘ{I2}. /2 width=2 by rex_inv_gref_bind_sn/ qed-. -lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} → - ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}. +lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2} → + ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ{I1}. /2 width=2 by rex_inv_gref_bind_dx/ qed-. (* Basic forward lemmas *****************************************************) lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T. - ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2. + ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,V] L2. /2 width=3 by rex_fwd_pair_sn/ qed-. lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T. - ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V. + ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V. /2 width=2 by rex_fwd_bind_dx/ qed-. lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T. - ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2. + ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2. /2 width=3 by rex_fwd_flat_dx/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fqup.ma index 8a71e9b6a..43f6f81b2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fqup.ma @@ -22,18 +22,18 @@ include "basic_2/rt_transition/rpx.ma". lemma rpx_refl: ∀h,G,T. reflexive … (rpx h G T). /2 width=1 by rex_refl/ qed. -lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → - ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2. +lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → + ∀I,T. ⦃G,L.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L.ⓑ{I}V2. /2 width=1 by rex_pair_refl/ qed. (* Advanced inversion lemmas ************************************************) -lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → - ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ. +lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → + ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ. /2 width=3 by rex_inv_bind_void/ qed-. (* Advanced forward lemmas **************************************************) lemma rpx_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T. - ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ. + ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ. /2 width=4 by rex_fwd_bind_dx_void/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fsle.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fsle.ma index fd4ddc459..6d9db8b65 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fsle.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_fsle.ma @@ -24,9 +24,9 @@ include "basic_2/rt_transition/rpx_fqup.ma". (* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *) (* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *) -(* Note: This invalidates rpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (rpxs h G)" *) -lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 → - ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄. +(* Note: This invalidates rpxs_cpx_conf: "∀h, G. s_r_confluent1 … (cpx h G) (rpxs h G)" *) +lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 → + ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄. #h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0 #G #L #T #IH #G0 #L0 * * [ #s #HG #HL #HT #X #HX #Y #HY destruct -IH @@ -133,6 +133,6 @@ lemma cpx_rex_conf (R) (h) (G): s_r_confluent1 … (cpx h G) (rex R). lemma rpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (rpx h G). /2 width=5 by cpx_rex_conf/ qed-. -lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 → - ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄. +lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 → + ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T1⦄. /3 width=5 by rpx_cpx_conf, rpx_fsge_comp/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_length.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_length.ma index 209b010ed..45a052c67 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_length.ma @@ -19,15 +19,15 @@ include "basic_2/rt_transition/rpx.ma". (* Forward lemmas with length for local environments ************************) -lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|. +lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → |L1| = |L2|. /2 width=3 by rex_fwd_length/ qed-. (* Inversion lemmas with length for local environments **********************) -lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 → +lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] Y2 → ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & - ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & + | ∃∃I,L1,L2,V1,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & + ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}. /2 width=1 by rex_inv_zero_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_lpx.ma index 24e840b1f..ba4a29ec5 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_lpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_lpx.ma @@ -20,16 +20,16 @@ include "basic_2/rt_transition/lpx.ma". (* Properties with syntactic equivalence for referred local environments ****) -lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2. +lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2. /2 width=1 by req_fwd_rex/ qed. (* Properties with unbound parallel rt-transition for full local envs *******) -lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2. +lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2. /2 width=1 by rex_lex/ qed. (* Inversion lemmas with unbound parallel rt-transition for full local envs *) -lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → - ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2. +lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → + ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h] L & L ≡[T] L2. /3 width=3 by rpx_fsge_comp, rex_inv_lex_req/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rdeq.ma index cb3c4b7a5..602285337 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rdeq.ma @@ -21,27 +21,27 @@ include "basic_2/rt_transition/rpx_fsle.ma". (* Properties with sort-irrelevant equivalence for local environments *******) -lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀I,T. - ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[V] L2. +lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → ∀I,T. + ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L & L ≛[V] L2. /3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-. -lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀I,V. - ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[T] L2. +lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ∀I,V. + ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L & L ≛[T] L2. /3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-. -lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀p. - ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V. +lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2 → ∀p. + ∃∃L,V. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V. /3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-. -lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀p,I,V. - ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2. +lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G,K1.ⓧ⦄ ⊢ ⬈[h,T] L2 → ∀p,I,V. + ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2. /3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-. lemma rpx_tdeq_conf: ∀h,G. s_r_confluent1 … cdeq (rpx h G). /2 width=5 by tdeq_rex_conf/ qed-. lemma rpx_tdeq_div: ∀h,T1,T2. T1 ≛ T2 → - ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2. + ∀G,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,T2] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T1] L2. /2 width=5 by tdeq_rex_div/ qed-. lemma cpx_tdeq_conf_sex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq. @@ -125,25 +125,25 @@ lemma cpx_tdeq_conf_sex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) c ] qed-. -lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 → +lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 → ∀T2. T0 ≛ T2 → - ∃∃T. T1 ≛ T & ⦃G, L⦄ ⊢ T2 ⬈[h] T. + ∃∃T. T1 ≛ T & ⦃G,L⦄ ⊢ T2 ⬈[h] T. #h #G #L #T0 #T1 #HT01 #T2 #HT02 elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02 /2 width=3 by rex_refl, ex2_intro/ qed-. lemma tdeq_cpx_trans: ∀h,G,L,T2. ∀T0:term. T2 ≛ T0 → - ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 → - ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛ T1. + ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 → + ∃∃T. ⦃G,L⦄ ⊢ T2 ⬈[h] T & T ≛ T1. #h #G #L #T2 #T0 #HT20 #T1 #HT01 elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/ qed-. (* Basic_2A1: uses: cpx_lleq_conf *) -lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 → +lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 → ∀L2. L0 ≛[T0] L2 → - ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T. + ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T. #h #G #L0 #T0 #T1 #HT01 #L2 #HL02 elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02 /2 width=3 by rex_refl, ex2_intro/ @@ -151,8 +151,8 @@ qed-. (* Basic_2A1: uses: lleq_cpx_trans *) lemma rdeq_cpx_trans: ∀h,G,L2,L0,T0. L2 ≛[T0] L0 → - ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 → - ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1. + ∀T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 → + ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1. #h #G #L2 #L0 #T0 #HL20 #T1 #HT01 elim (cpx_rdeq_conf … HT01 L2) -HT01 /3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/ @@ -161,9 +161,9 @@ qed-. lemma rpx_rdeq_conf: ∀h,G,T. confluent2 … (rpx h G T) (rdeq T). /3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-. -lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 → +lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G,L2⦄ ⊢ ⬈[h,T] K2 → ∀L1. L1 ≛[T] L2 → - ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[T] K2. + ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h,T] K1 & K1 ≛[T] K2. #h #G #T #L2 #K2 #HLK2 #L1 #HL12 elim (rpx_rdeq_conf … HLK2 L1) /3 width=3 by rdeq_sym, ex2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rpx.ma index 56899e45d..8b6716c10 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rpx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_rpx.ma @@ -19,16 +19,16 @@ include "basic_2/rt_transition/rpx.ma". (* Main properties **********************************************************) -theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → - ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2 → - ∀p. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L2. +theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 → + ∀I,V2,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V2 → + ∀p. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L2. /2 width=2 by rex_bind/ qed. -theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → - ∀I,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2. +theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → + ∀I,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2. /2 width=1 by rex_flat/ qed. -theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → - ∀T. ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ → - ∀p,I. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2. +theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → + ∀T. ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ → + ∀p,I. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2. /2 width=1 by rex_bind_void/ qed. diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/basic_2.ma b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/basic_2.ma index f18cd54f3..5a122b8ec 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/basic_2.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/basic_2.ma @@ -14,6 +14,6 @@ (* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) -notation "hvbox( 𝐁❴ break term 46 l, break term 46 h ❵ )" +notation "hvbox( 𝐁❴ term 46 l, break term 46 h ❵ )" non associative with precedence 90 for @{ 'Basic $l $h }. diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/cocompose_2.ma b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/cocompose_2.ma index 32741b07b..992179f98 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/cocompose_2.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/cocompose_2.ma @@ -14,6 +14,6 @@ (* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) -notation "hvbox(f2 ~ \circ break f1)" +notation "hvbox(f2 ~ \circ break f1)" (**) right associative with precedence 60 for @{ 'CoCompose $f2 $f1 }. diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/droppreds_2.ma b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/droppreds_2.ma index 1b10938b7..d74010073 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/droppreds_2.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/droppreds_2.ma @@ -14,6 +14,6 @@ (* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) -notation "hvbox( ⫱ * [ term 46 n ] term 46 T )" +notation "hvbox( ⫱ * [ term 46 n ] break term 46 T )" non associative with precedence 46 for @{ 'DropPreds $n $T }. diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/uparrowstar_2.ma b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/uparrowstar_2.ma index 553c4e494..a0ab933d8 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/uparrowstar_2.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/uparrowstar_2.ma @@ -14,6 +14,6 @@ (* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) -notation "hvbox( ↑ * [ term 46 n ] term 70 T )" +notation "hvbox( ↑ * [ term 46 n ] break term 70 T )" non associative with precedence 70 for @{ 'UpArrowStar $n $T }. diff --git a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/upspoonstar_2.ma b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/upspoonstar_2.ma index 12beff4ec..de75ff051 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/notation/functions/upspoonstar_2.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/notation/functions/upspoonstar_2.ma @@ -14,6 +14,6 @@ (* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) -notation "hvbox( ⫯ * [ term 46 n ] term 46 T )" +notation "hvbox( ⫯ * [ term 46 n ] break term 46 T )" non associative with precedence 46 for @{ 'UpSpoonStar $n $T }. diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_plus.ma index e1ef5d287..94c561032 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_plus.ma @@ -18,7 +18,7 @@ include "ground_2/relocation/mr2.ma". rec definition pluss (cs:mr2) (i:nat) on cs ≝ match cs with [ nil2 ⇒ ◊ -| cons2 l m cs ⇒ {l + i, m};pluss cs i +| cons2 l m cs ⇒ {l + i,m};pluss cs i ]. interpretation "plus (multiple relocation with pairs)" @@ -26,7 +26,7 @@ interpretation "plus (multiple relocation with pairs)" (* Basic properties *********************************************************) -lemma pluss_SO2: ∀l,m,cs. ({l, m};cs) + 1 = {↑l, m};cs + 1. +lemma pluss_SO2: ∀l,m,cs. ({l,m};cs) + 1 = {↑l,m};cs + 1. normalize // qed. (* Basic inversion lemmas ***************************************************) @@ -36,8 +36,8 @@ lemma pluss_inv_nil2: ∀i,cs. cs + i = ◊ → cs = ◊. #l #m #cs #H destruct qed. -lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l, m};cs2 → - ∃∃cs1. cs1 + i = cs2 & cs = {l - i, m};cs1. +lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l,m};cs2 → + ∃∃cs1. cs1 + i = cs2 & cs = {l - i,m};cs1. #i #l #m #cs2 * [ normalize #H destruct | #l1 #m1 #cs1 whd in ⊢ (??%?→?); #H destruct diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma index 4a03364b8..93980c881 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma @@ -27,20 +27,20 @@ interpretation "relational application (rtmap)" 'RAt i1 f i2 = (at f i1 i2). definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1. - ∀jf,jg,j. @⦃jf, f2⦄ ≘ j → @⦃jg, g2⦄ ≘ j → - ∃∃j0. @⦃j0, f1⦄ ≘ jf & @⦃j0, g1⦄ ≘ jg. + ∀jf,jg,j. @⦃jf,f2⦄ ≘ j → @⦃jg,g2⦄ ≘ j → + ∃∃j0. @⦃j0,f1⦄ ≘ jf & @⦃j0,g1⦄ ≘ jg. (* Basic inversion lemmas ***************************************************) -lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2. +lemma at_inv_ppx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2. #f #i1 #i2 * -f -i1 -i2 // [ #f #i1 #i2 #_ #g #j1 #j2 #_ * #_ #x #H destruct | #f #i1 #i2 #_ #g #j2 * #_ #x #_ #H elim (discr_push_next … H) ] qed-. -lemma at_inv_npx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f → - ∃∃j2. @⦃j1, g⦄ ≘ j2 & ↑j2 = i2. +lemma at_inv_npx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f → + ∃∃j2. @⦃j1,g⦄ ≘ j2 & ↑j2 = i2. #f #i1 #i2 * -f -i1 -i2 [ #f #g #j1 #j2 #_ * #_ #x #x1 #H destruct | #f #i1 #i2 #Hi #g #j1 #j2 * * * #x #x1 #H #Hf >(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/ @@ -48,8 +48,8 @@ lemma at_inv_npx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ] qed-. -lemma at_inv_xnx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ↑g = f → - ∃∃j2. @⦃i1, g⦄ ≘ j2 & ↑j2 = i2. +lemma at_inv_xnx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ↑g = f → + ∃∃j2. @⦃i1,g⦄ ≘ j2 & ↑j2 = i2. #f #i1 #i2 * -f -i1 -i2 [ #f #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H) | #f #i1 #i2 #_ #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H) @@ -59,43 +59,43 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma at_inv_ppn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → +lemma at_inv_ppn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j2. 0 = i1 → ⫯g = f → ↑j2 = i2 → ⊥. #f #i1 #i2 #Hf #g #j2 #H1 #H <(at_inv_ppx … Hf … H1 H) -f -g -i1 -i2 #H destruct qed-. -lemma at_inv_npp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → +lemma at_inv_npp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f → 0 = i2 → ⊥. #f #i1 #i2 #Hf #g #j1 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1 #x2 #Hg * -i2 #H destruct qed-. -lemma at_inv_npn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → - ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1, g⦄ ≘ j2. +lemma at_inv_npn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → + ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1,g⦄ ≘ j2. #f #i1 #i2 #Hf #g #j1 #j2 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1 #x2 #Hg * -i2 #H destruct // qed-. -lemma at_inv_xnp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → +lemma at_inv_xnp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ↑g = f → 0 = i2 → ⊥. #f #i1 #i2 #Hf #g #H elim (at_inv_xnx … Hf … H) -f #x2 #Hg * -i2 #H destruct qed-. -lemma at_inv_xnn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → - ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1, g⦄ ≘ j2. +lemma at_inv_xnn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → + ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1,g⦄ ≘ j2. #f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xnx … Hf … H) -f #x2 #Hg * -i2 #H destruct // qed-. -lemma at_inv_pxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f. +lemma at_inv_pxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f. #f elim (pn_split … f) * /2 width=2 by ex_intro/ #g #H #i1 #i2 #Hf #H1 #H2 cases (at_inv_xnp … Hf … H H2) qed-. -lemma at_inv_pxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 → - ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f. +lemma at_inv_pxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 → + ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f. #f elim (pn_split … f) * #g #H #i1 #i2 #Hf #j2 #H1 #H2 [ elim (at_inv_ppn … Hf … H1 H H2) @@ -103,7 +103,7 @@ lemma at_inv_pxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = ] qed-. -lemma at_inv_nxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → +lemma at_inv_nxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j1. ↑j1 = i1 → 0 = i2 → ⊥. #f elim (pn_split f) * #g #H #i1 #i2 #Hf #j1 #H1 #H2 @@ -112,37 +112,37 @@ lemma at_inv_nxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ] qed-. -lemma at_inv_nxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 → - (∃∃g. @⦃j1, g⦄ ≘ j2 & ⫯g = f) ∨ - ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f. +lemma at_inv_nxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 → + (∃∃g. @⦃j1,g⦄ ≘ j2 & ⫯g = f) ∨ + ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f. #f elim (pn_split f) * /4 width=7 by at_inv_xnn, at_inv_npn, ex2_intro, or_intror, or_introl/ qed-. (* Note: the following inversion lemmas must be checked *) -lemma at_inv_xpx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f → +lemma at_inv_xpx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f → (0 = i1 ∧ 0 = i2) ∨ - ∃∃j1,j2. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2. + ∃∃j1,j2. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2. #f * [2: #i1 ] #i2 #Hf #g #H [ elim (at_inv_npx … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/ | >(at_inv_ppx … Hf … H) -f /3 width=1 by conj, or_introl/ ] qed-. -lemma at_inv_xpp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1. +lemma at_inv_xpp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1. #f #i1 #i2 #Hf #g #H elim (at_inv_xpx … Hf … H) -f * // #j1 #j2 #_ #_ * -i2 #H destruct qed-. -lemma at_inv_xpn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 → - ∃∃j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1. +lemma at_inv_xpn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 → + ∃∃j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1. #f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xpx … Hf … H) -f * [ #_ * -i2 #H destruct | #x1 #x2 #Hg #H1 * -i2 #H destruct /2 width=3 by ex2_intro/ ] qed-. -lemma at_inv_xxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i2 → +lemma at_inv_xxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i2 → ∃∃g. 0 = i1 & ⫯g = f. #f elim (pn_split f) * #g #H #i1 #i2 #Hf #H2 @@ -151,9 +151,9 @@ lemma at_inv_xxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i2 → ] qed-. -lemma at_inv_xxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. ↑j2 = i2 → - (∃∃g,j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨ - ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f. +lemma at_inv_xxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. ↑j2 = i2 → + (∃∃g,j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨ + ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f. #f elim (pn_split f) * #g #H #i1 #i2 #Hf #j2 #H2 [ elim (at_inv_xpn … Hf … H H2) -i2 /3 width=5 by or_introl, ex3_2_intro/ @@ -163,7 +163,7 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma at_increasing: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → i1 ≤ i2. +lemma at_increasing: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → i1 ≤ i2. #i2 elim i2 -i2 [ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf // | #i2 #IH * // @@ -172,13 +172,13 @@ lemma at_increasing: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → i1 ≤ i2. ] qed-. -lemma at_increasing_strict: ∀g,i1,i2. @⦃i1, g⦄ ≘ i2 → ∀f. ↑f = g → - i1 < i2 ∧ @⦃i1, f⦄ ≘ ↓i2. +lemma at_increasing_strict: ∀g,i1,i2. @⦃i1,g⦄ ≘ i2 → ∀f. ↑f = g → + i1 < i2 ∧ @⦃i1,f⦄ ≘ ↓i2. #g #i1 #i2 #Hg #f #H elim (at_inv_xnx … Hg … H) -Hg -H /4 width=2 by conj, at_increasing, le_S_S/ qed-. -lemma at_fwd_id_ex: ∀f,i. @⦃i, f⦄ ≘ i → ∃g. ⫯g = f. +lemma at_fwd_id_ex: ∀f,i. @⦃i,f⦄ ≘ i → ∃g. ⫯g = f. #f elim (pn_split f) * /2 width=2 by ex_intro/ #g #H #i #Hf elim (at_inv_xnx … Hf … H) -Hf -H #j2 #Hg #H destruct lapply (at_increasing … Hg) -Hg @@ -187,7 +187,7 @@ qed-. (* Basic properties *********************************************************) -corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≘ i2). +corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1,f⦄ ≘ i2). #i1 #i2 #f1 #H1 cases H1 -f1 -i1 -i2 [ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /2 width=2 by at_refl/ | #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /3 width=7 by at_push/ @@ -195,12 +195,12 @@ corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≘ i2). ] qed-. -lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1, f⦄ ≘ i2). +lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1,f⦄ ≘ i2). #i1 #i2 @eq_repl_sym /2 width=3 by at_eq_repl_back/ qed-. -lemma at_le_ex: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀i1. i1 ≤ i2 → - ∃∃j1. @⦃i1, f⦄ ≘ j1 & j1 ≤ j2. +lemma at_le_ex: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀i1. i1 ≤ i2 → + ∃∃j1. @⦃i1,f⦄ ≘ j1 & j1 ≤ j2. #j2 elim j2 -j2 [2: #j2 #IH ] #i2 #f #Hf [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ] #g [ #x2 ] #Hg [ #H2 ] #H0 @@ -217,14 +217,14 @@ lemma at_le_ex: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀i1. i1 ≤ i2 → ] qed-. -lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≘ i2 → @⦃i1, f⦄ ≘ i1. +lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2,f⦄ ≘ i2 → @⦃i1,f⦄ ≘ i1. #i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ] #f #Hf elim (at_fwd_id_ex … Hf) /4 width=7 by at_inv_npn, at_push, at_refl/ qed-. (* Main properties **********************************************************) -theorem at_monotonic: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀j1,i1. @⦃i1, f⦄ ≘ j1 → +theorem at_monotonic: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀j1,i1. @⦃i1,f⦄ ≘ j1 → i1 < i2 → j1 < j2. #j2 elim j2 -j2 [ #i2 #f #H2f elim (at_inv_xxp … H2f) -H2f // @@ -240,7 +240,7 @@ theorem at_monotonic: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀j1,i1. @⦃i1, f⦄ ] qed-. -theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1, f⦄ ≘ j1 → ∀j2,i2. @⦃i2, f⦄ ≘ j2 → +theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1,f⦄ ≘ j1 → ∀j2,i2. @⦃i2,f⦄ ≘ j2 → j1 < j2 → i1 < i2. #j1 elim j1 -j1 [ #i1 #f #H1f elim (at_inv_xxp … H1f) -H1f // @@ -261,12 +261,12 @@ theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1, f⦄ ≘ j1 → ∀j2,i2. @⦃i2, ] qed-. -theorem at_mono: ∀f,i,i1. @⦃i, f⦄ ≘ i1 → ∀i2. @⦃i, f⦄ ≘ i2 → i2 = i1. +theorem at_mono: ∀f,i,i1. @⦃i,f⦄ ≘ i1 → ∀i2. @⦃i,f⦄ ≘ i2 → i2 = i1. #f #i #i1 #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) // #Hi elim (lt_le_false i i) /3 width=6 by at_inv_monotonic, eq_sym/ qed-. -theorem at_inj: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀i2. @⦃i2, f⦄ ≘ i → i1 = i2. +theorem at_inj: ∀f,i1,i. @⦃i1,f⦄ ≘ i → ∀i2. @⦃i2,f⦄ ≘ i → i1 = i2. #f #i1 #i #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) // #Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/ qed-. @@ -312,14 +312,14 @@ theorem at_div_pn: ∀f2,g2,f1,g1. (* Properties on tls ********************************************************) -lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≘ n → @⦃0, ⫱*[n]f⦄ ≘ 0. +lemma at_pxx_tls: ∀n,f. @⦃0,f⦄ ≘ n → @⦃0,⫱*[n]f⦄ ≘ 0. #n elim n -n // #n #IH #f #Hf cases (at_inv_pxn … Hf) -Hf [ |*: // ] #g #Hg #H0 destruct (H2f … Hf) in ⊢ (???%); -H2f // diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc.ma index 1e08d2474..482a6da39 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc.ma @@ -43,7 +43,7 @@ definition eq_f: relation rtc ≝ λc1,c2. ⊤. inductive eq_t: relation rtc ≝ | eq_t_intro: ∀ri1,ri2,rs1,rs2,ti,ts. - eq_t (〈ri1, rs1, ti, ts〉) (〈ri2, rs2, ti, ts〉) + eq_t (〈ri1,rs1,ti,ts〉) (〈ri2,rs2,ti,ts〉) . (* Basic properties *********************************************************) diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_isrt.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_isrt.ma index 9c988abea..45be4c4ad 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_isrt.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_isrt.ma @@ -18,47 +18,47 @@ include "ground_2/steps/rtc.ma". (* RT-TRANSITION COUNTER ****************************************************) definition isrt: relation2 nat rtc ≝ λts,c. - ∃∃ri,rs. 〈ri, rs, 0, ts〉 = c. + ∃∃ri,rs. 〈ri,rs,0,ts〉 = c. interpretation "test for costrained rt-transition counter (rtc)" 'IsRedType ts c = (isrt ts c). (* Basic properties *********************************************************) -lemma isr_00: 𝐑𝐓⦃0, 𝟘𝟘⦄. +lemma isr_00: 𝐑𝐓⦃0,𝟘𝟘⦄. /2 width=3 by ex1_2_intro/ qed. -lemma isr_10: 𝐑𝐓⦃0, 𝟙𝟘⦄. +lemma isr_10: 𝐑𝐓⦃0,𝟙𝟘⦄. /2 width=3 by ex1_2_intro/ qed. -lemma isrt_01: 𝐑𝐓⦃1, 𝟘𝟙⦄. +lemma isrt_01: 𝐑𝐓⦃1,𝟘𝟙⦄. /2 width=3 by ex1_2_intro/ qed. -lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n, c2⦄. +lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n,c2⦄. #n #c1 #c2 * #ri1 #rs1 #H destruct #H elim (eq_t_inv_dx … H) -H /2 width=3 by ex1_2_intro/ qed-. (* Basic inversion properties ***********************************************) -lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n, 𝟘𝟘⦄ → 0 = n. +lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n,𝟘𝟘⦄ → 0 = n. #n * #ri #rs #H destruct // qed-. -lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n, 𝟙𝟘⦄ → 0 = n. +lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n,𝟙𝟘⦄ → 0 = n. #n * #ri #rs #H destruct // qed-. -lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n, 𝟘𝟙⦄ → 1 = n. +lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n,𝟘𝟙⦄ → 1 = n. #n * #ri #rs #H destruct // qed-. (* Main inversion properties ************************************************) -theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1, c⦄ → 𝐑𝐓⦃n2, c⦄ → n1 = n2. +theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1,c⦄ → 𝐑𝐓⦃n2,c⦄ → n1 = n2. #n1 #n2 #c * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct // qed-. -theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → eq_t c1 c2. +theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → eq_t c1 c2. #n #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct // qed-. diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma index bfa170972..bf31a291f 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma @@ -18,7 +18,7 @@ include "ground_2/steps/rtc_shift.ma". definition max (c1:rtc) (c2:rtc): rtc ≝ match c1 with [ mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [ - mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 + mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 ] ]. @@ -28,7 +28,7 @@ interpretation "maximum (rtc)" (* Basic properties *********************************************************) lemma max_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2. - 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 = + 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 = (〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉). // qed. @@ -59,46 +59,46 @@ qed. (* Properties with test for constrained rt-transition counter ***************) -lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1∨n2, c1∨c2⦄. +lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1∨n2,c1∨c2⦄. #n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct /2 width=3 by ex1_2_intro/ qed. -lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄. +lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄. /2 width=1 by isrt_max/ qed. -lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄. +lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄. #n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/ qed. -lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄. +lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄. #n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/ qed. (* Inversion properties with test for constrained rt-transition counter *****) -lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → - ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & (n1 ∨ n2) = n. +lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → + ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & (n1 ∨ n2) = n. #n #c1 #c2 * #ri #rs #H elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4 elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/ qed-. -lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0, c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃0, c2⦄. +lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0,c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃0,c2⦄. #c1 #c2 #H elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H elim (max_inv_O3 … H) -H #H1 #H2 destruct /2 width=1 by conj/ qed-. -lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄. +lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄. #n #c1 #c2 #H #H2 elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct lapply (isrt_inj … Hn2 H2) -c2 #H destruct // qed-. -lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → eq_t c1 c2 → - ∧∧ 𝐑𝐓⦃n, c1⦄ & 𝐑𝐓⦃n, c2⦄. +lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → eq_t c1 c2 → + ∧∧ 𝐑𝐓⦃n,c1⦄ & 𝐑𝐓⦃n,c2⦄. #n #c1 #c2 #H #Hc12 elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H @@ -115,8 +115,8 @@ qed. (* Inversion lemmaswith shift ***********************************************) -lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n, ↕*c1 ∨ c2⦄ → - ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃n, c2⦄. +lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n,↕*c1 ∨ c2⦄ → + ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃n,c2⦄. #n #c1 #c2 #H elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1 diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma index 49ce55082..a5237af26 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma @@ -18,7 +18,7 @@ include "ground_2/steps/rtc_isrt.ma". definition plus (c1:rtc) (c2:rtc): rtc ≝ match c1 with [ mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [ - mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 + mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉 ] ]. @@ -29,7 +29,7 @@ interpretation "plus (rtc)" (**) (* plus is not disambiguated parentheses *) lemma plus_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2. - 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 = + 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉 = (〈ri1,rs1,ti1,ts1〉) + (〈ri2,rs2,ti2,ts2〉). // qed. @@ -56,38 +56,38 @@ qed. (* Properties with test for constrained rt-transition counter ***************) -lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1+n2, c1+c2⦄. +lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1+n2,c1+c2⦄. #n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct /2 width=3 by ex1_2_intro/ qed. -lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄. +lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄. /2 width=1 by isrt_plus/ qed. -lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄. +lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄. #n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by isrt_plus/ qed. -lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n, c⦄ → 𝐑𝐓⦃↑n, c+𝟘𝟙⦄. +lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n,c⦄ → 𝐑𝐓⦃↑n,c+𝟘𝟙⦄. /2 width=1 by isrt_plus/ qed. (* Inversion properties with test for constrained rt-transition counter *****) -lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → - ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & n1 + n2 = n. +lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → + ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & n1 + n2 = n. #n #c1 #c2 * #ri #rs #H elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4 elim (plus_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/ qed-. -lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄. +lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄. #n #c1 #c2 #H #H2 elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct lapply (isrt_inj … Hn2 H2) -c2 #H destruct // qed-. -lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃1, c2⦄ → - ∃∃m. 𝐑𝐓⦃m, c1⦄ & n = ↑m. +lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃1,c2⦄ → + ∃∃m. 𝐑𝐓⦃m,c1⦄ & n = ↑m. #n #c1 #c2 #H #H2 elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct lapply (isrt_inj … Hn2 H2) -c2 #H destruct diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_shift.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_shift.ma index e2f739d69..165e1eeda 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_shift.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_shift.ma @@ -18,14 +18,14 @@ include "ground_2/steps/rtc_isrt.ma". (* RT-TRANSITION COUNTER ****************************************************) definition shift (c:rtc): rtc ≝ match c with -[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs, 0, ti∨ts, 0〉 ]. +[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs,0,ti∨ts,0〉 ]. interpretation "shift (rtc)" 'UpDownArrowStar c = (shift c). (* Basic properties *********************************************************) -lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs, 0, ti∨ts, 0〉 = ↕*〈ri, rs, ti, ts〉. +lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs,0,ti∨ts,0〉 = ↕*〈ri,rs,ti,ts〉. normalize // qed. @@ -34,27 +34,27 @@ lemma shift_O: 𝟘𝟘 = ↕*𝟘𝟘. (* Basic inversion properties ***********************************************) -lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri, rs, ti, ts〉 = ↕*c → +lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri,rs,ti,ts〉 = ↕*c → ∃∃ri0,rs0,ti0,ts0. (ri0∨rs0) = ri & 0 = rs & (ti0∨ts0) = ti & 0 = ts & - 〈ri0, rs0, ti0, ts0〉 = c. + 〈ri0,rs0,ti0,ts0〉 = c. #ri #rs #ti #ts * #ri0 #rs0 #ti0 #ts0 ${SRC}.new +for SRC in `find ground_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do + sed "/$1/s!$2!$3!g" ${SRC} > ${SRC}.new if [ ! -s ${SRC}.new ] || diff ${SRC} ${SRC}.new > /dev/null; then rm -f ${SRC}.new; else echo ${SRC}; mv -f ${SRC} ${SRC}.old; mv -f ${SRC}.new ${SRC}; diff --git a/matita/matita/contribs/lambdadelta/static_2/etc/sh_lt.etc b/matita/matita/contribs/lambdadelta/static_2/etc/sh_lt.etc new file mode 100644 index 000000000..89eb9dde5 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/static_2/etc/sh_lt.etc @@ -0,0 +1,6 @@ +definition sh_N: sh ≝ mk_sh S …. +// defined. + +axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2). + +axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2. diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma index 8beb5bc3f..a53ee2c82 100644 --- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma +++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs.ma @@ -25,50 +25,50 @@ interpretation "iterated extension on referred entries (local environment)" (* Basic properties *********************************************************) -lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L → - ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2. +lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R,T] L → + ∀L2. L ⪤[R,T] L2 → L1 ⪤*[R,T] L2. #R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *) qed-. -lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L → - ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2. +lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R,T] L → + ∀L2. L ⪤*[R,T] L2 → L1 ⪤*[R,T] L2. #R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *) qed-. -lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆. +lemma rexs_atom: ∀R,I. ⋆ ⪤*[R,⓪{I}] ⋆. /2 width=1 by inj/ qed. lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s. - L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2. + L1 ⪤*[R,⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R,⋆s] L2.ⓑ{I}V2. #R #I #L1 #L2 #V1 #V2 #s #H elim H -L2 /3 width=4 by rex_sort, rexs_step_dx, inj/ qed. lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) → - ∀I,L1,L2,V. L1 ⪤*[R, V] L2 → - L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V. + ∀I,L1,L2,V. L1 ⪤*[R,V] L2 → + L1.ⓑ{I}V ⪤*[R,#0] L2.ⓑ{I}V. #R #HR #I #L1 #L2 #V #H elim H -L2 /3 width=5 by rex_pair, rexs_step_dx, inj/ qed. -lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 → - L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}. +lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 → + L1.ⓤ{I} ⪤*[R,#0] L2.ⓤ{I}. /3 width=3 by rex_unit, inj/ qed. lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i. - L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2. + L1 ⪤*[R,#i] L2 → L1.ⓑ{I}V1 ⪤*[R,#↑i] L2.ⓑ{I}V2. #R #I #L1 #L2 #V1 #V2 #i #H elim H -L2 /3 width=4 by rex_lref, rexs_step_dx, inj/ qed. lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l. - L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2. + L1 ⪤*[R,§l] L2 → L1.ⓑ{I}V1 ⪤*[R,§l] L2.ⓑ{I}V2. #R #I #L1 #L2 #V1 #V2 #l #H elim H -L2 /3 width=4 by rex_gref, rexs_step_dx, inj/ qed. lemma rexs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → - ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2. + ∀L1,L2,T. L1 ⪤*[R1,T] L2 → L1 ⪤*[R2,T] L2. #R1 #R2 #HR #L1 #L2 #T #H elim H -L2 /4 width=5 by rex_co, rexs_step_dx, inj/ qed-. @@ -76,19 +76,19 @@ qed-. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *) -lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆. +lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R,⓪{I}] Y2 → Y2 = ⋆. #R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/ qed-. (* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *) -lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆. +lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R,⓪{I}] ⋆ → Y1 = ⋆. #R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1 /3 width=3 by inj, rex_inv_atom_dx/ qed-. -lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 → +lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R,⋆s] Y2 → ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤*[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R #Y1 #Y2 #s #H elim H -Y2 [ #Y2 #H elim (rex_inv_sort … H) -H * @@ -104,9 +104,9 @@ lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 → ] qed-. -lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 → +lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R,§l] Y2 → ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤*[R,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R #Y1 #Y2 #l #H elim H -Y2 [ #Y2 #H elim (rex_inv_gref … H) -H * @@ -123,16 +123,16 @@ lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 → qed-. lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) → - ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 → - ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V. + ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 → + ∧∧ L1 ⪤*[R,V] L2 & L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V. #R #HR #p #I #L1 #L2 #V #T #H elim H -L2 [ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/ | #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/ ] qed-. -lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → - ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2. +lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 → + ∧∧ L1 ⪤*[R,V] L2 & L1 ⪤*[R,T] L2. #R #I #L1 #L2 #V #T #H elim H -L2 [ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/ | #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/ @@ -141,32 +141,32 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 → - ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}. +lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R,⋆s] Y2 → + ∃∃I2,L2. L1 ⪤*[R,⋆s] L2 & Y2 = L2.ⓘ{I2}. #R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H * [ #H destruct | #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} → - ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}. +lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R,⋆s] L2.ⓘ{I2} → + ∃∃I1,L1. L1 ⪤*[R,⋆s] L2 & Y1 = L1.ⓘ{I1}. #R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H * [ #_ #H destruct | #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 → - ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}. +lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R,§l] Y2 → + ∃∃I2,L2. L1 ⪤*[R,§l] L2 & Y2 = L2.ⓘ{I2}. #R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H * [ #H destruct | #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} → - ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}. +lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R,§l] L2.ⓘ{I2} → + ∃∃I1,L1. L1 ⪤*[R,§l] L2 & Y1 = L1.ⓘ{I1}. #R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H * [ #_ #H destruct | #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ @@ -175,18 +175,18 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2. +lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R,②{I}V.T] L2 → L1 ⪤*[R,V] L2. #R #I #L1 #L2 #V #T #H elim H -L2 /3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/ qed-. lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) → - ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 → - L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V. + ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 → + L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V. #R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H // qed-. -lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2. +lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 → L1 ⪤*[R,T] L2. #R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H // qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_drops.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_drops.ma index 0b83c6431..11025a4a1 100644 --- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_drops.ma @@ -19,19 +19,19 @@ include "static_2/i_static/rexs.ma". (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***) definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → - ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U → - ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → + ∀K2,T. K1 ⪤*[R,T] K2 → ∀U. ⬆*[f] T ≘ U → + ∃∃L2. L1 ⪤*[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2. definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → - ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U → - ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → + ∀L2,U. L1 ⪤*[R,U] L2 → ∀T. ⬆*[f] T ≘ U → + ∃∃K2. K1 ⪤*[R,T] K2 & ⬇*[b,f] L2 ≘ K2. definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝ - λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 → - ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U → - ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2. + λR. ∀L1,L2,U. L1 ⪤*[R,U] L2 → + ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U → + ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤*[R,T] K2. (* Properties with generic slicing for local environments *******************) diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_fqup.ma index c21d82017..fbb41879c 100644 --- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_fqup.ma @@ -25,13 +25,13 @@ lemma rexs_refl: ∀R. c_reflexive … R → (* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *) lemma rexs_pair_refl: ∀R. c_reflexive … R → - ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2. + ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R,T] L.ⓑ{I}V2. #R #HR #L #V1 #V2 #H elim H -V2 /3 width=3 by rexs_step_dx, rex_pair_refl, inj/ qed. lemma rexs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (sex cfull (cext2 R) f) L1 L2 → - L1 ⪤*[R, T] L2. + L1 ⪤*[R,T] L2. #R #L1 #L2 #T #f #Hf #H elim H -L2 [ elim (frees_total L1 T) | #L elim (frees_total L T) ] /5 width=7 by sex_sdj, rexs_step_dx, sdj_isid_sn, inj, ex2_intro/ @@ -41,16 +41,16 @@ qed. lemma rexs_ind_sn: ∀R. c_reflexive … R → ∀L1,T. ∀Q:predicate …. Q L1 → - (∀L,L2. L1 ⪤*[R, T] L → L ⪤[R, T] L2 → Q L → Q L2) → - ∀L2. L1 ⪤*[R, T] L2 → Q L2. + (∀L,L2. L1 ⪤*[R,T] L → L ⪤[R,T] L2 → Q L → Q L2) → + ∀L2. L1 ⪤*[R,T] L2 → Q L2. #R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by rex_refl/ qed-. lemma rexs_ind_dx: ∀R. c_reflexive … R → ∀L2,T. ∀Q:predicate …. Q L2 → - (∀L1,L. L1 ⪤[R, T] L → L ⪤*[R, T] L2 → Q L → Q L1) → - ∀L1. L1 ⪤*[R, T] L2 → Q L1. + (∀L1,L. L1 ⪤[R,T] L → L ⪤*[R,T] L2 → Q L → Q L1) → + ∀L1. L1 ⪤*[R,T] L2 → Q L1. #R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by rex_refl/ qed-. @@ -58,8 +58,8 @@ qed-. (* Advanced inversion lemmas ************************************************) lemma rexs_inv_bind_void: ∀R. c_reflexive … R → - ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 → - ∧∧ L1 ⪤*[R, V] L2 & L1.ⓧ ⪤*[R, T] L2.ⓧ. + ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 → + ∧∧ L1 ⪤*[R,V] L2 & L1.ⓧ ⪤*[R,T] L2.ⓧ. #R #HR #p #I #L1 #L2 #V #T #H @(rexs_ind_sn … HR … H) -L2 [ /3 width=1 by rexs_refl, conj/ | #L #L2 #_ #H * elim (rex_inv_bind_void … H) -H /3 width=3 by rexs_step_dx, conj/ @@ -69,7 +69,7 @@ qed-. (* Advanced forward lemmas **************************************************) lemma rexs_fwd_bind_dx_void: ∀R. c_reflexive … R → - ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 → - L1.ⓧ ⪤*[R, T] L2.ⓧ. + ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 → + L1.ⓧ ⪤*[R,T] L2.ⓧ. #R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind_void … H) -H // qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_length.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_length.ma index a5c82f9f5..7e0bc2e01 100644 --- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_length.ma @@ -20,7 +20,7 @@ include "static_2/i_static/rexs.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: uses: TC_lpx_sn_fwd_length *) -lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|. +lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R,T] L2 → |L1| = |L2|. #R #L1 #L2 #T #H elim H -L2 [ #L2 #HL12 >(rex_fwd_length … HL12) -HL12 // | #L #L2 #_ #HL2 #IHL1 diff --git a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_lex.ma b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_lex.ma index 4ac00b7a1..46d28f330 100644 --- a/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_lex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/i_static/rexs_lex.ma @@ -22,14 +22,14 @@ include "static_2/i_static/rexs_fqup.ma". (* Properties with generic extension of a context sensitive relation ********) lemma rexs_lex: ∀R. c_reflexive … R → - ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R, T] L2. + ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R,T] L2. #R #HR #L1 #L2 #T * /5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/ qed. lemma rexs_lex_req: ∀R. c_reflexive … R → ∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 → - L1 ⪤*[R, T] L2. + L1 ⪤*[R,T] L2. /3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed. (* Inversion lemmas with generic extension of a context sensitive relation **) @@ -39,7 +39,7 @@ lemma rexs_inv_lex_req: ∀R. c_reflexive … R → rex_fsge_compatible R → s_rs_transitive … R (λ_.lex R) → req_transitive R → - ∀L1,L2,T. L1 ⪤*[R, T] L2 → + ∀L1,L2,T. L1 ⪤*[R,T] L2 → ∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2. #R #H1R #H2R #H3R #H4R #L1 #L2 #T #H lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabbr_2.ma b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabbr_2.ma index 1eea5226e..1c31b90fc 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabbr_2.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabbr_2.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L. break ⓓ T1 )" +notation "hvbox( L. ⓓ break T1 )" left associative with precedence 50 for @{ 'DxAbbr $L $T1 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabst_2.ma b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabst_2.ma index 3bb334e86..6c385b47b 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabst_2.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxabst_2.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L. break ⓛ T1 )" +notation "hvbox( L. ⓛ break T1 )" left associative with precedence 51 for @{ 'DxAbst $L $T1 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind1_2.ma b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind1_2.ma index 5e78eeb18..a427352dd 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind1_2.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind1_2.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L. break ⓤ { term 46 I } )" +notation "hvbox( L. ⓤ { break term 46 I } )" non associative with precedence 47 for @{ 'DxBind1 $L $I }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind2_3.ma b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind2_3.ma index 6800d8e5c..3b9e43699 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind2_3.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/functions/dxbind2_3.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L. break ⓑ { term 46 I } break term 49 T1 )" +notation "hvbox( L. ⓑ { break term 46 I } break term 49 T1 )" non associative with precedence 48 for @{ 'DxBind2 $L $I $T1 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/functions/upspoon_2.ma b/matita/matita/contribs/lambdadelta/static_2/notation/functions/upspoon_2.ma new file mode 100644 index 000000000..c04bf2278 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/static_2/notation/functions/upspoon_2.ma @@ -0,0 +1,19 @@ +(**************************************************************************) +(* ___ *) +(* ||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( ⫯[ term 46 h ] break term 46 s )" + non associative with precedence 46 + for @{ 'UpSpoon $h $s }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_6.ma b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_6.ma index 57bdc3d94..8a8a95876 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_6.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_6.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )" +notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" non associative with precedence 45 for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_7.ma b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_7.ma index 3bbe31a83..49b6472fe 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_7.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermplus_7.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )" +notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" non associative with precedence 45 for @{ 'SupTermPlus $b $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_6.ma b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_6.ma index a0bf55faa..279b363e7 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_6.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_6.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" +notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" non associative with precedence 45 for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_7.ma b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_7.ma index 865019860..8b0ac452c 100644 --- a/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_7.ma +++ b/matita/matita/contribs/lambdadelta/static_2/notation/relations/suptermstar_7.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" +notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )" non associative with precedence 45 for @{ 'SupTermStar $b $G1 $L1 $T1 $G2 $L2 $T2 }. diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma index 939aeb534..dc4d6d75c 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma @@ -39,68 +39,68 @@ interpretation "generic slicing (local environment)" 'RDropStar b f L1 L2 = (drops b f L1 L2). definition d_liftable1: predicate (relation2 lenv term) ≝ - λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → + λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U → R L U. definition d_liftable1_isuni: predicate (relation2 lenv term) ≝ - λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ → + λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ → ∀U. ⬆*[f] T ≘ U → R L U. definition d_deliftable1: predicate (relation2 lenv term) ≝ - λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → + λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U → R K T. definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝ - λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ → + λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U → R K T. definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C. predicate (lenv → relation C) ≝ - λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K → + λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K → ∀U1. S f T1 U1 → ∃∃U2. S f T2 U2 & R L U1 U2. definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C. predicate (lenv → relation C) ≝ - λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K → + λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. S f T1 U1 → ∃∃T2. S f T2 U2 & R K T1 T2. definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C. predicate (lenv → relation C) ≝ - λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K → + λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K → ∀U1. S f T1 U1 → ∀U2. S f T2 U2 → R L U1 U2. definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C. predicate (lenv → relation C) ≝ - λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K → + λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. S f T1 U1 → ∀T2. S f T2 U2 → R K T1 T2. definition co_dropable_sn: predicate (rtmap → relation lenv) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 → - ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≘ K2. + ∃∃K2. R f1 K1 K2 & ⬇*[b,f] L2 ≘ K2. definition co_dropable_dx: predicate (rtmap → relation lenv) ≝ λR. ∀f2,L1,L2. R f2 L1 L2 → - ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → + ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀f1. f ~⊚ f1 ≘ f2 → - ∃∃K1. ⬇*[b, f] L1 ≘ K1 & R f1 K1 K2. + ∃∃K1. ⬇*[b,f] L1 ≘ K1 & R f1 K1 K2. definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 → + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 → ∀f2. f ~⊚ f1 ≘ f2 → - ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2. + ∃∃L2. R f2 L1 L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2. (* Basic properties *********************************************************) -lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆. +lemma drops_atom_F: ∀f. ⬇*[Ⓕ,f] ⋆ ≘ ⋆. #f @drops_atom #H destruct qed. -lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L2). +lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b,f] L1 ≘ L2). #b #L1 #L2 #f1 #H elim H -f1 -L1 -L2 [ /4 width=3 by drops_atom, isid_eq_repl_back/ | #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H @@ -110,23 +110,23 @@ lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L ] qed-. -lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≘ L2). +lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b,f] L1 ≘ L2). #b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *) qed-. (* Basic_2A1: includes: drop_FT *) -lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2. +lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2. #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by drops_atom, drops_drop, drops_skip/ qed. (* Basic_2A1: includes: drop_gen *) -lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2. +lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[b,f] L1 ≘ L2. * /2 width=1 by drops_TF/ qed-. (* Basic_2A1: includes: drop_T *) -lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2. +lemma drops_F: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2. * /2 width=1 by drops_TF/ qed-. @@ -146,7 +146,7 @@ qed-. (* Basic inversion lemmas ***************************************************) -fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ → +fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → X = ⋆ → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄). #b #f #X #Y * -f -X -Y [ /3 width=1 by conj/ @@ -157,11 +157,11 @@ qed-. (* Basic_1: includes: drop_gen_sort *) (* Basic_2A1: includes: drop_inv_atom1 *) -lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄). +lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b,f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄). /2 width=3 by drops_inv_atom1_aux/ qed-. -fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g → - ⬇*[b, g] K ≘ Y. +fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g → + ⬇*[b,g] K ≘ Y. #b #f #X #Y * -f -X -Y [ #f #Hf #g #J #K #H destruct | #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct // @@ -171,11 +171,11 @@ qed-. (* Basic_1: includes: drop_gen_drop *) (* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *) -lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b, ↑f] K.ⓘ{I} ≘ Y → ⬇*[b, f] K ≘ Y. +lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b,↑f] K.ⓘ{I} ≘ Y → ⬇*[b,f] K ≘ Y. /2 width=6 by drops_inv_drop1_aux/ qed-. -fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g → - ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}. +fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g → + ∃∃I2,K2. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}. #b #f #X #Y * -f -X -Y [ #f #Hf #g #J1 #K1 #H destruct | #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2) @@ -186,12 +186,12 @@ qed-. (* Basic_1: includes: drop_gen_skip_l *) (* Basic_2A1: includes: drop_inv_skip1 *) -lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b, ⫯f] K1.ⓘ{I1} ≘ Y → - ∃∃I2,K2. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}. +lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b,⫯f] K1.ⓘ{I1} ≘ Y → + ∃∃I2,K2. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}. /2 width=5 by drops_inv_skip1_aux/ qed-. -fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g → - ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}. +fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g → + ∃∃I1,K1. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}. #b #f #X #Y * -f -X -Y [ #f #Hf #g #J2 #K2 #H destruct | #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2) @@ -202,14 +202,14 @@ qed-. (* Basic_1: includes: drop_gen_skip_r *) (* Basic_2A1: includes: drop_inv_skip2 *) -lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b, ⫯f] X ≘ K2.ⓘ{I2} → - ∃∃I1,K1. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}. +lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b,⫯f] X ≘ K2.ⓘ{I2} → + ∃∃I1,K1. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}. /2 width=5 by drops_inv_skip2_aux/ qed-. (* Basic forward lemmas *****************************************************) -fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} → - ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K. +fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} → + ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K. #b #f2 #X #Y #H elim H -f2 -X -Y [ #f2 #Hf2 #J #K #H destruct | #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL @@ -219,14 +219,14 @@ fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ ] qed-. -lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} → - ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K. +lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} → + ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K. /2 width=4 by drops_fwd_drop2_aux/ qed-. (* Properties with test for identity ****************************************) (* Basic_2A1: includes: drop_refl *) -lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≘ L. +lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b,f] L ≘ L. #b #L elim L -L /2 width=1 by drops_atom/ #L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf /3 width=1 by drops_skip, liftsb_refl/ @@ -236,15 +236,15 @@ qed. (* Basic_1: includes: drop_gen_refl *) (* Basic_2A1: includes: drop_inv_O2 *) -lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2. +lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2. #b #f #L1 #L2 #H elim H -f -L1 -L2 // [ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) // | /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/ ] qed-. -lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} → - ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b, f] X ≘ K. +lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} → + ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b,f] X ≘ K. #b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H #g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf /3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/ @@ -252,14 +252,14 @@ qed-. (* Forward lemmas with test for finite colength *****************************) -lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐅⦃f⦄. +lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐅⦃f⦄. #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by isfin_next, isfin_push, isfin_isid/ qed-. (* Properties with test for uniformity **************************************) -lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≘ K. +lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ,f] L ≘ K. #f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/ #f #_ #g #H #IH destruct * /2 width=2 by ex_intro/ #L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/ @@ -267,9 +267,9 @@ qed-. (* Inversion lemmas with test for uniformity ********************************) -lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ → +lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐔⦃f⦄ → (𝐈⦃f⦄ ∧ L1 = L2) ∨ - ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g. + ∃∃g,I,K. ⬇*[Ⓣ,g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g. #f #L1 #L2 * -f -L1 -L2 [ /4 width=1 by or_introl, conj/ | /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/ @@ -278,9 +278,9 @@ lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ → qed-. (* Basic_2A1: was: drop_inv_O1_pair1 *) -lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 → +lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b,f] K.ⓘ{I} ≘ L2 → (𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨ - ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≘ L2 & f = ↑g. + ∃∃g. 𝐔⦃g⦄ & ⬇*[b,g] K ≘ L2 & f = ↑g. #b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct [ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct <(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z @@ -290,9 +290,9 @@ lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} qed-. (* Basic_2A1: was: drop_inv_O1_pair2 *) -lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} → +lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K.ⓘ{I} → (𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨ - ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g. + ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b,g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g. #b #f #I #K * [ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct | #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H * @@ -302,16 +302,16 @@ lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K. ] qed-. -lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{I} → - ∃∃I1,K1. ⬇*[b, f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}. +lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,↑f] L1 ≘ K.ⓘ{I} → + ∃∃I1,K1. ⬇*[b,f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}. #b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf * [ #H elim (isid_inv_next … H) -H // | /2 width=4 by ex2_2_intro/ ] qed-. -fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ → - ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}. +fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ,f] L1 ≘ L2 → 𝐔⦃f⦄ → + ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ,f] L1 ≘ K.ⓘ{I}. #f #L1 #L2 #H elim H -f -L1 -L2 [ #f #_ #_ #J #K #H destruct | #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct @@ -324,16 +324,16 @@ fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ → qed-. (* Basic_2A1: includes: drop_inv_FT *) -lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}. +lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}. /2 width=3 by drops_inv_TF_aux/ qed-. (* Basic_2A1: includes: drop_inv_gen *) -lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}. +lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}. * /2 width=1 by drops_inv_TF/ qed-. (* Basic_2A1: includes: drop_inv_T *) -lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b, f] L ≘ K.ⓘ{I}. +lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b,f] L ≘ K.ⓘ{I}. * /2 width=1 by drops_inv_TF/ qed-. @@ -341,13 +341,13 @@ qed-. (* Basic_1: was: drop_S *) (* Basic_2A1: was: drop_fwd_drop2 *) -lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{I} → ⬇*[b, ↑f] X ≘ K. +lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b,f] X ≘ K.ⓘ{I} → ⬇*[b,↑f] X ≘ K. /3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-. (* Inversion lemmas with uniform relocations ********************************) -lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ → - ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f. +lemma drops_inv_atom2: ∀b,L,f. ⬇*[b,f] L ≘ ⋆ → + ∃∃n,f1. ⬇*[b,𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f. #b #L elim L -L [ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/ | #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct @@ -368,7 +368,7 @@ qed-. (* Properties with uniform relocations **************************************) -lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}. +lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}. #L elim L -L /2 width=1 by or_introl/ #L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/ #i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/ @@ -376,8 +376,8 @@ lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[ qed-. (* Basic_2A1: includes: drop_split *) -lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ → - ∃∃L. ⬇*[b, f1] L1 ≘ L & ⬇*[b, f2] L ≘ L2. +lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ → + ∃∃L. ⬇*[b,f1] L1 ≘ L & ⬇*[b,f2] L ≘ L2. #b #f #L1 #L2 #H elim H -f -L1 -L2 [ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom #H lapply (H0f H) -b @@ -396,8 +396,8 @@ lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ ] qed-. -lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ → - ∃∃L2. ⬇*[Ⓕ, f2] L ≘ L2 & ⬇*[Ⓕ, f] L1 ≘ L2. +lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b,f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ → + ∃∃L2. ⬇*[Ⓕ,f2] L ≘ L2 & ⬇*[Ⓕ,f] L1 ≘ L2. #b #f1 #L1 #L #H elim H -f1 -L1 -L [ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct | #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ] @@ -421,8 +421,8 @@ lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ⬇*[b,⫯⫱*[↑i2]f] L1 ≘ L2. /3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-. -lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≘ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≘ i → - ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b, ⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J. +lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b,f] L ≘ K0.ⓘ{I} → ∀i. @⦃O,f⦄ ≘ i → + ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b,⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J. #b #f #I #L #K0 #H #i #Hf elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H lapply (drops_tls_at … Hf … H) -H #H diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_drops.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_drops.ma index 4b52fd651..4e241edfd 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_drops.ma @@ -20,9 +20,9 @@ include "static_2/relocation/drops_weight.ma". (* Main properties **********************************************************) (* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *) -theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L → - ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 → - ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2. +theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L → + ∀b2,f,L2. ⬇*[b2,f] L1 ≘ L2 → + ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2,f2] L ≘ L2. #b1 #f1 #L1 #L #H elim H -f1 -L1 -L [ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2 #H #Hf destruct @drops_atom @@ -41,9 +41,9 @@ qed-. (* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm drops_drop_trans *) -theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L → - ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 → - ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2. +theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L → + ∀b2,f2,L2. ⬇*[b2,f2] L ≘ L2 → + ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2,f] L1 ≘ L2. #b1 #f1 #L1 #L #H elim H -f1 -L1 -L [ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2 #H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H @@ -85,13 +85,13 @@ qed-. (* Advanced properties ******************************************************) (* Basic_2A1: includes: drop_mono *) -lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 → - ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2. +lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1,f] L ≘ L1 → + ∀b2,L2. ⬇*[b2,f] L ≘ L2 → L1 = L2. #b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f) /3 width=8 by drops_conf, drops_fwd_isid/ qed-. -lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥. +lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥. #L #i #H1 #I #K #H2 lapply (drops_F … H2) -H2 #H2 lapply (drops_mono … H2 … H1) -L -i #H destruct @@ -106,21 +106,21 @@ lemma drops_ldec_dec: ∀L,i. Decidable (∃∃K,W. ⬇*[i] L ≘ K.ⓛW). qed-. (* Basic_2A1: includes: drop_conf_lt *) -lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 → - ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} → +lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2,f] L ≘ L2 → + ∀b1,f1,I1,K1. ⬇*[b1,f1] L ≘ K1.ⓘ{I1} → ∀f2. f1 ⊚ ⫯f2 ≘ f → ∃∃I2,K2. L2 = K2.ⓘ{I2} & - ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1. + ⬇*[b2,f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1. #b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf #H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/ qed-. (* Basic_2A1: includes: drop_trans_lt *) -lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L → - ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} → +lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L → + ∀b2,f2,I2,K2. ⬇*[b2,f2] L ≘ K2.ⓘ{I2} → ∀f. f1 ⊚ f2 ≘ ⫯f → ∃∃I1,K1. L1 = K1.ⓘ{I1} & - ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1. + ⬇*[b1∧b2,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1. #b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf lapply (drops_trans … H1 … H2 … Hf) -L -Hf #H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/ @@ -128,7 +128,7 @@ qed-. (* Basic_2A1: includes: drops_conf_div *) lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K. - ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} → + ⬇*[Ⓣ,f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ,f2] L ≘ K.ⓘ{I2} → 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2. #f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2 lapply (drops_isuni_fwd_drop2 … Hf1) // #H1 diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_length.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_length.ma index 94085a331..1a20d722f 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_length.ma @@ -20,13 +20,13 @@ include "static_2/relocation/drops.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: includes: drop_fwd_length_le4 *) -lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → |L2| ≤ |L1|. +lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → |L2| ≤ |L1|. #b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/ qed-. (* Basic_2A1: includes: drop_fwd_length_eq1 *) -theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1, f] L1 ≘ K1 → - ∀L2,K2. ⬇*[b2, f] L2 ≘ K2 → +theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1,f] L1 ≘ K1 → + ∀L2,K2. ⬇*[b2,f] L2 ≘ K2 → |L1| = |L2| → |K1| = |K2|. #b1 #b2 #f #L1 #K1 #HLK1 elim HLK1 -f -L1 -K1 [ #f #_ #L2 #K2 #HLK2 #H lapply (length_inv_zero_sn … H) -H @@ -43,7 +43,7 @@ qed-. (* forward lemmas with finite colength assignment ***************************) -lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → +lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ∃∃n. 𝐂⦃f⦄ ≘ n & |L1| = |L2| + n. #f #L1 #L2 #H elim H -f -L1 -L2 [ /4 width=3 by fcla_isid, ex2_intro/ @@ -53,25 +53,25 @@ lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → qed-. (* Basic_2A1: includes: drop_fwd_length *) -lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → +lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → |L1| = |L2| + n. #f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf #k #Hm #H <(fcla_mono … Hm … Hn) -f // qed-. -lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → +lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ∃∃n. 𝐂⦃f⦄ ≘ n & n ≤ |L1|. #f #L1 #L2 #H elim (drops_fwd_fcla … H) -H /2 width=3 by ex2_intro/ qed-. (* Basic_2A1: includes: drop_fwd_length_le2 *) -lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → +lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → n ≤ |L1|. #f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H #k #Hm #H <(fcla_mono … Hm … Hn) -f // qed-. -lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → +lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} → ∃∃n. 𝐂⦃f⦄ ≘ n & n < |L1|. #f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H #n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/ @@ -79,27 +79,27 @@ qed-. (* Basic_2A1: includes: drop_fwd_length_lt2 *) lemma drops_fcla_fwd_lt2: ∀f,L1,I2,K2,n. - ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n → + ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n → n < |L1|. #f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H #k #Hm #H <(fcla_mono … Hm … Hn) -f // qed-. (* Basic_2A1: includes: drop_fwd_length_lt4 *) -lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n → +lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n → |L2| < |L1|. #f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f /2 width=1 by lt_minus_to_plus_r/ qed-. (* Basic_2A1: includes: drop_inv_length_eq *) -lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄. +lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄. #f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H #n #Hn H1 -L1 @@ -107,7 +107,7 @@ elim (drops_fwd_fcla … HLK2) -HLK2 #n2 #Hn2 #H2 >H2 -L2 <(fcla_mono … Hn2 … Hn1) -f // qed-. -theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → +theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 → ∃∃n. 𝐂⦃f1⦄ ≘ n & 𝐂⦃f2⦄ ≘ n. #f1 #f2 #L1 #L2 #H1 #H2 elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1 @@ -116,7 +116,7 @@ lapply (injective_plus_r … H) -L2 #H destruct /2 width=3 by ex2_intro/ qed-. theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2. - ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 → + ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 → n1 = n2. #f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2 lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1 diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_lex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_lex.ma index 1e33fc3d8..8db2c8830 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_lex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_lex.ma @@ -19,16 +19,16 @@ include "static_2/relocation/drops_sex.ma". (* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************) definition dedropable_sn: predicate … ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 → - ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 → + ∃∃L2. L1 ⪤[R] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2. definition dropable_sn: predicate … ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 → - ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 → + ∃∃K2. K1 ⪤[R] K2 & ⬇*[b,f] L2 ≘ K2. definition dropable_dx: predicate … ≝ - λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → - ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2. + λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → + ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R] K2. (* Properties with generic extension ****************************************) @@ -58,8 +58,8 @@ qed-. (* Basic_2A1: includes: lpx_sn_drop_conf *) lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 → - ∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ → - ∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2. + ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ → + ∃∃K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2. #R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf elim (sex_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf [ #Z2 #K2 #HLK2 #HK12 #H @@ -71,8 +71,8 @@ qed-. (* Basic_2A1: includes: lpx_sn_drop_trans *) lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 → - ∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ → - ∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2. + ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ → + ∃∃K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2. #R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf elim (sex_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf [ #Z1 #K1 #HLK1 #HK12 #H diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_seq.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_seq.ma index 94beb0a71..e214e9480 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_seq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_seq.ma @@ -30,9 +30,9 @@ lemma seq_co_dropable_dx: co_dropable_dx seq. (* Basic_2A1: includes: lreq_drop_trans_be *) lemma seq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 → - ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ → + ∀b,f,I,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2. + ∃∃K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2. #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2 elim (sex_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf #I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2 @@ -41,18 +41,18 @@ qed-. (* Basic_2A1: includes: lreq_drop_conf_be *) lemma seq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 → - ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ → + ∀b,f,I,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2. + ∃∃K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2. #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2 elim (seq_drops_trans_next … (seq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf /3 width=3 by seq_sym, ex2_intro/ qed-. lemma drops_seq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 → - ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 → + ∀b,f,I,L1. ⬇*[b,f] L1.ⓘ{I} ≘ K1 → ∀f2. f ~⊚ f1 ≘ ↑f2 → - ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}. + ∃∃L2. ⬇*[b,f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}. #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2 elim (drops_sex_trans_next … HK12 … HLK1 … Hf2) -f1 -K1 /2 width=6 by cfull_lift_sn, ceq_lift_sn/ diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma index e1263a2fd..8f3058cb9 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma @@ -42,9 +42,9 @@ lemma sex_co_dropable_sn: ∀RN,RP. co_dropable_sn (sex RN RP). qed-. lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f2,L1,L2. L1 ⪤[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 → - ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 → - f ~⊚ f1 ≘ f2 → L1 ⪤[RN, RP, f2] L2. + ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 → + f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2. #RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 // #g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H [ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct @@ -87,9 +87,9 @@ lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → ] qed-. -fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → - ∀f2,L1. L1 ⪤[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 → - ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[RN, RP, f1] K2. +fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → + ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 → + ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2. #RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H #H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/ @@ -113,10 +113,10 @@ lemma sex_co_dropable_dx: ∀RN,RP. co_dropable_dx (sex RN RP). /2 width=5 by sex_dropable_dx_aux/ qed-. lemma sex_drops_conf_next: ∀RN,RP. - ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ → + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2. + ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (sex_inv_next1 … HX) -HX @@ -124,30 +124,30 @@ elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf qed-. lemma sex_drops_conf_push: ∀RN,RP. - ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ → + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ⫯f1 ≘ f2 → - ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2. + ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (sex_inv_push1 … HX) -HX #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ → +lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≘ f2 → - ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2. + ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (sex_inv_next2 … HX) -HX #I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ → +lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ → ∀f1. f ~⊚ ⫯f1 ≘ f2 → - ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2. + ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (sex_inv_push2 … HX) -HX @@ -156,10 +156,10 @@ qed-. lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 → - ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 → + ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 → ∀f2. f ~⊚ f1 ≘ ↑f2 → - ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}. + ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX @@ -168,19 +168,19 @@ qed-. lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP → - ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 → - ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 → + ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 → + ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 → ∀f2. f ~⊚ f1 ≘ ⫯f2 → - ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}. + ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/ qed-. -lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ → - ∀f,L2. L1 ⪤[RN, RP, f] L2 → - ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆. +lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b,f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ → + ∀f,L2. L1 ⪤[RN,RP,f] L2 → + ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b,f1] L2 ≘ ⋆. #RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3 elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1 #L #H #HL2 lapply (sex_inv_atom1 … H) -H // diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_vector.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_vector.ma index 8c43b6404..4f5eef017 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_vector.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_vector.ma @@ -19,7 +19,7 @@ include "static_2/relocation/drops.ma". definition d_liftable1_all: predicate (relation2 lenv term) ≝ λR. ∀K,Ts. all … (R K) Ts → - ∀b,f,L. ⬇*[b, f] L ≘ K → + ∀b,f,L. ⬇*[b,f] L ≘ K → ∀Us. ⬆*[f] Ts ≘ Us → all … (R L) Us. (* Properties with generic relocation for term vectors **********************) diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_weight.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_weight.ma index dfcb44500..8c94cce14 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_weight.ma @@ -21,7 +21,7 @@ include "static_2/relocation/drops.ma". (* Forward lemmas with weight for local environments ************************) (* Basic_2A1: includes: drop_fwd_lw *) -lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}. +lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}. #b #f #L1 #L2 #H elim H -f -L1 -L2 // [ /2 width=3 by transitive_le/ | #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 normalize @@ -30,7 +30,7 @@ lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}. qed-. (* Basic_2A1: includes: drop_fwd_lw_lt *) -lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → +lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → (𝐈⦃f⦄ → ⊥) → ♯{L2} < ♯{L1}. #f #L1 #L2 #H elim H -f -L1 -L2 [ #f #Hf #Hnf elim Hnf -Hnf /2 width=1 by/ @@ -43,14 +43,14 @@ qed-. (* Forward lemmas with restricted weight for closures ***********************) (* Basic_2A1: includes: drop_fwd_rfw *) -lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b, f] L ≘ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}. +lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b,f] L ≘ K.ⓑ{I}V → ∀T. ♯{K,V} < ♯{L,T}. #b #f #I #L #K #V #HLK lapply (drops_fwd_lw … HLK) -HLK normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt, monotonic_lt_plus_r/ qed-. (* Advanced inversion lemma *************************************************) -lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b, f] L ≘ L.ⓘ{I} → ⊥. +lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b,f] L ≘ L.ⓘ{I} → ⊥. #b #f #I #L #H lapply (drops_fwd_lw … H) -b -f /2 width=4 by lt_le_false/ (**) (* full auto is a bit slow: 19s *) qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lex.ma index 1900d273d..76381a9fa 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/lex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lex.ma @@ -22,7 +22,7 @@ include "static_2/relocation/sex.ma". (* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************) definition lex (R): relation lenv ≝ - λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull, cext2 R, f] L2. + λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull,cext2 R,f] L2. interpretation "generic extension (local environment)" 'Relation R L1 L2 = (lex R L1 L2). diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma index 681363f03..4ec47cc91 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/lifts.ma @@ -24,7 +24,7 @@ include "static_2/syntax/term.ma". *) inductive lifts: rtmap → relation term ≝ | lifts_sort: ∀f,s. lifts f (⋆s) (⋆s) -| lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → lifts f (#i1) (#i2) +| lifts_lref: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → lifts f (#i1) (#i2) | lifts_gref: ∀f,l. lifts f (§l) (§l) | lifts_bind: ∀f,p,I,V1,V2,T1,T2. lifts f V1 V2 → lifts (⫯f) T1 T2 → @@ -80,7 +80,7 @@ lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s. /2 width=4 by lifts_inv_sort1_aux/ qed-. fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 → - ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2. + ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2. #f #X #Y * -f -X -Y [ #f #s #x #H destruct | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/ @@ -93,7 +93,7 @@ qed-. (* Basic_1: was: lift1_lref *) (* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *) lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y → - ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2. + ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2. /2 width=3 by lifts_inv_lref1_aux/ qed-. fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l. @@ -162,7 +162,7 @@ lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s. /2 width=4 by lifts_inv_sort2_aux/ qed-. fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 → - ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1. + ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1. #f #X #Y * -f -X -Y [ #f #s #x #H destruct | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/ @@ -175,7 +175,7 @@ qed-. (* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *) (* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *) lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 → - ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1. + ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1. /2 width=3 by lifts_inv_lref2_aux/ qed-. fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l. @@ -234,7 +234,7 @@ lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ⓕ{I}V2.T2 → lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y → ∨∨ ∃∃s. I = Sort s & Y = ⋆s - | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j + | ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j | ∃∃l. I = GRef l & Y = §l. #f * #n #Y #H [ lapply (lifts_inv_sort1 … H) @@ -245,7 +245,7 @@ qed-. lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} → ∨∨ ∃∃s. X = ⋆s & I = Sort s - | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j + | ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j | ∃∃l. X = §l & I = GRef l. #f * #n #X #H [ lapply (lifts_inv_sort2 … H) diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma index df09463c8..6247b8686 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma @@ -38,7 +38,7 @@ definition R_pw_confluent2_sex: relation3 lenv bind bind → relation3 lenv bind relation3 rtmap lenv bind ≝ λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0. ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 → - ∀L1. L0 ⪤[RN1, RP1, f] L1 → ∀L2. L0 ⪤[RN2, RP2, f] L2 → + ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 → ∃∃I. R2 L1 I1 I & R1 L2 I2 I. definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind → @@ -46,22 +46,22 @@ definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind relation3 lenv bind bind → relation3 lenv bind bind → relation3 rtmap lenv bind ≝ λR1,R2,R3,RN,RP,f,L1,I1. - ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN, RP, f] L2 → + ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 → ∀I2. R2 L2 I I2 → R3 L1 I1 I2. (* Basic inversion lemmas ***************************************************) -fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆. +fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I1 #I2 #L1 #L2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom1 *) -lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆. +lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆. /2 width=6 by sex_inv_atom1_aux/ qed-. -fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. +fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct | #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct @@ -71,12 +71,12 @@ fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K qed-. (* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ↑g] Y → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. +lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,↑g] Y → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. /2 width=7 by sex_inv_next1_aux/ qed-. -fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. +fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct | #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H) @@ -85,21 +85,21 @@ fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K ] qed-. -lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ⫯g] Y → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. +lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,⫯g] Y → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. /2 width=7 by sex_inv_push1_aux/ qed-. -fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆. +fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I1 #I2 #L1 #L2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom2 *) -lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆. +lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆. /2 width=6 by sex_inv_atom2_aux/ qed-. -fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. +fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J2 #K2 #H destruct | #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct @@ -109,12 +109,12 @@ fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K qed-. (* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ↑g] K2.ⓘ{J2} → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. +lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ{J2} → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. /2 width=7 by sex_inv_next2_aux/ qed-. -fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. +fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. #RN #RP #f #X #Y * -f -X -Y [ #f #J2 #K2 #g #H destruct | #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H) @@ -123,28 +123,28 @@ fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K ] qed-. -lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ⫯g] K2.ⓘ{J2} → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. +lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ{J2} → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. /2 width=7 by sex_inv_push2_aux/ qed-. (* Basic_2A1: includes lpx_sn_inv_pair *) lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2. + L1.ⓘ{I1} ⪤[RN,RP,↑f] L2.ⓘ{I2} → + L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2. + L1.ⓘ{I1} ⪤[RN,RP,⫯f] L2.ⓘ{I2} → + L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. -lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 → +lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫱f] L2 → RN L1 I1 I2 → RP L1 I1 I2 → - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}. + L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2}. #RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) * /2 width=1 by sex_next, sex_push/ qed-. @@ -152,8 +152,8 @@ qed-. (* Basic forward lemmas *****************************************************) lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, ⫱f] L2. + L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} → + L1 ⪤[RN,RP,⫱f] L2. #RN #RP #f #I1 #I2 #L1 #L2 #Hf elim (pn_split f) * #g #H destruct [ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf // @@ -161,7 +161,7 @@ qed-. (* Basic properties *********************************************************) -lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2). +lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2). #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H [ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/ @@ -169,7 +169,7 @@ lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] ] qed-. -lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2). +lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2). #RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *) qed-. @@ -189,20 +189,20 @@ lemma sex_sym: ∀RN,RP. qed-. lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → + L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} → ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 → - L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}. + L1.ⓘ{J1} ⪤[RN,RP,f] L2.ⓘ{J2}. /3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-. lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 → - ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2. + ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → L1 ⪤[RN2,RP2,f] L2. #RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by sex_atom, sex_next, sex_push/ qed-. lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 → - ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ → - L1 ⪤[RN2, RP2, f] L2. + ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → 𝐈⦃f⦄ → + L1 ⪤[RN2,RP2,f] L2. #RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 // #f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H [ elim (isid_inv_next … H) -H // @@ -211,8 +211,8 @@ lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 → qed-. lemma sex_sdj: ∀RN,RP. RP ⊆ RN → - ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 → - ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2. + ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 → + ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2. #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 [ elim (sdj_inv_nx … H12) -H12 [2,3: // ] @@ -223,8 +223,8 @@ lemma sex_sdj: ∀RN,RP. RP ⊆ RN → qed-. lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP → - ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2. + ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 → + ∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2. #RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 // #f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12 [ elim (pn_split f1) * ] @@ -236,8 +236,8 @@ lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP → qed-. lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN → - ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 → - ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2. + ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 → + ∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2. #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 [2: elim (pn_split f2) * ] @@ -249,8 +249,8 @@ lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN → qed-. lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → - ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ⊆ g → - ∃∃L. L1 ⪤[R1, RP, g] L & L ⪤[R2, cfull, f] L2. + ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ⊆ g → + ∃∃L. L1 ⪤[R1,RP,g] L & L ⪤[R2,cfull,f] L2. #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2 [ /2 width=3 by sex_atom, ex2_intro/ ] #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H @@ -262,8 +262,8 @@ lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → qed-. lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → - ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ∥ g → - ∃∃L. L1 ⪤[RP, R1, g] L & L ⪤[R2, cfull, f] L2. + ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g → + ∃∃L. L1 ⪤[RP,R1,g] L & L ⪤[R2,cfull,f] L2. #R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2 [ /2 width=3 by sex_atom, ex2_intro/ ] #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H @@ -277,7 +277,7 @@ qed-. lemma sex_dec: ∀RN,RP. (∀L,I1,I2. Decidable (RN L I1 I2)) → (∀L,I1,I2. Decidable (RP L I1 I2)) → - ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2). + ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2). #RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ] [ /2 width=1 by sex_atom, or_introl/ | #L2 #I2 #f @or_intror #H diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_length.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_length.ma index a7b2a5d71..4a2379889 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_length.ma @@ -20,13 +20,13 @@ include "static_2/relocation/sex.ma". (* Forward lemmas with length for local environments ************************) (* Note: "#f #I1 #I2 #L1 #L2 >length_bind >length_bind //" was needed to conclude *) -lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → |L1| = |L2|. +lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → |L1| = |L2|. #RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 // qed-. (* Properties with length for local environments ****************************) -lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull, cfull, f] L2. +lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull,cfull,f] L2. #L1 elim L1 -L1 [ #Y2 #H >(length_inv_zero_sn … H) -Y2 // | #L1 #I1 #IH #Y2 #H #f @@ -36,7 +36,7 @@ lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull, cfull, f] qed. lemma sex_length_isid: ∀R,L1,L2. |L1| = |L2| → - ∀f. 𝐈⦃f⦄ → L1 ⪤[R, cfull, f] L2. + ∀f. 𝐈⦃f⦄ → L1 ⪤[R,cfull,f] L2. #R #L1 elim L1 -L1 [ #Y2 #H >(length_inv_zero_sn … H) -Y2 // | #L1 #I1 #IH #Y2 #H #f #Hf diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma index 571379e2f..c1954b1a7 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma @@ -23,9 +23,9 @@ theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP): ∀L1,f. (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) → (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) → - ∀L0. L1 ⪤[RN1, RP1, f] L0 → - ∀L2. L0 ⪤[RN2, RP2, f] L2 → - L1 ⪤[RN, RP, f] L2. + ∀L0. L1 ⪤[RN1,RP1,f] L0 → + ∀L2. L0 ⪤[RN2,RP2,f] L2 → + L1 ⪤[RN,RP,f] L2. #RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1 [ #f #_ #_ #L0 #H1 #L2 #H2 lapply (sex_inv_atom1 … H1) -H1 #H destruct @@ -50,8 +50,8 @@ theorem sex_trans (RN) (RP) (f): (∀g,I,K. sex_transitive RN RN RN RN RP g K I) Transitive … (sex RN RP f). /2 width=9 by sex_trans_gen/ qed-. -theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1, cfull, f] L → 𝐈⦃f⦄ → - ∀L2. L ⪤[R2, cfull, f] L2 → L1 ⪤[R3, cfull, f] L2. +theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈⦃f⦄ → + ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2. #R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f [ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ] #f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H @@ -93,9 +93,9 @@ theorem sex_canc_dx: ∀RN,RP,f. Transitive … (sex RN RP f) → /3 width=3 by/ qed-. lemma sex_meet: ∀RN,RP,L1,L2. - ∀f1. L1 ⪤[RN, RP, f1] L2 → - ∀f2. L1 ⪤[RN, RP, f2] L2 → - ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN, RP, f] L2. + ∀f1. L1 ⪤[RN,RP,f1] L2 → + ∀f2. L1 ⪤[RN,RP,f2] L2 → + ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct @@ -106,9 +106,9 @@ try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H qed-. lemma sex_join: ∀RN,RP,L1,L2. - ∀f1. L1 ⪤[RN, RP, f1] L2 → - ∀f2. L1 ⪤[RN, RP, f2] L2 → - ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN, RP, f] L2. + ∀f1. L1 ⪤[RN,RP,f1] L2 → + ∀f2. L1 ⪤[RN,RP,f2] L2 → + ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2. #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_tc.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_tc.ma index 339ac98a7..8d1cf5538 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_tc.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_tc.ma @@ -34,7 +34,7 @@ lemma sex_tc_next_sn: ∀RN,RP. c_reflexive … RN → qed. lemma sex_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → - ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 → + ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 → TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}). #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2 /4 width=5 by sex_refl, sex_next, step, inj/ @@ -48,18 +48,18 @@ lemma sex_tc_push_sn: ∀RN,RP. c_reflexive … RP → qed. lemma sex_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → - ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 → + ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 → TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}). #RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2 /4 width=5 by sex_refl, sex_push, step, inj/ qed. -lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[CTC … RN, RP, f] L2. +lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[CTC … RN,RP,f] L2. #RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by sex_push, sex_next, inj/ qed. -lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[RN, CTC … RP, f] L2. +lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[RN,CTC … RP,f] L2. #RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 /3 width=1 by sex_push, sex_next, inj/ qed. @@ -82,8 +82,8 @@ qed. (* Basic_2A1: uses: TC_lpx_sn_ind *) theorem sex_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP → - ∀f,L1,L. L1 ⪤[RN, RP, f] L → 𝐈⦃f⦄ → - ∀L2. L ⪤[RN, CTC … RP, f] L2 → L1⪤ [RN, CTC … RP, f] L2. + ∀f,L1,L. L1 ⪤[RN,RP,f] L → 𝐈⦃f⦄ → + ∀L2. L ⪤[RN,CTC … RP,f] L2 → L1⪤ [RN,CTC … RP,f] L2. #RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L [ #f #_ #Y #H -HRP >(sex_inv_atom1 … H) -Y // ] #f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H @@ -99,7 +99,7 @@ qed-. (* Advanced properties ******************************************************) lemma sex_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP → - ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN, CTC … RP, f] L2. + ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN,CTC … RP,f] L2. #RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1 /3 width=3 by sex_tc_step_dx, sex_tc_inj_dx/ qed. @@ -107,13 +107,13 @@ qed. (* Advanced inversion lemmas ************************************************) lemma sex_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → - ∀f,L1,L2. L1 ⪤[CTC … RN, RP, f] L2 → TC … (sex RN RP f) L1 L2. + ∀f,L1,L2. L1 ⪤[CTC … RN,RP,f] L2 → TC … (sex RN RP f) L1 L2. #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by sex_tc_next, sex_tc_push_sn, sex_atom, inj/ qed-. lemma sex_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → - ∀f,L1,L2. L1 ⪤[RN, CTC … RP, f] L2 → TC … (sex RN RP f) L1 L2. + ∀f,L1,L2. L1 ⪤[RN,CTC … RP,f] L2 → TC … (sex RN RP f) L1 L2. #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by sex_tc_push, sex_tc_next_sn, sex_atom, inj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup.ma index e3dae5bf4..df964c8e9 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup.ma @@ -30,55 +30,55 @@ interpretation "plus-iterated structural successor (closure)" (* Basic properties *********************************************************) -lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. +lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. /2 width=1 by tri_inj/ qed. lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_step/ qed. lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. - ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. + ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_TC_strap/ qed. -lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄. +lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G,L,②{I}V.T⦄ ⊐+[b] ⦃G,L,V⦄. /2 width=1 by fqu_pair_sn, fqu_fqup/ qed. -lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄. +lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G,L.ⓑ{I}V,T⦄. /2 width=1 by fqu_bind_dx, fqu_fqup/ qed. -lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄. +lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G,L.ⓧ,T⦄. /3 width=1 by fqu_clear, fqu_fqup/ qed. -lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄. +lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G,L,ⓕ{I}V.T⦄ ⊐+[b] ⦃G,L,T⦄. /2 width=1 by fqu_flat_dx, fqu_fqup/ qed. -lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄. +lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G,L,V2⦄. /2 width=5 by fqu_pair_sn, fqup_strap1/ qed. -lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄. +lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G,L,ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I1}V1,T⦄. /2 width=5 by fqu_flat_dx, fqup_strap1/ qed. -lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄. +lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G,L.ⓑ{I2}V2,T⦄. /2 width=5 by fqu_bind_dx, fqup_strap1/ qed. (* Basic eliminators ********************************************************) lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. - (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2. + (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → + (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2. #b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H @(tri_TC_ind … IH1 IH2 G2 L2 T2 H) qed-. lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. - (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1. + (∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1) → + (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1. #b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H @(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H) qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_drops.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_drops.ma index 4eafa94d2..37bd15ada 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_drops.ma @@ -20,7 +20,7 @@ include "static_2/s_computation/fqup.ma". (* Properties with generic slicing for local environments *******************) lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ U → - ⦃G, L, U⦄ ⊐+[b] ⦃G, K, T⦄. + ⦃G,L,U⦄ ⊐+[b] ⦃G,K,T⦄. #b #G #K #T #i elim i -i [ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK #I #Y #HY #H destruct <(drops_fwd_isid … HY) -K // @@ -33,7 +33,7 @@ lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ qed. lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ⬆*[i] T1 ≘ U1 → - ⦃G1, K1, T1⦄ ⊐[b] ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, K2, T2⦄. + ⦃G1,K1,T1⦄ ⊐[b] ⦃G2,K2,T2⦄ → ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,K2,T2⦄. #b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 * [ #HLK1 #HTU1 #HT12 >(drops_fwd_isid … HLK1) -L1 // @@ -42,5 +42,5 @@ lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ] qed-. -lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+[b] ⦃G, K, V⦄. +lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L,#i⦄ ⊐+[b] ⦃G,K,V⦄. /2 width=6 by fqup_drops_strap1/ qed. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_weight.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_weight.ma index 25a81dfa8..c92721f84 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqup_weight.ma @@ -19,8 +19,8 @@ include "static_2/s_computation/fqup.ma". (* Forward lemmas with weight for closures **********************************) -lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ♯{G2, L2, T2} < ♯{G1, L1, T1}. +lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ♯{G2,L2,T2} < ♯{G1,L1,T1}. #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 /3 width=3 by fqu_fwd_fw, transitive_lt/ qed-. @@ -28,7 +28,7 @@ qed-. (* Advanced eliminators *****************************************************) lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. ( - ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → ∀G1,L1,T1. Q G1 L1 T1. #b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct @@ -36,7 +36,7 @@ lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. ( qed-. lemma fqup_wf_ind_eq: ∀b. ∀Q:relation3 …. ( - ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2 ) → ∀G1,L1,T1. Q G1 L1 T1. #b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma index fb6a7ea7d..1289e8487 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus.ma @@ -31,15 +31,15 @@ interpretation "star-iterated structural successor (closure)" (* Basic eliminators ********************************************************) lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 → - (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) → - ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2. + (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) → + ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2. #b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H @(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) // qed-. lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 → - (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) → - ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1. + (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) → + ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1. #b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) // qed-. @@ -49,56 +49,56 @@ qed-. lemma fqus_refl: ∀b. tri_reflexive … (fqus b). /2 width=1 by tri_inj/ qed. -lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=1 by tri_inj/ qed. -lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_step/ qed-. -lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. /2 width=5 by tri_TC_strap/ qed-. (* Basic inversion lemmas ***************************************************) -lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨ - ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄. + ∃∃G,L,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/ #G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/ * #HG #HL #HT #_ destruct // qed-. -lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨ - ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. + ∃∃J,L. ⦃G1,L,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}. #b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ qed-. -lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2 - | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0 - | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j. + | ∃∃J,L,V. ⦃G1,L,V⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓑ{J}V & i = 0 + | ∃∃J,L,j. ⦃G1,L,#j⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J} & i = ↑j. #b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/ #G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/ qed-. -lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ → (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨ - ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}. + ∃∃J,L. ⦃G1,L,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}. #b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/ qed-. -lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓧ,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ ∧ b = Ⓕ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. #b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/ #G #L #T #H elim (fqu_inv_bind1 … H) -H * [4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct @@ -106,21 +106,21 @@ lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ qed-. -lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ → +lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄ - | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐* ⦃G2,L2,T2⦄ + | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⊐* ⦃G2,L2,T2⦄ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐* ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}. #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ] /3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/ #_ #H destruct qed-. -lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2 - | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ - | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}. + | ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ + | ∃∃J,L,T. ⦃G1,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}. #b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/ #G #L #T #H elim (fqu_inv_flat1 … H) -H * [3: #J ] #H1 #H2 #H3 #H destruct @@ -129,35 +129,35 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ → +lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2. #b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/ #G #L #T #H elim (fqu_inv_atom1 … H) qed-. -lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1,L1.ⓘ{I},⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1,L1,⋆s⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_sort1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1,L1.ⓑ{I}V1,#0⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1,L1,V1⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_zero1_pair … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1,L1.ⓘ{I},#↑i⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1,L1,#i⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_lref1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ qed-. -lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1,L1.ⓘ{I},§l⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1,L1,§l⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/ #G #L #T #H elim (fqu_inv_gref1_bind … H) -H #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/ diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_drops.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_drops.ma index 0d2797139..7c0f79cba 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_drops.ma @@ -20,7 +20,7 @@ include "static_2/s_computation/fqus_fqup.ma". (* Properties with generic slicing for local environments *******************) lemma fqus_drops: ∀b,G,L,K,T,U,i. ⬇*[i] L ≘ K → ⬆*[i] T ≘ U → - ⦃G, L, U⦄ ⊐*[b] ⦃G, K, T⦄. + ⦃G,L,U⦄ ⊐*[b] ⦃G,K,T⦄. #b #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/ #HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K // qed. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_fqup.ma index 05a450dae..f5f521dd5 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_fqup.ma @@ -19,14 +19,14 @@ include "static_2/s_computation/fqus.ma". (* Alternative definition with plus-iterated supclosure *********************) -lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 /3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/ qed. (* Basic_2A1: was: fqus_inv_gen *) -lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2). +lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2). #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 // #G #G2 #L #L2 #T #T2 #_ * [ #H2 * /3 width=5 by fqup_strap1, or_introl/ @@ -37,38 +37,38 @@ qed-. (* Advanced properties ******************************************************) -lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. +lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1 [ /2 width=5 by fqup_strap1/ | * /2 width=1 by fqu_fqup/ ] qed-. -lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. +lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2 [ /2 width=5 by fqup_strap2/ | * /2 width=1 by fqu_fqup/ ] qed-. -lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. +lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2 /2 width=5 by fqus_strap1_fqu, fqup_strap1/ qed-. -lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → - ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄. +lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G,L,T⦄ → + ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄. #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1 /3 width=5 by fqus_strap2_fqu, fqup_strap2/ qed-. (* Advanced inversion lemmas for plus-iterated supclosure *******************) -lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → - ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄. +lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+[b] ⦃G2,L2,T2⦄ → + ∃∃G,L,T. ⦃G1,L1,T1⦄ ⊐[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄. #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/ #G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_weight.ma b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_weight.ma index f94366706..02be24fc7 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_weight.ma @@ -19,15 +19,15 @@ include "static_2/s_computation/fqus.ma". (* Forward lemmas with weight for closures **********************************) -lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}. +lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}. #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2 /3 width=3 by fquq_fwd_fw, transitive_le/ qed-. (* Advanced inversion lemmas ************************************************) -lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G, L, ⓪{I}⦄ ⊐*[b] ⦃G, L, X⦄ → ⓪{I} = X. +lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G,L,⓪{I}⦄ ⊐*[b] ⦃G,L,X⦄ → ⓪{I} = X. #b #I #G #L #X #H elim (fqus_inv_fqu_sn … H) -H * // #G0 #L0 #T0 #H1 #H2 lapply (fqu_fwd_fw … H1) lapply (fqus_fwd_fw … H2) -H2 -H1 #H2 #H1 lapply (le_to_lt_to_lt … H2 H1) -G0 -L0 -T0 diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu.ma index b834566d3..bff7b2fe9 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu.ma @@ -44,18 +44,18 @@ interpretation (* Basic properties *********************************************************) -lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄. +lemma fqu_sort: ∀b,I,G,L,s. ⦃G,L.ⓘ{I},⋆s⦄ ⊐[b] ⦃G,L,⋆s⦄. /2 width=1 by fqu_drop/ qed. -lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄. +lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G,L.ⓘ{I},#↑i⦄ ⊐[b] ⦃G,L,#i⦄. /2 width=1 by fqu_drop/ qed. -lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄. +lemma fqu_gref: ∀b,I,G,L,l. ⦃G,L.ⓘ{I},§l⦄ ⊐[b] ⦃G,L,§l⦄. /2 width=1 by fqu_drop/ qed. (* Basic inversion lemmas ***************************************************) -fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀s. T1 = ⋆s → ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s. #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 @@ -69,11 +69,11 @@ fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L ] qed-. -lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s. /2 width=4 by fqu_inv_sort1_aux/ qed-. -fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀i. T1 = #i → (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨ ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j. @@ -88,12 +88,12 @@ fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L ] qed-. -lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐[b] ⦃G2,L2,T2⦄ → (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨ ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j. /2 width=4 by fqu_inv_lref1_aux/ qed-. -fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀l. T1 = §l → ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l. #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 @@ -107,11 +107,11 @@ fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L ] qed-. -lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l. /2 width=4 by fqu_inv_gref1_aux/ qed-. -fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 @@ -127,14 +127,14 @@ fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L ] qed-. -lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1. /2 width=4 by fqu_inv_bind1_aux/ qed-. -lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ → +lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1. @@ -143,7 +143,7 @@ lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1 * #_ #_ #_ #H destruct qed-. -fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀I,V1,U1. T1 = ⓕ{I}V1.U1 → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1 = L2 & U1 = T2 @@ -158,7 +158,7 @@ fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L ] qed-. -lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2 | ∧∧ G1 = G2 & L1 = L2 & U1 = T2 | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1. @@ -166,31 +166,31 @@ lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[ (* Advanced inversion lemmas ************************************************) -lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥. +lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⊐[b] ⦃G2,L2,T2⦄ → ⊥. #b * #x #G1 #G2 #L2 #T2 #H [ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H #I [2: #V |3: #i ] #_ #H destruct qed-. -lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1,K.ⓘ{I},⋆s⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = ⋆s. #b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H #Z #X #H1 #H2 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1,K.ⓑ{I}V,#0⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = V. #b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H * #Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1,K.ⓘ{I},#(↑i)⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = #i. #b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H * #Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/ qed-. -lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1,K.ⓘ{I},§l⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L2 = K & T2 = §l. #b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H #Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/ diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_length.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_length.ma index 2cfea06b8..c2aa1a829 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_length.ma @@ -19,13 +19,13 @@ include "static_2/s_transition/fqu.ma". (* Forward lemmas with length for local environments ************************) -fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ∀i. T1 = #i → |L2| < |L1|. #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2,3: #p] #I #G #L #V #T [2: #_ ] #j #H destruct qed-. -lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ → +lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐[b] ⦃G2,L2,T2⦄ → |L2| < |L1|. /2 width=8 by fqu_fwd_length_lref1_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_tdeq.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_tdeq.ma index d08dc97d2..7d5e2e87d 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_tdeq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_tdeq.ma @@ -19,7 +19,7 @@ include "static_2/s_transition/fqu_length.ma". (* Inversion lemmas with context-free sort-irrelevant equivalence for terms *) -fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → +fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → G1 = G2 → |L1| = |L2| → T1 ≛ T2 → ⊥. #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 [1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H) @@ -29,7 +29,7 @@ fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2 qed-. (* Basic_2A1: uses: fqu_inv_eq *) -lemma fqu_inv_tdeq: ∀b,G,L1,L2,T1,T2. ⦃G, L1, T1⦄ ⊐[b] ⦃G, L2, T2⦄ → +lemma fqu_inv_tdeq: ∀b,G,L1,L2,T1,T2. ⦃G,L1,T1⦄ ⊐[b] ⦃G,L2,T2⦄ → |L1| = |L2| → T1 ≛ T2 → ⊥. #b #G #L1 #L2 #T1 #T2 #H @(fqu_inv_tdeq_aux … H) // (**) (* full auto fails *) diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_weight.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_weight.ma index 365f5c2db..2c19e9244 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fqu_weight.ma @@ -20,8 +20,8 @@ include "static_2/s_transition/fqu.ma". (* Forward lemmas with weight for closures **********************************) -lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → - ♯{G2, L2, T2} < ♯{G1, L1, T1}. +lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → + ♯{G2,L2,T2} < ♯{G1,L1,T1}. #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // #I #I1 #I2 #G #L #HI12 normalize in ⊢ (?%%); -I1 <(lifts_fwd_tw … HI12) /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/ @@ -30,7 +30,7 @@ qed-. (* Advanced eliminators *****************************************************) lemma fqu_wf_ind: ∀b. ∀Q:relation3 …. ( - ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) → + ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) → Q G1 L1 T1 ) → ∀G1,L1,T1. Q G1 L1 T1. #b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=2 by fqu_fwd_fw/ diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq.ma index fc36281aa..426c81518 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq.ma @@ -37,7 +37,7 @@ interpretation lemma fquq_refl: ∀b. tri_reflexive … (fquq b). // qed. -lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄. +lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄. /2 width=1 by or_introl/ qed. (* Basic_2A1: removed theorems 8: diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_length.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_length.ma index 08a70b7c3..2e11ace14 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_length.ma @@ -19,7 +19,7 @@ include "static_2/s_transition/fquq.ma". (* Forward lemmas with length for local environments ************************) -lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → +lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → |L2| ≤ |L1|. #b #G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ] /3 width=6 by fqu_fwd_length_lref1, lt_to_le/ diff --git a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_weight.ma b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_weight.ma index 8a8ef78a7..f9293ec68 100644 --- a/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/s_transition/fquq_weight.ma @@ -19,8 +19,8 @@ include "static_2/s_transition/fquq.ma". (* Forward lemmas with weight for closures **********************************) -lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → - ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}. +lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮[b] ⦃G2,L2,T2⦄ → + ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}. #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ] /3 width=2 by fqu_fwd_fw, lt_to_le/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa.ma index b7e9ca046..d204bdc21 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa.ma @@ -37,7 +37,7 @@ interpretation "atomic arity assignment (term)" (* Basic inversion lemmas ***************************************************) -fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪. +fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪. #G #L #T #A * -G -L -T -A // [ #I #G #L #V #B #_ #s #H destruct | #I #G #L #A #i #_ #s #H destruct @@ -48,11 +48,11 @@ fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → ] qed-. -lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪. +lemma aaa_inv_sort: ∀G,L,A,s. ⦃G,L⦄ ⊢ ⋆s ⁝ A → A = ⓪. /2 width=6 by aaa_inv_sort_aux/ qed-. -fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 → - ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A. +fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → T = #0 → + ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A. #G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/ [ #G #L #s #H destruct | #I #G #L #A #i #_ #H destruct @@ -63,12 +63,12 @@ fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 → ] qed-. -lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A → - ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A. +lemma aaa_inv_zero: ∀G,L,A. ⦃G,L⦄ ⊢ #0 ⁝ A → + ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A. /2 width=3 by aaa_inv_zero_aux/ qed-. -fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) → - ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A. +fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) → + ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A. #G #L #T #A * -G -L -T -A [ #G #L #s #j #H destruct | #I #G #L #V #B #_ #j #H destruct @@ -80,11 +80,11 @@ fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) ] qed-. -lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A → - ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A. +lemma aaa_inv_lref: ∀G,L,A,i. ⦃G,L⦄ ⊢ #↑i ⁝ A → + ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A. /2 width=3 by aaa_inv_lref_aux/ qed-. -fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥. +fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥. #G #L #T #A * -G -L -T -A [ #G #L #s #k #H destruct | #I #G #L #V #B #_ #k #H destruct @@ -96,11 +96,11 @@ fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ] qed-. -lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥. +lemma aaa_inv_gref: ∀G,L,A,l. ⦃G,L⦄ ⊢ §l ⁝ A → ⊥. /2 width=7 by aaa_inv_gref_aux/ qed-. -fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U → - ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A. +fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U → + ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L.ⓓW⦄ ⊢ U ⁝ A. #G #L #T #A * -G -L -T -A [ #G #L #s #q #W #U #H destruct | #I #G #L #V #B #_ #q #W #U #H destruct @@ -112,12 +112,12 @@ fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{ ] qed-. -lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A → - ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A. +lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G,L⦄ ⊢ ⓓ{p}V.T ⁝ A → + ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L.ⓓV⦄ ⊢ T ⁝ A. /2 width=4 by aaa_inv_abbr_aux/ qed-. -fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U → - ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2. +fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U → + ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2. #G #L #T #A * -G -L -T -A [ #G #L #s #q #W #U #H destruct | #I #G #L #V #B #_ #q #W #U #H destruct @@ -129,12 +129,12 @@ fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{ ] qed-. -lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A → - ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2. +lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G,L⦄ ⊢ ⓛ{p}W.T ⁝ A → + ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2. /2 width=4 by aaa_inv_abst_aux/ qed-. -fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U → - ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A. +fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U → + ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L⦄ ⊢ U ⁝ ②B.A. #G #L #T #A * -G -L -T -A [ #G #L #s #W #U #H destruct | #I #G #L #V #B #_ #W #U #H destruct @@ -146,12 +146,12 @@ fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U ] qed-. -lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A → - ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A. +lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G,L⦄ ⊢ ⓐV.T ⁝ A → + ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L⦄ ⊢ T ⁝ ②B.A. /2 width=3 by aaa_inv_appl_aux/ qed-. -fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U → - ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A. +fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U → + ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ U ⁝ A. #G #L #T #A * -G -L -T -A [ #G #L #s #W #U #H destruct | #I #G #L #V #B #_ #W #U #H destruct @@ -163,6 +163,6 @@ fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U ] qed-. -lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A → - ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A. +lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G,L⦄ ⊢ ⓝW.T ⁝ A → + ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ T ⁝ A. /2 width=3 by aaa_inv_cast_aux/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_aaa.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_aaa.ma index 15b4fe4e1..bd1eb91a2 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa_aaa.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa_aaa.ma @@ -18,7 +18,7 @@ include "static_2/static/aaa.ma". (* Main inversion lemmas ****************************************************) -theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2. +theorem aaa_mono: ∀G,L,T,A1. ⦃G,L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T ⁝ A2 → A1 = A2. #G #L #T #A1 #H elim H -G -L -T -A1 [ #G #L #s #A2 #H >(aaa_inv_sort … H) -H // | #I1 #G #L #V1 #B #_ #IH #A2 #H diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_drops.ma index 1cc311ddf..b006c2d21 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa_drops.ma @@ -22,7 +22,7 @@ include "static_2/static/aaa.ma". (* Advanced properties ******************************************************) (* Basic_2A1: was: aaa_lref *) -lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B. +lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ B → ⦃G,L⦄ ⊢ #i ⁝ B. #I #G #K #V #B #i elim i -i [ #L #H lapply (drops_fwd_isid … H ?) -H // #H destruct /2 width=1 by aaa_zero/ @@ -34,8 +34,8 @@ qed. (* Advanced inversion lemmas ************************************************) (* Basic_2A1: was: aaa_inv_lref *) -lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A → - ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A. +lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G,L⦄ ⊢ #i ⁝ A → + ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A. #G #A #i elim i -i [ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/ | #i #IH #L #H elim (aaa_inv_lref … H) -H @@ -47,8 +47,8 @@ qed-. (* Basic_2A1: includes: aaa_lift *) (* Note: it should use drops_split_trans_pair2 *) -lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 → - ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A. +lemma aaa_lifts: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b,f] L2 ≘ L1 → + ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G,L2⦄ ⊢ T2 ⁝ A. @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * * [ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH lapply (aaa_inv_sort … H) -H #H destruct @@ -86,8 +86,8 @@ qed-. (* Inversion lemmas with generic slicing for local environments *************) (* Basic_2A1: includes: aaa_inv_lift *) -lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 → - ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A. +lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G,L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b,f] L2 ≘ L1 → + ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G,L1⦄ ⊢ T1 ⁝ A. @(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * * [ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH lapply (aaa_inv_sort … H) -H #H destruct diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_fdeq.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_fdeq.ma index 777183562..d002095bb 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa_fdeq.ma @@ -19,7 +19,7 @@ include "static_2/static/aaa_rdeq.ma". (* Properties with sort-irrelevant equivalence on referred entries **********) -lemma aaa_fdeq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → - ∀A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → ⦃G2, L2⦄ ⊢ T2 ⁝ A. +lemma aaa_fdeq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → + ∀A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → ⦃G2,L2⦄ ⊢ T2 ⁝ A. #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=5 by aaa_tdeq_conf_rdeq/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_fqus.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_fqus.ma index f02455f9f..d3b9d83cb 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa_fqus.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa_fqus.ma @@ -19,8 +19,8 @@ include "static_2/static/aaa_drops.ma". (* Properties on supclosure *************************************************) -lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐ ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 [ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H #J #K #V #H #HA destruct /2 width=2 by ex_intro/ @@ -43,21 +43,21 @@ lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ] qed-. -lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐⸮ ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/ * #H1 #H2 #H3 destruct /2 width=2 by ex_intro/ qed-. -lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐+ ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 [2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ] /2 width=6 by aaa_fqu_conf/ qed-. -lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → - ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2. +lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⊐* ⦃G2,L2,T2⦄ → + ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2. #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/ * #H1 #H2 #H3 destruct /2 width=2 by ex_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_rdeq.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_rdeq.ma index 265ed7f5c..c08a02487 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/aaa_rdeq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/aaa_rdeq.ma @@ -19,8 +19,8 @@ include "static_2/static/aaa.ma". (* Properties with sort-irrelevant equivalence on referred entries **********) -lemma aaa_tdeq_conf_rdeq: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛ T2 → - ∀L2. L1 ≛[T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A. +lemma aaa_tdeq_conf_rdeq: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛ T2 → + ∀L2. L1 ≛[T1] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A. #G #L1 #T1 #A #H elim H -G -L1 -T1 -A [ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 // | #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1 diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fdeq.ma b/matita/matita/contribs/lambdadelta/static_2/static/fdeq.ma index be82b9302..1c5b47ba1 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fdeq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fdeq.ma @@ -30,17 +30,17 @@ interpretation (* Basic_properties *********************************************************) lemma fdeq_intro_dx (G): ∀L1,L2,T2. L1 ≛[T2] L2 → - ∀T1. T1 ≛ T2 → ⦃G, L1, T1⦄ ≛ ⦃G, L2, T2⦄. + ∀T1. T1 ≛ T2 → ⦃G,L1,T1⦄ ≛ ⦃G,L2,T2⦄. /3 width=3 by fdeq_intro_sn, tdeq_rdeq_div/ qed. (* Basic inversion lemmas ***************************************************) -lemma fdeq_inv_gen_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → +lemma fdeq_inv_gen_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L1 ≛[T1] L2 & T1 ≛ T2. #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/ qed-. -lemma fdeq_inv_gen_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → +lemma fdeq_inv_gen_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ∧∧ G1 = G2 & L1 ≛[T2] L2 & T1 ≛ T2. #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /3 width=3 by tdeq_rdeq_conf, and3_intro/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fdeq.ma b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fdeq.ma index d337f7054..1edc17d50 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fdeq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fdeq.ma @@ -32,18 +32,18 @@ theorem fdeq_trans: tri_transitive … fdeq. /4 width=5 by fdeq_intro_sn, rdeq_trans, tdeq_rdeq_div, tdeq_trans/ qed-. -theorem fdeq_canc_sn: ∀G,G1,L,L1,T,T1. ⦃G, L, T⦄ ≛ ⦃G1, L1, T1⦄→ - ∀G2,L2,T2. ⦃G, L, T⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄. +theorem fdeq_canc_sn: ∀G,G1,L,L1,T,T1. ⦃G,L,T⦄ ≛ ⦃G1,L1,T1⦄→ + ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄. /3 width=5 by fdeq_trans, fdeq_sym/ qed-. -theorem fdeq_canc_dx: ∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G2, L2, T2⦄ ≛ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄. +theorem fdeq_canc_dx: ∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G2,L2,T2⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄. /3 width=5 by fdeq_trans, fdeq_sym/ qed-. (* Main inversion lemmas with degree-based equivalence on terms *************) -theorem fdeq_tdneq_repl_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄ → - ∀U1,U2. ⦃G1, L1, U1⦄ ≛ ⦃G2, L2, U2⦄ → +theorem fdeq_tdneq_repl_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → + ∀U1,U2. ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄ → (T2 ≛ U2 → ⊥) → (T1 ≛ U1 → ⊥). #G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1 elim (fdeq_inv_gen_sn … HT) -HT #_ #_ #HT diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqup.ma index 19fe848f7..333a0f787 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqup.ma @@ -20,7 +20,7 @@ include "static_2/static/fdeq.ma". (* Properties with sort-irrelevant equivalence for terms ********************) lemma tdeq_fdeq: ∀T1,T2. T1 ≛ T2 → - ∀G,L. ⦃G, L, T1⦄ ≛ ⦃G, L, T2⦄. + ∀G,L. ⦃G,L,T1⦄ ≛ ⦃G,L,T2⦄. /2 width=1 by fdeq_intro_sn/ qed. (* Advanced properties ******************************************************) diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqus.ma b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqus.ma index 9662b6b32..26ae320e7 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqus.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_fqus.ma @@ -19,9 +19,9 @@ include "static_2/static/fdeq.ma". (* Properties with star-iterated structural successor for closures **********) -lemma fdeq_fqus_trans: ∀b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛ ⦃G, L, T⦄ → - ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → - ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛ ⦃G2, L2, T2⦄. +lemma fdeq_fqus_trans: ∀b,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → + ∀G2,L2,T2. ⦃G,L,T⦄ ⊐*[b] ⦃G2,L2,T2⦄ → + ∃∃G,L0,T0. ⦃G1,L1,T1⦄ ⊐*[b] ⦃G,L0,T0⦄ & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄. #b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2 elim(fdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct elim (rdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2 diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_req.ma b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_req.ma index c430e7aae..5dbf18b5e 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fdeq_req.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fdeq_req.ma @@ -20,7 +20,7 @@ include "static_2/static/fdeq.ma". (* Properties with syntactic equivalence on referred entries ****************) lemma req_rdeq_trans: ∀L1,L,T1. L1 ≡[T1] L → - ∀G1,G2,L2,T2. ⦃G1, L, T1⦄ ≛ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛ ⦃G2, L2, T2⦄. + ∀G1,G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄. #L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H elim (fdeq_inv_gen_sn … H) -H #H #HL2 #T12 destruct /3 width=3 by fdeq_intro_sn, req_rdeq_trans/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma index 3c6466086..8a9fc3ce0 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma @@ -20,7 +20,7 @@ include "static_2/static/frees_fqup.ma". (* Advanced properties ******************************************************) -lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ → +lemma frees_atom_drops: ∀b,L,i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f. #b #L elim L -L /2 width=1 by frees_atom/ #L #I #IH * @@ -74,7 +74,7 @@ qed. (* Advanced inversion lemmas ************************************************) lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f → - ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g + ∨∨ ∃∃g. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g & ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g. @@ -98,7 +98,7 @@ qed-. (* Properties with generic slicing for local environments *******************) lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 → - ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U → + ∀f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U → ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2. #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T [ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3 @@ -154,7 +154,7 @@ lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 → ] qed-. -lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U → +lemma frees_lifts_SO: ∀b,L,K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U → ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f. #b #L #K #HLK #T #U #HTU #f #Hf @(frees_lifts b … Hf … HTU) // (**) (* auto fails *) @@ -163,21 +163,21 @@ qed. (* Forward lemmas with generic slicing for local environments ***************) lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → - ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U → ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2. /4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-. (* Inversion lemmas with generic slicing for local environments *************) lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → - ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U → ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1. #b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T) /3 width=9 by frees_fwd_coafter, ex2_intro/ qed-. lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f → - ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U → + ∀K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U → K ⊢ 𝐅*⦃T⦄ ≘ ⫱f. #b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U #f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf @@ -185,7 +185,7 @@ lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f → qed-. lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 → - ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U → + ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U → ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1. #b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U /3 width=7 by frees_eq_repl_back, coafter_inj/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fsle.ma b/matita/matita/contribs/lambdadelta/static_2/static/fsle.ma index bfafabbdb..51c334d10 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fsle.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fsle.ma @@ -21,7 +21,7 @@ include "static_2/static/frees.ma". definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2. ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 & - L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2. + L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2. interpretation "free variables inclusion (restricted closure)" 'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2). @@ -31,8 +31,8 @@ interpretation "free variables inclusion (term)" (* Basic properties *********************************************************) -lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄. +lemma fsle_sort: ∀L,s1,s2. ⦃L,⋆s1⦄ ⊆ ⦃L,⋆s2⦄. /3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed. -lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄. +lemma fsle_gref: ∀L,l1,l2. ⦃L,§l1⦄ ⊆ ⦃L,§l2⦄. /3 width=8 by frees_gref, sle_refl, ex4_4_intro/ qed. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fsle_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/fsle_drops.ma index 542d8b10e..1eee58094 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fsle_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fsle_drops.ma @@ -20,7 +20,7 @@ include "static_2/static/fsle_length.ma". (* Advanced properties ******************************************************) lemma fsle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| → - ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄. + ∀T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ⦃L1.ⓧ,U1⦄ ⊆ ⦃L2,T2⦄. #T1 #U1 #HTU1 #L1 #L2 #H1L #T2 * #n #m #f #g #Hf #Hg #H2L #Hfg lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct @@ -40,8 +40,8 @@ lapply (frees_lifts_SO (Ⓣ) (L2.ⓧ) … HTU2 … Hg) @(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_dx/ (**) (* explict constructor *) qed-. -lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ → - ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1}, W1⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄. +lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ → + ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1},W1⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄. #K1 #K2 #HK #V1 #V2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12 #W1 #HVW1 #I1 #I2 @@ -49,9 +49,9 @@ elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct /5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/ qed. -lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ → +lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1,T1⦄ ⊆ ⦃K2,T2⦄ → ∀U1,U2. ⬆*[1] T1 ≘ U1 → ⬆*[1] T2 ≘ U2 → - ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄. + ∀I1,I2. ⦃K1.ⓘ{I1},U1⦄ ⊆ ⦃K2.ⓘ{I2},U2⦄. #K1 #K2 #HK #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12 #U1 #U2 #HTU1 #HTU2 #I1 #I2 @@ -62,8 +62,8 @@ qed. (* Advanced inversion lemmas ************************************************) lemma fsle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → - ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2, U2⦄ → - ∀p. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.U2⦄. + ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2,U2⦄ → + ∀p. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.U2⦄. #T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2 * #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fsle_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/static/fsle_fqup.ma index 8056bf243..04162cc18 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/fsle_fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/fsle_fqup.ma @@ -26,8 +26,8 @@ elim (frees_total L T) #f #Hf qed. lemma fsle_shift: ∀L1,L2. |L1| = |L2| → - ∀I,T1,T2,V. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ → - ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄. + ∀I,T1,T2,V. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I}V,T2⦄ → + ∀p. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V.T2⦄. #L1 #L2 #H1L #I #T1 #T2 #V * #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct @@ -38,8 +38,8 @@ lapply (sor_inv_sle_dx … Hg) #H0g /4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/ qed. -lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ → - ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄. +lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ → + ∀p,I,T2. ⦃L1,V1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄. #L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2 elim (frees_total (L2.ⓧ) T2) #g2 #Hg2 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_ @@ -47,8 +47,8 @@ elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg # /4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/ qed. -lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| → - ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄. +lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2.ⓧ,T2⦄ → |L1| ≤ |L2| → + ∀p,I,V2. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄. #L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2 elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct (tdeq_inv_lref1 … H) -X /2 width=5 by fqu_lref_O, ex3_2_intro/ @@ -45,18 +45,18 @@ lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, ] qed-. -lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ → +lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐[b] ⦃G2,L2,T1⦄ → ∀U2. U2 ≛ U1 → - ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2. + ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2. #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21 elim (fqu_tdeq_conf … H12 U2) -H12 /3 width=5 by rdeq_sym, tdeq_sym, ex3_2_intro/ qed-. (* Basic_2A1: uses: lleq_fqu_trans *) -lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ → +lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐[b] ⦃G2,K2,U⦄ → ∀L1. L1 ≛[T] L2 → - ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2. + ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2. #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U [ #I #G #L2 #V2 #L1 #H elim (rdeq_inv_zero_pair_dx … H) -H #K1 #V1 #HV1 #HV12 #H destruct @@ -80,9 +80,9 @@ qed-. (* Properties with optional structural successor for closures ***************) -lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ → +lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐⸮[b] ⦃G2,L2,T1⦄ → ∀U2. U2 ≛ U1 → - ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2. + ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐⸮[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2. #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H [ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1 /3 width=5 by fqu_fquq, ex3_2_intro/ @@ -91,9 +91,9 @@ lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, qed-. (* Basic_2A1: was just: lleq_fquq_trans *) -lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ → +lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐⸮[b] ⦃G2,K2,U⦄ → ∀L1. L1 ≛[T] L2 → - ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2. + ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐⸮[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2. #b #G1 #G2 #L2 #K2 #T #U #H elim H -H [ #H #L1 #HL12 elim (rdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/ | * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/ @@ -103,9 +103,9 @@ qed-. (* Properties with plus-iterated structural successor for closures **********) (* Basic_2A1: was just: lleq_fqup_trans *) -lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ → +lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐+[b] ⦃G2,K2,U⦄ → ∀L1. L1 ≛[T] L2 → - ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2. + ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐+[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2. #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U [ #G2 #K2 #U #HTU #L1 #HL12 elim (rdeq_fqu_trans … HTU … HL12) -L2 /3 width=5 by fqu_fqup, ex3_2_intro/ @@ -118,9 +118,9 @@ lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, ] qed-. -lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ → +lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐+[b] ⦃G2,L2,T1⦄ → ∀U2. U2 ≛ U1 → - ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2. + ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐+[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2. #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1 [ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1 /3 width=5 by fqu_fqup, ex3_2_intro/ @@ -136,9 +136,9 @@ qed-. (* Properties with star-iterated structural successor for closures **********) -lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ → +lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⊐*[b] ⦃G2,L2,T1⦄ → ∀U2. U2 ≛ U1 → - ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛ T1 & L ≛[T1] L2. + ∃∃L,T2. ⦃G1,L1,U2⦄ ⊐*[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2. #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H [ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/ | * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/ @@ -146,9 +146,9 @@ lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L qed-. (* Basic_2A1: was just: lleq_fqus_trans *) -lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ → +lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⊐*[b] ⦃G2,K2,U⦄ → ∀L1. L1 ≛[T] L2 → - ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛ U & K1 ≛[U] K2. + ∃∃K1,U0. ⦃G1,L1,T⦄ ⊐*[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2. #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H [ #H elim (rdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/ | * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rdeq_length.ma b/matita/matita/contribs/lambdadelta/static_2/static/rdeq_length.ma index c5b0e5f31..967c4c0e6 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rdeq_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rdeq_length.ma @@ -44,7 +44,7 @@ lemma rdeq_unit_length: ∀L1,L2. |L1| = |L2| → (* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *) lemma rdeq_lifts_bi: ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[T] K2 → - ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 → + ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 → ∀U. ⬆*[f] T ≘ U → L1 ≛[U] L2. /3 width=9 by rex_lifts_bi, tdeq_lifts_sn/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/req.ma b/matita/matita/contribs/lambdadelta/static_2/static/req.ma index 13965b92d..46e436d68 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/req.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/req.ma @@ -69,7 +69,7 @@ lemma req_inv_lref_bind_dx: ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ{I2} → (* Basic_2A1: was: llpx_sn_lrefl *) (* Basic_2A1: this should have been lleq_fwd_llpx_sn *) lemma req_fwd_rex: ∀R. c_reflexive … R → - ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R, T] L2. + ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R,T] L2. #R #HR #L1 #L2 #T * #f #Hf #HL12 /4 width=7 by sex_co, cext2_co, ex2_intro/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/req_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/req_drops.ma index 0eae38683..76f25cd40 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/req_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/req_drops.ma @@ -21,6 +21,6 @@ include "static_2/static/req.ma". (* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *) lemma req_inv_lifts_bi: ∀L1,L2,U. L1 ≡[U] L2 → ∀b,f. 𝐔⦃f⦄ → - ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 → + ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 → ∀T. ⬆*[f] T ≘ U → K1 ≡[T] K2. /2 width=10 by rex_inv_lifts_bi/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/req_fsle.ma b/matita/matita/contribs/lambdadelta/static_2/static/req_fsle.ma index d91c9905b..efafbb903 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/req_fsle.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/req_fsle.ma @@ -29,5 +29,5 @@ qed. (* Forward lemmas with free variables inclusion for restricted closures *****) lemma req_rex_trans: ∀R. req_transitive R → - ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R, T] L2 → L1 ⪤[R, T] L2. + ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R,T] L2 → L1 ⪤[R,T] L2. /4 width=16 by req_fsle_comp, rex_trans_fsle, rex_trans_next/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma index 515e24bef..0848e8503 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma @@ -21,7 +21,7 @@ include "static_2/static/frees.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) definition rex (R) (T): relation lenv ≝ - λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R, cfull, f] L2. + λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R,cfull,f] L2. interpretation "generic extension on referred entries (local environment)" 'Relation R T L1 L2 = (rex R T L1 L2). @@ -30,32 +30,32 @@ definition R_confluent2_rex: relation4 (relation3 lenv term term) (relation3 lenv term term) … ≝ λR1,R2,RP1,RP2. ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⪤[RP1, T0] L1 → ∀L2. L0 ⪤[RP2, T0] L2 → + ∀L1. L0 ⪤[RP1,T0] L1 → ∀L2. L0 ⪤[RP2,T0] L2 → ∃∃T. R2 L1 T1 T & R1 L2 T2 T. definition rex_confluent: relation … ≝ λR1,R2. - ∀K1,K,V1. K1 ⪤[R1, V1] K → ∀V. R1 K1 V1 V → - ∀K2. K ⪤[R2, V] K2 → K ⪤[R2, V1] K2. + ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V → + ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2. definition rex_transitive: relation3 ? (relation3 ?? term) … ≝ λR1,R2,R3. - ∀K1,K,V1. K1 ⪤[R1, V1] K → + ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2. (* Basic inversion lemmas ***************************************************) -lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R, T] Y2 → Y2 = ⋆. +lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R,T] Y2 → Y2 = ⋆. #R #Y2 #T * /2 width=4 by sex_inv_atom1/ qed-. -lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R, T] ⋆ → Y1 = ⋆. +lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R,T] ⋆ → Y1 = ⋆. #R #I #Y1 * /2 width=4 by sex_inv_atom2/ qed-. -lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 → +lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R,⋆s] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, ⋆s] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -66,11 +66,11 @@ lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 → ] qed-. -lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 → +lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R,#0] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 & + | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 - | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R, cfull, f] L2 & + | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R,cfull,f] L2 & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}. #R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/ @@ -84,9 +84,9 @@ lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 → ] qed-. -lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 → +lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R,#↑i] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, #i] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,#i] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -96,9 +96,9 @@ lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 → ] qed-. -lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 → +lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R,§l] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, §l] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -110,39 +110,39 @@ lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 → qed-. (* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *) -lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → - ∧∧ L1 ⪤[R, V1] L2 & L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2. +lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → + ∧∧ L1 ⪤[R,V1] L2 & L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2. #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf /6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Basic_2A1: uses: llpx_sn_inv_flat *) -lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → - ∧∧ L1 ⪤[R, V] L2 & L1 ⪤[R, T] L2. +lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → + ∧∧ L1 ⪤[R,V] L2 & L1 ⪤[R,T] L2. #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf /5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Advanced inversion lemmas ************************************************) -lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R, ⋆s] L2 → - ∃∃I2,K2. K1 ⪤[R, ⋆s] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R,⋆s] L2 → + ∃∃I2,K2. K1 ⪤[R,⋆s] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R, ⋆s] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, ⋆s] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R,⋆s] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,⋆s] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 → - ∃∃K2,V2. K1 ⪤[R, V1] K2 & R K1 V1 V2 & +lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R,#0] L2 → + ∃∃K2,V2. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2. #R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H * [ #H destruct @@ -152,8 +152,8 @@ lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 → ] qed-. -lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 → - ∃∃K1,V1. K1 ⪤[R, V1] K2 & R K1 V1 V2 & +lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R,#0] K2.ⓑ{I}V2 → + ∃∃K1,V1. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1. #R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H * [ #_ #H destruct @@ -163,8 +163,8 @@ lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 → ] qed-. -lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 → - ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 & +lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R,#0] L2 → + ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L2 = K2.ⓤ{I}. #R #I #K1 #L2 #H elim (rex_inv_zero … H) -H * [ #H destruct @@ -173,8 +173,8 @@ lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 → ] qed-. -lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} → - ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 & +lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R,#0] K2.ⓤ{I} → + ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L1 = K1.ⓤ{I}. #R #I #L1 #K2 #H elim (rex_inv_zero … H) -H * [ #_ #H destruct @@ -183,32 +183,32 @@ lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} → ] qed-. -lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R, #↑i] L2 → - ∃∃I2,K2. K1 ⪤[R, #i] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R,#↑i] L2 → + ∃∃I2,K2. K1 ⪤[R,#i] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R, #↑i] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, #i] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R,#↑i] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,#i] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R, §l] L2 → - ∃∃I2,K2. K1 ⪤[R, §l] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R,§l] L2 → + ∃∃I2,K2. K1 ⪤[R,§l] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R, §l] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, §l] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R,§l] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,§l] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ @@ -218,30 +218,30 @@ qed-. (* Basic forward lemmas *****************************************************) lemma rex_fwd_zero_pair (R): ∀I,K1,K2,V1,V2. - K1.ⓑ{I}V1 ⪤[R, #0] K2.ⓑ{I}V2 → K1 ⪤[R, V1] K2. + K1.ⓑ{I}V1 ⪤[R,#0] K2.ⓑ{I}V2 → K1 ⪤[R,V1] K2. #R #I #K1 #K2 #V1 #V2 #H elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct // qed-. (* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *) -lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R, ②{I}V.T] L2 → L1 ⪤[R, V] L2. +lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R,②{I}V.T] L2 → L1 ⪤[R,V] L2. #R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL [ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf /4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/ qed-. (* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *) -lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2. +lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2. #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV // qed-. (* Basic_2A1: uses: llpx_sn_fwd_flat_dx *) -lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → L1 ⪤[R, T] L2. +lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → L1 ⪤[R,T] L2. #R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H // qed-. -lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R, T] K2.ⓘ{I2} → +lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ{I2} → ∃∃I1,K1. L1 = K1.ⓘ{I1}. #R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct [ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct @@ -250,63 +250,63 @@ qed-. (* Basic properties *********************************************************) -lemma rex_atom (R): ∀I. ⋆ ⪤[R, ⓪{I}] ⋆. +lemma rex_atom (R): ∀I. ⋆ ⪤[R,⓪{I}] ⋆. #R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/ qed. lemma rex_sort (R): ∀I1,I2,L1,L2,s. - L1 ⪤[R, ⋆s] L2 → L1.ⓘ{I1} ⪤[R, ⋆s] L2.ⓘ{I2}. + L1 ⪤[R,⋆s] L2 → L1.ⓘ{I1} ⪤[R,⋆s] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #s * #f #Hf #H12 lapply (frees_inv_sort … Hf) -Hf /4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/ qed. -lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, #0] L2.ⓑ{I}V2. +lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,#0] L2.ⓑ{I}V2. #R #I1 #I2 #L1 #L2 #V1 * /4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/ qed. -lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 → - L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}. +lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 → + L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}. /4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed. lemma rex_lref (R): ∀I1,I2,L1,L2,i. - L1 ⪤[R, #i] L2 → L1.ⓘ{I1} ⪤[R, #↑i] L2.ⓘ{I2}. + L1 ⪤[R,#i] L2 → L1.ⓘ{I1} ⪤[R,#↑i] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/ qed. lemma rex_gref (R): ∀I1,I2,L1,L2,l. - L1 ⪤[R, §l] L2 → L1.ⓘ{I1} ⪤[R, §l] L2.ⓘ{I2}. + L1 ⪤[R,§l] L2 → L1.ⓘ{I1} ⪤[R,§l] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #l * #f #Hf #H12 lapply (frees_inv_gref … Hf) -Hf /4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/ qed. lemma rex_bind_repl_dx (R): ∀I,I1,L1,L2,T. - L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I1} → + L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I1} → ∀I2. cext2 R L1 I I2 → - L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I2}. + L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I2}. #R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR /3 width=5 by sex_pair_repl, ex2_intro/ qed-. (* Basic_2A1: uses: llpx_sn_co *) lemma rex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → - ∀L1,L2,T. L1 ⪤[R1, T] L2 → L1 ⪤[R2, T] L2. + ∀L1,L2,T. L1 ⪤[R1,T] L2 → L1 ⪤[R2,T] L2. #R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/ qed-. lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2. (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) → (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) → - L1 ⪤[R1, T1] L2 → L1 ⪤[R2, T2] L2. + L1 ⪤[R1,T1] L2 → L1 ⪤[R2,T2] L2. #R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 * /4 width=7 by sex_co_isid, ex2_intro/ qed-. lemma rex_unit_sn (R1) (R2): - ∀I,K1,L2. K1.ⓤ{I} ⪤[R1, #0] L2 → K1.ⓤ{I} ⪤[R2, #0] L2. + ∀I,K1,L2. K1.ⓤ{I} ⪤[R1,#0] L2 → K1.ⓤ{I} ⪤[R2,#0] L2. #R1 #R2 #I #K1 #L2 #H elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct /3 width=7 by rex_unit, sex_co_isid/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_drops.ma index 60e335d57..5a0466800 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_drops.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_drops.ma @@ -20,19 +20,19 @@ include "static_2/static/rex.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) definition f_dedropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → - ∀K2,T. K1 ⪤[R, T] K2 → ∀U. ⬆*[f] T ≘ U → - ∃∃L2. L1 ⪤[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → + ∀K2,T. K1 ⪤[R,T] K2 → ∀U. ⬆*[f] T ≘ U → + ∃∃L2. L1 ⪤[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2. definition f_dropable_sn: predicate (relation3 lenv term term) ≝ - λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → - ∀L2,U. L1 ⪤[R, U] L2 → ∀T. ⬆*[f] T ≘ U → - ∃∃K2. K1 ⪤[R, T] K2 & ⬇*[b, f] L2 ≘ K2. + λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → + ∀L2,U. L1 ⪤[R,U] L2 → ∀T. ⬆*[f] T ≘ U → + ∃∃K2. K1 ⪤[R,T] K2 & ⬇*[b,f] L2 ≘ K2. definition f_dropable_dx: predicate (relation3 lenv term term) ≝ - λR. ∀L1,L2,U. L1 ⪤[R, U] L2 → - ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U → - ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R, T] K2. + λR. ∀L1,L2,U. L1 ⪤[R,U] L2 → + ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U → + ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R,T] K2. definition f_transitive_next: relation3 … ≝ λR1,R2,R3. ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → @@ -87,48 +87,48 @@ elim (sex_co_dropable_dx … HL12 … HLK2 … H2f) -L2 qed-. (* Basic_2A1: uses: llpx_sn_inv_lift_O *) -lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R, U] L2 → ∀b,f. 𝐔⦃f⦄ → - ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 → - ∀T. ⬆*[f] T ≘ U → K1 ⪤[R, T] K2. +lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R,U] L2 → ∀b,f. 𝐔⦃f⦄ → + ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 → + ∀T. ⬆*[f] T ≘ U → K1 ⪤[R,T] K2. #R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU elim (rex_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct // qed-. -lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 → - ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R, V1] K2 & R K1 V1 V2. +lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 → + ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R,V1] K2 & R K1 V1 V2. #R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // #Y #HY #HLK2 elim (rex_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 → - ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R, V1] K2 & R K1 V1 V2. +lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 → + ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R,V1] K2 & R K1 V1 V2. #R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // #Y #HLK1 #HY elim (rex_inv_zero_pair_dx … HY) -HY #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ qed-. lemma rex_inv_lref_pair_bi (R) (L1) (L2) (i): - L1 ⪤[R, #i] L2 → + L1 ⪤[R,#i] L2 → ∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 → ∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 → - ∧∧ K1 ⪤[R, V1] K2 & R K1 V1 V2 & I1 = I2. + ∧∧ K1 ⪤[R,V1] K2 & R K1 V1 V2 & I1 = I2. #R #L1 #L2 #i #H12 #I1 #K1 #V1 #H1 #I2 #K2 #V2 #H2 elim (rex_inv_lref_pair_sn … H12 … H1) -L1 #Y2 #X2 #HLY2 #HK12 #HV12 lapply (drops_mono … HLY2 … H2) -HLY2 -H2 #H destruct /2 width=1 by and3_intro/ qed-. -lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} → - ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄. +lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} → + ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄. #R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // #Y #HY #HLK2 elim (rex_inv_zero_unit_sn … HY) -HY #f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} → - ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄. +lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} → + ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄. #R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // #Y #HLK1 #HY elim (rex_inv_zero_unit_dx … HY) -HY #f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_fqup.ma index c45914f34..5c7e82b33 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_fqup.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_fqup.ma @@ -20,13 +20,13 @@ include "static_2/static/rex.ma". (* Advanced properties ******************************************************) (* Basic_2A1: uses: llpx_sn_refl *) -lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R, T] L. +lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R,T] L. #R #HR #L #T elim (frees_total L T) /4 width=3 by sex_refl, ext2_refl, ex2_intro/ qed. lemma rex_pair_refl: ∀R. (∀L. reflexive … (R L)) → - ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R, T] L.ⓑ{I}V2. + ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R,T] L.ⓑ{I}V2. #R #HR #L #V1 #V2 #HV12 #I #T elim (frees_total (L.ⓑ{I}V1) T) #f #Hf elim (pn_split f) * #g #H destruct @@ -35,15 +35,15 @@ qed. (* Advanced inversion lemmas ************************************************) -lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 → - L1 ⪤[R, V] L2 ∧ L1.ⓧ ⪤[R, T] L2.ⓧ. +lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 → + L1 ⪤[R,V] L2 ∧ L1.ⓧ ⪤[R,T] L2.ⓧ. #R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind_void … Hf) -Hf /6 width=6 by sle_sex_trans, sex_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Advanced forward lemmas **************************************************) -lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 → - L1.ⓧ ⪤[R, T] L2.ⓧ. +lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 → + L1.ⓧ ⪤[R,T] L2.ⓧ. #R #p #I #L1 #L2 #V #T #H elim (rex_inv_bind_void … H) -H // qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_fsle.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_fsle.ma index 24f68014d..1a2829da3 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_fsle.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_fsle.ma @@ -20,19 +20,19 @@ include "static_2/static/rex_rex.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) definition R_fsge_compatible: predicate (relation3 …) ≝ λRN. - ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄. + ∀L,T1,T2. RN L T1 T2 → ⦃L,T2⦄ ⊆ ⦃L,T1⦄. definition rex_fsge_compatible: predicate (relation3 …) ≝ λRN. - ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄. + ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L2,T⦄ ⊆ ⦃L1,T⦄. definition rex_fsle_compatible: predicate (relation3 …) ≝ λRN. - ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄. + ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L1,T⦄ ⊆ ⦃L2,T⦄. (* Basic inversions with free variables inclusion for restricted closures ***) lemma frees_sex_conf: ∀R. rex_fsge_compatible R → ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → - ∀L2. L1 ⪤[cext2 R, cfull, f1] L2 → + ∀L2. L1 ⪤[cext2 R,cfull,f1] L2 → ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 & f2 ⊆ f1. #R #HR #L1 #T #f1 #Hf1 #L2 #H1L lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L @@ -41,9 +41,9 @@ qed-. (* Properties with free variables inclusion for restricted closures *********) -(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *) -lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ → - ∀L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2. +(* Note: we just need lveq_inv_refl: ∀L, n1, n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *) +lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L1,T2⦄ → + ∀L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2. #R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12 elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct /4 width=5 by rex_inv_frees, sle_sex_trans, ex2_intro/ @@ -60,8 +60,8 @@ qed-. lemma rex_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → rex_fsge_compatible R1 → - ∀L1,L2,V. L1 ⪤[R1, V] L2 → ∀I,T. - ∃∃L. L1 ⪤[R1, ②{I}V.T] L & L ⪤[R2, V] L2. + ∀L1,L2,V. L1 ⪤[R1,V] L2 → ∀I,T. + ∃∃L. L1 ⪤[R1,②{I}V.T] L & L ⪤[R2,V] L2. #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T [ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy @@ -79,8 +79,8 @@ qed-. lemma rex_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → rex_fsge_compatible R1 → - ∀L1,L2,T. L1 ⪤[R1, T] L2 → ∀I,V. - ∃∃L. L1 ⪤[R1, ⓕ{I}V.T] L & L ⪤[R2, T] L2. + ∀L1,L2,T. L1 ⪤[R1,T] L2 → ∀I,V. + ∃∃L. L1 ⪤[R1,ⓕ{I}V.T] L & L ⪤[R2,T] L2. #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy @@ -95,8 +95,8 @@ qed-. lemma rex_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → rex_fsge_compatible R1 → - ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1, T] L2 → ∀p. - ∃∃L,V. L1 ⪤[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2, T] L2 & R1 L1 V1 V. + ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1,T] L2 → ∀p. + ∃∃L,V. L1 ⪤[R1,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2,T] L2 & R1 L1 V1 V. #R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy @@ -115,8 +115,8 @@ qed-. lemma rex_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → rex_fsge_compatible R1 → - ∀L1,L2,T. L1.ⓧ ⪤[R1, T] L2 → ∀p,I,V. - ∃∃L. L1 ⪤[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2, T] L2. + ∀L1,L2,T. L1.ⓧ ⪤[R1,T] L2 → ∀p,I,V. + ∃∃L. L1 ⪤[R1,ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2,T] L2. #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy @@ -166,8 +166,8 @@ qed-. theorem rex_trans_fsle: ∀R1,R2,R3. rex_fsle_compatible R1 → f_transitive_next R1 R2 R3 → - ∀L1,L,T. L1 ⪤[R1, T] L → - ∀L2. L ⪤[R2, T] L2 → L1 ⪤[R3, T] L2. + ∀L1,L,T. L1 ⪤[R1,T] L → + ∀L2. L ⪤[R2,T] L2 → L1 ⪤[R3,T] L2. #R1 #R2 #R3 #H1R #H2R #L1 #L #T #H lapply (H1R … H) -H1R #H0 cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2 diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_length.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_length.ma index e279e159d..e91111142 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_length.ma @@ -20,14 +20,14 @@ include "static_2/static/rex_drops.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: uses: llpx_sn_fwd_length *) -lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R, T] L2 → |L1| = |L2|. +lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R,T] L2 → |L1| = |L2|. #R #L1 #L2 #T * /2 width=4 by sex_fwd_length/ qed-. (* Properties with length for local environments ****************************) (* Basic_2A1: uses: llpx_sn_sort *) -lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R, ⋆s] L2. +lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R,⋆s] L2. #R #L1 elim L1 -L1 [ #Y #H #s >(length_inv_zero_sn … H) -H // | #K1 #I1 #IH #Y #H #s @@ -37,7 +37,7 @@ lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R, ⋆s] L2. qed. (* Basic_2A1: uses: llpx_sn_gref *) -lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R, §l] L2. +lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R,§l] L2. #R #L1 elim L1 -L1 [ #Y #H #s >(length_inv_zero_sn … H) -H // | #K1 #I1 #IH #Y #H #s @@ -46,14 +46,14 @@ lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R, §l] L2. ] qed. -lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}. +lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}. /3 width=3 by rex_unit, sex_length_isid/ qed. (* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *) lemma rex_lifts_bi (R): d_liftable2_sn … lifts R → - ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R, T] K2 → - ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 → - ∀U. ⬆*[f] T ≘ U → L1 ⪤[R, U] L2. + ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R,T] K2 → + ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 → + ∀U. ⬆*[f] T ≘ U → L1 ⪤[R,U] L2. #R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU elim (frees_total L1 U) #f2 #Hf2 lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf @@ -62,9 +62,9 @@ qed-. (* Inversion lemmas with length for local environment ***********************) -lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 → +lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R,#0] Y2 → ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 & + | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}. #R #Y1 #Y2 #H elim (rex_inv_zero … H) -H * diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_lex.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_lex.ma index ffaebbc2e..d031d93c4 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_lex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_lex.ma @@ -20,7 +20,7 @@ include "static_2/static/req.ma". (* Properties with generic extension of a context-sensitive relation ********) -lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R, T] L2. +lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R,T] L2. #R #L1 #L2 * #f #Hf #HL12 #T elim (frees_total L1 T) #g #Hg /4 width=5 by sex_sdj, sdj_isid_sn, ex2_intro/ @@ -30,7 +30,7 @@ qed. lemma rex_inv_lex_req: ∀R. c_reflexive … R → rex_fsge_compatible R → - ∀L1,L2,T. L1 ⪤[R, T] L2 → + ∀L1,L2,T. L1 ⪤[R,T] L2 → ∃∃L. L1 ⪤[R] L & L ≡[T] L2. #R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL elim (sex_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex_rex.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex_rex.ma index 08a4f192c..7bbf5d895 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex_rex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex_rex.ma @@ -20,8 +20,8 @@ include "static_2/static/rex.ma". (* Advanced inversion lemmas ************************************************) -lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 → - ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2. +lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R,T] L2 → + ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R,cfull,f] L2. #R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/ qed-. @@ -29,7 +29,7 @@ qed-. (* Basic_2A1: uses: llpx_sn_dec *) lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2). + ∀L1,L2,T. Decidable (L1 ⪤[R,T] L2). #R #HR #L1 #L2 #T elim (frees_total L1 T) #f #Hf elim (sex_dec (cext2 R) cfull … L1 L2 f) @@ -40,8 +40,8 @@ qed-. (* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *) theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T. - L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 → - L1 ⪤[R, ⓑ{p,I}V1.T] L2. + L1 ⪤[R,V1] L2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2 → + L1 ⪤[R,ⓑ{p,I}V1.T] L2. #R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2)) /3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/ @@ -49,15 +49,15 @@ qed. (* Basic_2A1: llpx_sn_flat *) theorem rex_flat: ∀R,I,L1,L2,V,T. - L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 → - L1 ⪤[R, ⓕ{I}V.T] L2. + L1 ⪤[R,V] L2 → L1 ⪤[R,T] L2 → + L1 ⪤[R,ⓕ{I}V.T] L2. #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2) /3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/ qed. theorem rex_bind_void: ∀R,p,I,L1,L2,V,T. - L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ → - L1 ⪤[R, ⓑ{p,I}V.T] L2. + L1 ⪤[R,V] L2 → L1.ⓧ ⪤[R,T] L2.ⓧ → + L1 ⪤[R,ⓑ{p,I}V.T] L2. #R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2)) /3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/ @@ -67,23 +67,23 @@ qed. (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *) lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) → - (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥). + ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) → + (L1 ⪤[R,V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R,T] L2.ⓑ{I}V → ⊥). #R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V) /4 width=2 by rex_bind, or_intror, or_introl/ qed-. (* Basic_2A1: uses: nllpx_sn_inv_flat *) lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) → - (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥). + ∀I,L1,L2,V,T. (L1 ⪤[R,ⓕ{I}V.T] L2 → ⊥) → + (L1 ⪤[R,V] L2 → ⊥) ∨ (L1 ⪤[R,T] L2 → ⊥). #R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V) /4 width=1 by rex_flat, or_intror, or_introl/ qed-. lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) → - (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥). + ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) → + (L1 ⪤[R,V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R,T] L2.ⓧ → ⊥). #R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V) /4 width=2 by rex_bind_void, or_intror, or_introl/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/cl_restricted_weight.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/cl_restricted_weight.ma index 31c3f7adc..bc5b92945 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/cl_restricted_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/cl_restricted_weight.ma @@ -24,27 +24,27 @@ interpretation "weight (restricted closure)" 'Weight L T = (rfw L T). (* Basic properties *********************************************************) (* Basic_1: was: flt_shift *) -lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{p,I}V.T}. +lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V,T} < ♯{K,ⓑ{p,I}V.T}. normalize /2 width=1 by monotonic_le_plus_r/ qed. -lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1}, T} < ♯{K, ⓑ{p,I2}V.T}. +lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1},T} < ♯{K,ⓑ{p,I2}V.T}. normalize /4 width=1 by monotonic_le_plus_r, le_S_S/ qed. -lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}. +lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L,②{I}V.T}. normalize in ⊢ (?→?→?→?→?%%); // qed. -lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}. +lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L,②{I}V.T}. normalize in ⊢ (?→?→?→?→?%%); // qed. -lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}. +lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L.ⓑ{I}V,T}. normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/ qed. -lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}. +lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L.ⓑ{I}V,T}. normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/ qed. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/cl_weight.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/cl_weight.ma index a51094adc..af52e35b1 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/cl_weight.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/cl_weight.ma @@ -26,23 +26,23 @@ interpretation "weight (closure)" 'Weight G L T = (fw G L T). (* Basic properties *********************************************************) (* Basic_1: was: flt_shift *) -lemma fw_shift: ∀p,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{p,I}V.T}. +lemma fw_shift: ∀p,I,G,K,V,T. ♯{G,K.ⓑ{I}V,T} < ♯{G,K,ⓑ{p,I}V.T}. normalize /2 width=1 by monotonic_le_plus_r/ qed. -lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G, K.ⓤ{I1}, T} < ♯{G, K, ⓑ{p,I2}V.T}. +lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G,K.ⓤ{I1},T} < ♯{G,K,ⓑ{p,I2}V.T}. normalize /4 width=1 by monotonic_le_plus_r, le_S_S/ qed. -lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}. +lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L,②{I}V.T}. normalize in ⊢ (?→?→?→?→?→?%%); // qed. -lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}. +lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G,L,T} < ♯{G,L,②{I}V.T}. normalize in ⊢ (?→?→?→?→?→?%%); // qed. -lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}. +lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L.ⓑ{I}V,T}. normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/ qed. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma index 855978783..e6d2fcbfc 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma @@ -32,7 +32,7 @@ interpretation "equivalence up to exclusion binders (local environment)" (* Basic properties *********************************************************) -lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L. +lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L. #L elim L -L /2 width=1 by lveq_atom, lveq_bind/ qed. @@ -43,10 +43,10 @@ qed-. (* Basic inversion lemmas ***************************************************) -fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → 0 = n1 → 0 = n2 → ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2 - | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2. + | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 [1: /3 width=1 by or_introl, conj/ |2: /3 width=7 by ex3_4_intro, or_intror/ @@ -54,14 +54,14 @@ fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → ] qed-. -lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → +lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 → ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2 - | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2. + | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2. /2 width=5 by lveq_inv_zero_aux/ qed-. -fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∀m1. ↑m1 = n1 → - ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2. + ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 [1: #m #H destruct |2: #I1 #I2 #K1 #K2 #_ #m #H destruct @@ -69,18 +69,18 @@ fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → ] qed-. -lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 → - ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2. +lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 → + ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2. /2 width=3 by lveq_inv_succ_sn_aux/ qed-. -lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 → - ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1. +lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 → + ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1. #K1 #L2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/ qed-. -fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 [1: #m1 #m2 #H1 #H2 destruct @@ -89,17 +89,17 @@ fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → ] qed-. -lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥. +lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1,↑n2] L2 → ⊥. /2 width=9 by lveq_inv_succ_aux/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2. +lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0,0] K2.ⓘ{I2} → K1 ≋ⓧ*[0,0] K2. #I1 #I2 #K1 #K2 #H elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct // qed-. -lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2. +lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2. * [2: #n1 ] * [2,4: #n2 ] #H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct @@ -108,8 +108,8 @@ lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & ] qed-. -lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ → - ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2. +lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1,n2] ⋆ → + ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2. #I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct @@ -121,16 +121,16 @@ lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ → ] qed-. -lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} → - ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2. +lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ{I2} → + ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2. #I2 #K2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_bind_atom … H) -H /3 width=3 by lveq_sym, ex4_intro/ qed-. -lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → - ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2. +lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 → + ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2. #I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct @@ -142,14 +142,14 @@ lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n ] qed-. -lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 → - ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2. +lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 → + ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2. #L1 #L2 #n1 #n2 #H elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/ qed-. -lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ → - ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1. +lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ → + ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1. #L1 #L2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_void_succ_sn … H) -H @@ -158,19 +158,19 @@ qed-. (* Advanced forward lemmas **************************************************) -lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∨∨ 0 = n1 | 0 = n2. #L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H [ elim (lveq_inv_succ … H) ] /2 width=1 by or_introl, or_intror/ qed-. -lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1. +lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2 → 0 = n1. #I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct ] qed-. -lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2. +lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 → 0 = n2. /3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-. diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma index 104d2b8b9..8ae0a99be 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_length.ma @@ -19,7 +19,7 @@ include "static_2/syntax/lveq.ma". (* Properties with length for local environments ****************************) -lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0, 0] L2. +lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0,0] L2. #L1 elim L1 -L1 [ #Y2 #H >(length_inv_zero_sn … H) -Y2 /2 width=3 by lveq_atom, ex_intro/ | #K1 #I1 #IH #Y2 #H @@ -30,69 +30,69 @@ qed. (* Forward lemmas with length for local environments ************************) -lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n1 ≤ |L1|. +lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n1 ≤ |L1|. #L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize /2 width=1 by le_S_S/ qed-. -lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n2 ≤ |L2|. +lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n2 ≤ |L2|. #L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize /2 width=1 by le_S_S/ qed-. -lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∧∧ |L1|-|L2| = n1 & |L2|-|L1| = n2. #L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 /2 width=1 by conj/ #K1 #K2 #n #_ * #H1 #H2 >length_bind /3 width=1 by minus_Sn_m, conj/ qed-. -lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L1| ≤ |L2| → 0 = n1. +lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| ≤ |L2| → 0 = n1. #L1 #L2 #n1 #n2 #H #HL elim (lveq_fwd_length … H) -H >(eq_minus_O … HL) // qed-. -lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → 0 = n2. +lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| → 0 = n2. #L1 #L2 #n1 #n2 #H #HL elim (lveq_fwd_length … H) -H >(eq_minus_O … HL) // qed-. -lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| = |L2| → ∧∧ 0 = n1 & 0 = n2. #L1 #L2 #n1 #n2 #H #HL elim (lveq_fwd_length … H) -H >HL -HL /2 width=1 by conj/ qed-. -lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| + n2 = |L2| + n1. #L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize /2 width=2 by injective_plus_r/ qed-. -lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → |L1| = |L2|. +lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 → |L1| = |L2|. /3 width=2 by lveq_fwd_length_plus, injective_plus_l/ qed-. -lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → +lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| - n1 = |L2| - n2. /3 width=3 by lveq_fwd_length_plus, lveq_fwd_length_le_dx, lveq_fwd_length_le_sn, plus_to_minus_2/ qed-. lemma lveq_fwd_abst_bind_length_le: ∀I1,I2,L1,L2,V1,n1,n2. - L1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2.ⓘ{I2} → |L1| ≤ |L2|. + L1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2.ⓘ{I2} → |L1| ≤ |L2|. #I1 #I2 #L1 #L2 #V1 #n1 #n2 #HL lapply (lveq_fwd_pair_sn … HL) #H destruct elim (lveq_fwd_length … HL) -HL >length_bind >length_bind // qed-. lemma lveq_fwd_bind_abst_length_le: ∀I1,I2,L1,L2,V2,n1,n2. - L1.ⓘ{I1} ≋ⓧ*[n1, n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|. + L1.ⓘ{I1} ≋ⓧ*[n1,n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|. /3 width=6 by lveq_fwd_abst_bind_length_le, lveq_sym/ qed-. (* Inversion lemmas with length for local environments **********************) -lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2.ⓧ → |L1| ≤ |L2| → - ∃∃m2. L1 ≋ ⓧ*[n1, m2] L2 & 0 = n1 & ↑m2 = n2. +lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2.ⓧ → |L1| ≤ |L2| → + ∃∃m2. L1 ≋ ⓧ*[n1,m2] L2 & 0 = n1 & ↑m2 = n2. #L1 #L2 #n1 #n2 #H #HL12 lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0 lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0 @@ -100,8 +100,8 @@ elim (le_inv_S1 … H0) -H0 #m2 #_ #H0 destruct elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/ qed-. -lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → - ∃∃m1. L1 ≋ ⓧ*[m1, n2] L2 & ↑m1 = n1 & 0 = n2. +lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| → + ∃∃m1. L1 ≋ ⓧ*[m1,n2] L2 & ↑m1 = n1 & 0 = n2. #L1 #L2 #n1 #n2 #H #HL lapply (lveq_sym … H) -H #H elim (lveq_inv_void_dx_length … H HL) -H -HL diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_lveq.ma index 8ac40a55f..c0c8cf1ab 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_lveq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq_lveq.ma @@ -18,16 +18,16 @@ include "static_2/syntax/lveq_length.ma". (* Main inversion lemmas ****************************************************) -theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 → - ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} → +theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0,0] K2 → + ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1,m2] K2.ⓘ{I2} → ∧∧ 0 = m1 & 0 = m2. #K1 #K2 #HK #I1 #I2 #m1 #m2 #H lapply (lveq_fwd_length_eq … HK) -HK #HK elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/ qed-. -theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → - ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 → +theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → + ∀m1,m2. L1 ≋ⓧ*[m1,m2] L2 → ∧∧ n1 = m1 & n2 = m2. #L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct @@ -36,8 +36,8 @@ elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct qed-. theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| → - ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 → - ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 → + ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 → + ∀m1,m2. K1.ⓧ ≋ⓧ*[m1,m2] K2 → ∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2. #L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct @@ -47,7 +47,7 @@ elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct qed-. theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| → - ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 → - ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ → + ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 → + ∀m1,m2. K1 ≋ⓧ*[m1,m2] K2.ⓧ → ∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1. /3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *) diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/item_sd.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma similarity index 100% rename from matita/matita/contribs/lambdadelta/static_2/syntax/item_sd.ma rename to matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/sh.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/sh.ma new file mode 100644 index 000000000..bb7aacd2c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/sh.ma @@ -0,0 +1,26 @@ +(**************************************************************************) +(* ___ *) +(* ||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/lib/arith.ma". +include "static_2/notation/functions/upspoon_2.ma". + +(* SORT HIERARCHY ***********************************************************) + +(* sort hierarchy specification *) +record sh: Type[0] ≝ { + next: nat → nat (* next sort in the hierarchy *) +}. + +interpretation "next sort (sort hierarchy)" + 'UpSpoon h s = (next h s). diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/item_sh.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma similarity index 65% rename from matita/matita/contribs/lambdadelta/static_2/syntax/item_sh.ma rename to matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma index 9e0b03f73..459d4d16b 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/item_sh.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.ma @@ -12,33 +12,25 @@ (* *) (**************************************************************************) -include "ground_2/lib/arith.ma". +include "static_2/syntax/sort.ma". (* SORT HIERARCHY ***********************************************************) -(* sort hierarchy specification *) -record sh: Type[0] ≝ { - next : nat → nat; (* next sort in the hierarchy *) - next_lt: ∀s. s < next s (* strict monotonicity condition *) +record is_lt (h): Prop ≝ +{ + next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *) }. -definition sh_N: sh ≝ mk_sh S …. -// defined. - (* Basic properties *********************************************************) -lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s. -#h #s #n elim n -n // normalize #n #IH -lapply (next_lt h ((next h)^n s)) #H +lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s. +#h #Hh #s #n elim n -n [ // ] normalize #n #IH +lapply (next_lt … Hh ((next h)^n s)) #H lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/ qed. -lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s. -#h #s #n normalize -lapply (nexts_le h s n) #H -@(le_to_lt_to_lt … H) // +lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s. +#h #Hh #s #n normalize +lapply (nexts_le … Hh s n) #H +@(le_to_lt_to_lt … H) /2 width=1 by next_lt/ qed. - -axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2). - -axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2. -- 2.39.2