From 981599dd384b3424c60297ea3a64ab0af9788ea2 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 3 May 2017 09:56:08 +0000 Subject: [PATCH] notational change for lexs --- .../lambdadelta/basic_2/i_static/tc_lfxs.ma | 66 ++++++------- .../basic_2/i_static/tc_lfxs_drops.ma | 12 +-- .../basic_2/i_static/tc_lfxs_fqup.ma | 10 +- .../basic_2/i_static/tc_lfxs_length.ma | 2 +- .../notation/relations/relationstar_4.ma | 2 +- .../notation/relations/relationstar_5.ma | 2 +- .../notation/relations/relationstarstar_4.ma | 2 +- .../basic_2/relocation/drops_lexs.ma | 30 +++--- .../lambdadelta/basic_2/relocation/lexs.ma | 88 ++++++++--------- .../basic_2/relocation/lexs_length.ma | 2 +- .../basic_2/relocation/lexs_lexs.ma | 18 ++-- .../basic_2/rt_transition/lfpx_frees.ma | 2 +- .../lambdadelta/basic_2/static/lfxs.ma | 96 +++++++++---------- .../lambdadelta/basic_2/static/lfxs_drops.ma | 24 ++--- .../lambdadelta/basic_2/static/lfxs_fqup.ma | 4 +- .../lambdadelta/basic_2/static/lfxs_length.ma | 2 +- .../lambdadelta/basic_2/static/lfxs_lfxs.ma | 34 +++---- matita/matita/predefined_virtuals.ml | 4 +- 18 files changed, 200 insertions(+), 200 deletions(-) diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs.ma index b8976cbd9..e1dd22673 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs.ma @@ -25,40 +25,40 @@ interpretation "iterated extension on referred entries (local environment)" (* Basic properties *********************************************************) -lemma tc_lfxs_step_dx: ∀R,L1,L,T. L1 ⦻**[R, T] L → - ∀L2. L ⦻*[R, T] L2 → L1 ⦻**[R, T] L2. +lemma tc_lfxs_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 tc_lfxs_step_sn: ∀R,L1,L,T. L1 ⦻*[R, T] L → - ∀L2. L ⦻**[R, T] L2 → L1 ⦻**[R, T] L2. +lemma tc_lfxs_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 tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆. +lemma tc_lfxs_atom: ∀R,I. ⋆ ⪤**[R, ⓪{I}] ⋆. /2 width=1 by inj/ qed. lemma tc_lfxs_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 lfxs_sort, tc_lfxs_step_dx, inj/ qed. lemma tc_lfxs_zero: ∀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 lfxs_zero, tc_lfxs_step_dx, inj/ qed. lemma tc_lfxs_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 lfxs_lref, tc_lfxs_step_dx, inj/ qed. lemma tc_lfxs_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 lfxs_gref, tc_lfxs_step_dx, inj/ qed. @@ -71,7 +71,7 @@ lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull → qed-. lemma tc_lfxs_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 lfxs_co, tc_lfxs_step_dx, inj/ qed-. @@ -79,19 +79,19 @@ qed-. (* Basic inversion lemmas ***************************************************) (* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *) -lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆. +lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤**[R, ⓪{I}] Y2 → Y2 = ⋆. #R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/ qed-. (* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *) -lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆. +lemma tc_lfxs_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, lfxs_inv_atom_dx/ qed-. -lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻**[R, ⋆s] Y2 → +lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤**[R, ⋆s] Y2 → (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 & + ∃∃I,L1,L2,V1,V2. L1 ⪤**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 #s #H elim H -Y2 [ #Y2 #H elim (lfxs_inv_sort … H) -H * @@ -107,9 +107,9 @@ lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻**[R, ⋆s] Y2 → ] qed-. -lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻**[R, §l] Y2 → +lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤**[R, §l] Y2 → (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, §l] L2 & + ∃∃I,L1,L2,V1,V2. L1 ⪤**[R, §l] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 #l #H elim H -Y2 [ #Y2 #H elim (lfxs_inv_gref … H) -H * @@ -126,16 +126,16 @@ lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻**[R, §l] Y2 → qed-. lemma tc_lfxs_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 (lfxs_inv_bind … V ? H) -H /3 width=1 by inj, conj/ | #L #L2 #_ #H * elim (lfxs_inv_bind … V ? H) -H /3 width=3 by tc_lfxs_step_dx, conj/ ] qed-. -lemma tc_lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ⓕ{I}V.T] L2 → - L1 ⦻**[R, V] L2 ∧ L1 ⦻**[R, T] L2. +lemma tc_lfxs_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 (lfxs_inv_flat … H) -H /3 width=1 by inj, conj/ | #L #L2 #_ #H * elim (lfxs_inv_flat … H) -H /3 width=3 by tc_lfxs_step_dx, conj/ @@ -144,32 +144,32 @@ qed-. (* Advanced inversion lemmas ************************************************) -lemma tc_lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻**[R, ⋆s] Y2 → - ∃∃L2,V2. L1 ⦻**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. +lemma tc_lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⪤**[R, ⋆s] Y2 → + ∃∃L2,V2. L1 ⪤**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #s #H elim (tc_lfxs_inv_sort … H) -H * [ #H destruct | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma tc_lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻**[R, ⋆s] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. +lemma tc_lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⪤**[R, ⋆s] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #s #H elim (tc_lfxs_inv_sort … H) -H * [ #_ #H destruct | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma tc_lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻**[R, §l] Y2 → - ∃∃L2,V2. L1 ⦻**[R, §l] L2 & Y2 = L2.ⓑ{I}V2. +lemma tc_lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⪤**[R, §l] Y2 → + ∃∃L2,V2. L1 ⪤**[R, §l] L2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #l #H elim (tc_lfxs_inv_gref … H) -H * [ #H destruct | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma tc_lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻**[R, §l] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻**[R, §l] L2 & Y1 = L1.ⓑ{I}V1. +lemma tc_lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⪤**[R, §l] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤**[R, §l] L2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #l #H elim (tc_lfxs_inv_gref … H) -H * [ #_ #H destruct | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ @@ -178,18 +178,18 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma tc_lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ②{I}V.T] L2 → L1 ⦻**[R, V] L2. +lemma tc_lfxs_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 lfxs_fwd_pair_sn, tc_lfxs_step_dx, inj/ qed-. lemma tc_lfxs_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 (tc_lfxs_inv_bind … H) -H // qed-. -lemma tc_lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ⓕ{I}V.T] L2 → L1 ⦻**[R, T] L2. +lemma tc_lfxs_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 (tc_lfxs_inv_flat … H) -H // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_drops.ma index 64bb1f3f1..13aec62e4 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_drops.ma @@ -20,18 +20,18 @@ include "basic_2/i_static/tc_lfxs.ma". definition tc_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. + ∀K2,T. K1 ⪤**[R, T] K2 → ∀U. ⬆*[f] T ≡ U → + ∃∃L2. L1 ⪤**[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. definition tc_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. + ∀L2,U. L1 ⪤**[R, U] L2 → ∀T. ⬆*[f] T ≡ U → + ∃∃K2. K1 ⪤**[R, T] K2 & ⬇*[b, f] L2 ≡ K2. definition tc_dropable_dx: predicate (relation3 lenv term term) ≝ - λR. ∀L1,L2,U. L1 ⦻**[R, U] L2 → + 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. + ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⪤**[R, T] K2. (* Properties with generic slicing for local environments *******************) diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.ma index e17a1c85d..e69f071f2 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.ma @@ -24,7 +24,7 @@ lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (t (* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *) lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) → - ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⦻**[R, T] L.ⓑ{I}V2. + ∀L,V1,V2. LTC … 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 tc_lfxs_step_dx, lfxs_pair, inj/ qed. @@ -33,16 +33,16 @@ qed. lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) → ∀L1,T. ∀R0:predicate …. R0 L1 → - (∀L,L2. L1 ⦻**[R, T] L → L ⦻*[R, T] L2 → R0 L → R0 L2) → - ∀L2. L1 ⦻**[R, T] L2 → R0 L2. + (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → R0 L → R0 L2) → + ∀L2. L1 ⪤**[R, T] L2 → R0 L2. #R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/ qed-. lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) → ∀L2,T. ∀R0:predicate …. R0 L2 → - (∀L1,L. L1 ⦻*[R, T] L → L ⦻**[R, T] L2 → R0 L → R0 L1) → - ∀L1. L1 ⦻**[R, T] L2 → R0 L1. + (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → R0 L → R0 L1) → + ∀L1. L1 ⪤**[R, T] L2 → R0 L1. #R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_length.ma index 414acd7da..8280cd80c 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_length.ma @@ -20,7 +20,7 @@ include "basic_2/i_static/tc_lfxs.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: uses: TC_lpx_sn_fwd_length *) -lemma tc_lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻**[R, T] L2 → |L1| = |L2|. +lemma tc_lfxs_fwd_length: ∀R,L1,L2,T. L1 ⪤**[R, T] L2 → |L1| = |L2|. #R #L1 #L2 #T #H elim H -L2 [ #L2 #HL12 >(lfxs_fwd_length … HL12) -HL12 // | #L #L2 #_ #HL2 #IHL1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma index 9e3581baa..141856a66 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_4.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L1 ⦻ * [ break term 46 R , break term 46 T ] break term 46 L2 )" +notation "hvbox( L1 ⪤ * [ break term 46 R , break term 46 T ] break term 46 L2 )" non associative with precedence 45 for @{ 'RelationStar $R $T $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_5.ma index f6df0bec5..135111ba3 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_5.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstar_5.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L1 ⦻ * [ break term 46 R1 , break term 46 R2 , break term 46 f ] break term 46 L2 )" +notation "hvbox( L1 ⪤ * [ break term 46 R1 , break term 46 R2 , break term 46 f ] break term 46 L2 )" non associative with precedence 45 for @{ 'RelationStar $R1 $R2 $f $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstarstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstarstar_4.ma index 145078d07..f03b54a77 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstarstar_4.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/relationstarstar_4.ma @@ -14,6 +14,6 @@ (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************) -notation "hvbox( L1 ⦻ * * [ break term 46 R , break term 46 T ] break term 46 L2 )" +notation "hvbox( L1 ⪤ * * [ break term 46 R , break term 46 T ] break term 46 L2 )" non associative with precedence 45 for @{ 'RelationStarStar $R $T $L1 $L2 }. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma index 1e22db645..a42aaa465 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma @@ -60,8 +60,8 @@ lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive ? (RN L)) → ( qed-. fact lexs_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. + ∀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 (lexs_inv_atom2 … H) -H #H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/ @@ -87,10 +87,10 @@ lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP). (* Basic_2A1: includes: lpx_sn_drop_conf *) (**) lemma lexs_drops_conf_next: ∀RN,RP. - ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 → + ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 → ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ → ∀f1. f ~⊚ ⫯f1 ≡ f2 → - ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2. + ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2 elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX @@ -98,10 +98,10 @@ elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf qed-. lemma lexs_drops_conf_push: ∀RN,RP. - ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 → + ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 → ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≡ f2 → - ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2. + ∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2 elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf #X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX @@ -109,20 +109,20 @@ elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf qed-. (* Basic_2A1: includes: lpx_sn_drop_trans *) -lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 → +lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 → ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ → ∀f1. f ~⊚ ⫯f1 ≡ f2 → - ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2. + ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2 elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma lexs_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 → +lemma lexs_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 → ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ → ∀f1. f ~⊚ ↑f1 ≡ f2 → - ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2. + ∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2. #RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2 elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf #X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX @@ -131,10 +131,10 @@ qed-. lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → d_liftable2_sn RN → d_liftable2_sn RP → - ∀f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 → + ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 → ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 → ∀f2. f ~⊚ f1 ≡ ⫯f2 → - ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2. + ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2 elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX @@ -143,10 +143,10 @@ qed-. lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) → d_liftable2_sn RN → d_liftable2_sn RP → - ∀f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 → + ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 → ∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 → ∀f2. f ~⊚ f1 ≡ ↑f2 → - ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2. + ∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2. #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2 elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP #X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX @@ -154,7 +154,7 @@ elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 qed-. lemma drops_atom2_lexs_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≡ ⋆ → 𝐔⦃f1⦄ → - ∀f,L2. L1 ⦻*[RN, RP, f] L2 → + ∀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 (lexs_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma index e4e501f09..0b8465271 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -38,28 +38,28 @@ definition R_pw_confluent2_lexs: relation3 lenv term term → relation3 lenv ter relation3 rtmap lenv term ≝ λR1,R2,RN1,RP1,RN2,RP2,f,L0,T0. ∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 → + ∀L1. L0 ⪤*[RN1, RP1, f] L1 → ∀L2. L0 ⪤*[RN2, RP2, f] L2 → ∃∃T. R2 L1 T1 T & R1 L2 T2 T. definition lexs_transitive: relation5 (relation3 lenv term term) (relation3 lenv term term) … ≝ λR1,R2,R3,RN,RP. - ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⦻*[RN, RP, f] L2 → + ∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⪤*[RN, RP, f] L2 → ∀T2. R2 L2 T T2 → R3 L1 T1 T2. (* Basic inversion lemmas ***************************************************) -fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆. +fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom1 *) -lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⦻*[RN, RP, f] Y → Y = ⋆. +lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤*[RN, RP, f] Y → Y = ⋆. /2 width=6 by lexs_inv_atom1_aux/ qed-. -fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. +fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g → + ∃∃K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J #K1 #W1 #H destruct | #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_next … H2) -g destruct @@ -69,13 +69,13 @@ fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X qed-. (* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma lexs_inv_next1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. +lemma lexs_inv_next1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⪤*[RN, RP, ⫯g] Y → + ∃∃K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2. /2 width=7 by lexs_inv_next1_aux/ qed-. -fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. +fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g → + ∃∃K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J #K1 #W1 #H destruct | #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_next_push … H) @@ -84,21 +84,21 @@ fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X ] qed-. -lemma lexs_inv_push1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y → - ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. +lemma lexs_inv_push1: ∀RN,RP,g,J,K1,Y,W1. K1.ⓑ{J}W1 ⪤*[RN, RP, ↑g] Y → + ∃∃K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2. /2 width=7 by lexs_inv_push1_aux/ qed-. -fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆. +fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom2 *) -lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⦻*[RN, RP, f] ⋆ → X = ⋆. +lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆. /2 width=6 by lexs_inv_atom2_aux/ qed-. -fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. +fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g → + ∃∃K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J #K2 #W2 #H destruct | #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_next … H2) -g destruct @@ -108,12 +108,12 @@ fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y qed-. (* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma lexs_inv_next2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. +lemma lexs_inv_next2: ∀RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ⫯g] K2.ⓑ{J}W2 → + ∃∃K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1. /2 width=7 by lexs_inv_next2_aux/ qed-. -fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. +fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g → + ∃∃K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. #RN #RP #f #X #Y * -f -X -Y [ #f #J #K2 #W2 #g #H destruct | #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_next_push … H) @@ -122,28 +122,28 @@ fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y ] qed-. -lemma lexs_inv_push2: ∀RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 → - ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. +lemma lexs_inv_push2: ∀RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ↑g] K2.ⓑ{J}W2 → + ∃∃K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1. /2 width=7 by lexs_inv_push2_aux/ qed-. (* Basic_2A1: includes lpx_sn_inv_pair *) lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. - L1.ⓑ{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2. + L1.ⓑ{I1}V1 ⪤*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) → + ∧∧ L1 ⪤*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2. #RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_next1 … H) -H #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ qed-. lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. - L1.ⓑ{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) → - ∧∧ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2. + L1.ⓑ{I1}V1 ⪤*[RN, RP, ↑f] (L2.ⓑ{I2}V2) → + ∧∧ L1 ⪤*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2. #RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push1 … H) -H #L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/ qed-. -lemma lexs_inv_tl: ∀RN,RP,f,I,L1,L2,V1,V2. L1 ⦻*[RN, RP, ⫱f] L2 → +lemma lexs_inv_tl: ∀RN,RP,f,I,L1,L2,V1,V2. L1 ⪤*[RN, RP, ⫱f] L2 → RN L1 V1 V2 → RP L1 V1 V2 → - L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2. + L1.ⓑ{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2. #RN #RP #f #I #L2 #L2 #V1 #V2 elim (pn_split f) * /2 width=1 by lexs_next, lexs_push/ qed-. @@ -151,8 +151,8 @@ qed-. (* Basic forward lemmas *****************************************************) lemma lexs_fwd_pair: ∀RN,RP,f,I1,I2,L1,L2,V1,V2. - L1.ⓑ{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 → - L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2. + L1.ⓑ{I1}V1 ⪤*[RN, RP, f] L2.ⓑ{I2}V2 → + L1 ⪤*[RN, RP, ⫱f] L2 ∧ I1 = I2. #RN #RP #f #I1 #I2 #L2 #L2 #V1 #V2 #Hf elim (pn_split f) * #g #H destruct [ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf @@ -161,7 +161,7 @@ qed-. (* Basic properties *********************************************************) -lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP, f] L2). +lemma lexs_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 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H [ elim (eq_inv_nx … H) -H /3 width=3 by lexs_next/ @@ -169,7 +169,7 @@ lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP, ] qed-. -lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⦻*[RN, RP, f] L2). +lemma lexs_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 lexs_eq_repl_back/ (**) (* full auto fails *) qed-. @@ -191,9 +191,9 @@ lemma lexs_sym: ∀RN,RP. qed-. lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2. - L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 → + L1.ⓑ{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2 → ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 → - L1.ⓑ{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2. + L1.ⓑ{I}W1 ⪤*[RN, RP, f] L2.ⓑ{I}W2. #RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/ qed-. @@ -201,15 +201,15 @@ qed-. lemma lexs_co: ∀RN1,RP1,RN2,RP2. (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) → (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) → - ∀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 lexs_atom, lexs_next, lexs_push/ qed-. lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) → - ∀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 #I #K1 #K2 #V1 #V2 #_ #HV12 #IH #H [ elim (isid_inv_next … H) -H // @@ -218,8 +218,8 @@ lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. qed-. lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) → - ∀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 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH [ * * [2: #n1 ] ] #f1 #H @@ -231,8 +231,8 @@ lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) → qed-. lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) → - ∀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 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH [2: * * [2: #n2 ] ] #f2 #H @@ -244,8 +244,8 @@ lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) → qed-. lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → - ∀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 lexs_atom, ex2_intro/ ] #f #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #y #H @@ -259,7 +259,7 @@ qed-. lemma lexs_dec: ∀RN,RP. (∀L,T1,T2. Decidable (RN L T1 T2)) → (∀L,T1,T2. Decidable (RP L T1 T2)) → - ∀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 #V1 #IH * ] [ /2 width=1 by lexs_atom, or_introl/ | #L2 #I2 #V2 #f @or_intror #H diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma index 972c773ed..af1eb9177 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_length.ma @@ -20,7 +20,7 @@ include "basic_2/relocation/lexs.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: includes: lpx_sn_fwd_length *) -lemma lexs_fwd_length: ∀RN,RP,f,L1,L2. L1 ⦻*[RN, RP, f] L2 → |L1| = |L2|. +lemma lexs_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → |L1| = |L2|. #RM #RP #f #L1 #L2 #H elim H -f -L1 -L2 // #f #I #L1 #L2 #V1 #V2 >length_pair >length_pair // qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma index c0d7c6a12..c8edf5e53 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma @@ -22,9 +22,9 @@ include "basic_2/relocation/drops.ma". theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f): lexs_transitive RN1 RN2 RN RN1 RP1 → lexs_transitive RP1 RP2 RP RN1 RP1 → - ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 → - ∀L2. L0 ⦻*[RN2, RP2, f] L2 → - L1 ⦻*[RN, RP, f] L2. + ∀L1,L0. L1 ⪤*[RN1, RP1, f] L0 → + ∀L2. L0 ⪤*[RN2, RP2, f] L2 → + L1 ⪤*[RN, RP, f] L2. #RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0 [ #f #L2 #H >(lexs_inv_atom1 … H) -L2 // | #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H @@ -74,9 +74,9 @@ theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) → /3 width=3 by/ qed-. lemma lexs_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 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct @@ -87,9 +87,9 @@ try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H qed-. lemma lexs_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 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf elim (pn_split f2) * #g2 #H2 destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma index 1ec892cbf..d949c0a19 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lfpx_frees.ma @@ -24,7 +24,7 @@ include "basic_2/rt_transition/cpx_drops.ma". (* Basic_2A1: uses: lpx_cpx_frees_trans *) lemma cpx_frees_conf_lfpx: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → - ∀L2. L1 ⦻*[cpx h G, cfull, f1] L2 → + ∀L2. L1 ⪤*[cpx h G, cfull, f1] L2 → ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 → ∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1. #h #G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1 diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index 367b19d32..157a12038 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -20,7 +20,7 @@ include "basic_2/static/frees.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) definition lfxs (R) (T): relation lenv ≝ - λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2. + λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⪤*[R, cfull, f] L2. interpretation "generic extension on referred entries (local environment)" 'RelationStar R T L1 L2 = (lfxs R T L1 L2). @@ -33,48 +33,48 @@ definition R_frees_confluent: predicate (relation3 lenv term term) ≝ definition lexs_frees_confluent: relation (relation3 lenv term term) ≝ λRN,RP. ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 → - ∀L2. L1 ⦻*[RN, RP, f1] L2 → + ∀L2. L1 ⪤*[RN, RP, f1] L2 → ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1. definition R_confluent2_lfxs: 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. (* Basic properties *********************************************************) -lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. +lemma lfxs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆. /3 width=3 by lexs_atom, frees_atom, ex2_intro/ qed. (* Basic_2A1: uses: llpx_sn_sort *) lemma lfxs_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 * /3 width=3 by lexs_push, frees_sort, ex2_intro/ qed. -lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2. +lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2. #R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/ qed. lemma lfxs_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 * /3 width=3 by lexs_push, frees_lref, ex2_intro/ qed. (* Basic_2A1: uses: llpx_sn_gref *) lemma lfxs_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 * /3 width=3 by lexs_push, frees_gref, ex2_intro/ qed. lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1. - L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 → + L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V1 → ∀V2. R L1 V V2 → - L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2. + L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V2. #R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR /3 width=5 by lexs_pair_repl, ex2_intro/ qed-. @@ -88,31 +88,31 @@ qed-. (* Basic_2A1: uses: llpx_sn_co *) lemma lfxs_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 * /4 width=7 by lexs_co, ex2_intro/ qed-. lemma lfxs_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 lexs_co_isid, ex2_intro/ qed-. (* Basic inversion lemmas ***************************************************) -lemma lfxs_inv_atom_sn: ∀R,Y2,T. ⋆ ⦻*[R, T] Y2 → Y2 = ⋆. +lemma lfxs_inv_atom_sn: ∀R,Y2,T. ⋆ ⪤*[R, T] Y2 → Y2 = ⋆. #R #Y2 #T * /2 width=4 by lexs_inv_atom1/ qed-. -lemma lfxs_inv_atom_dx: ∀R,Y1,T. Y1 ⦻*[R, T] ⋆ → Y1 = ⋆. +lemma lfxs_inv_atom_dx: ∀R,Y1,T. Y1 ⪤*[R, T] ⋆ → Y1 = ⋆. #R #I #Y1 * /2 width=4 by lexs_inv_atom2/ qed-. -lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → +lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 → (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 & + ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -123,9 +123,9 @@ lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 → ] qed-. -lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → +lemma lfxs_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. #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 * [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ @@ -134,9 +134,9 @@ lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 → ] qed-. -lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → +lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⪤*[R, #⫯i] Y2 → (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 & + ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, #i] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 * [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/ @@ -145,9 +145,9 @@ lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → ] qed-. -lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → +lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 → (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 & + ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. #R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -159,39 +159,39 @@ lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → qed-. (* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *) -lemma lfxs_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 lfxs_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_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Basic_2A1: uses: llpx_sn_inv_flat *) -lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → - L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2. +lemma lfxs_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_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Advanced inversion lemmas ************************************************) -lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 → - ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. +lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⪤*[R, ⋆s] Y2 → + ∃∃L2,V2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * [ #H destruct | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. +lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⪤*[R, ⋆s] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * [ #_ #H destruct | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → - ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & +lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⪤*[R, #0] Y2 → + ∃∃L2,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H * [ #H destruct @@ -200,8 +200,8 @@ lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 → ] qed-. -lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 & +lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⪤*[R, #0] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤*[R, V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H * [ #_ #H destruct @@ -210,32 +210,32 @@ lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 → ] qed-. -lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 → - ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2. +lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⪤*[R, #⫯i] Y2 → + ∃∃L2,V2. L1 ⪤*[R, #i] L2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H * [ #H destruct | #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1. +lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⪤*[R, #⫯i] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤*[R, #i] L2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H * [ #_ #H destruct | #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 → - ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. +lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⪤*[R, §l] Y2 → + ∃∃L2,V2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓑ{I}V2. #R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H * [ #H destruct | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 → - ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. +lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⪤*[R, §l] L2.ⓑ{I}V2 → + ∃∃L1,V1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓑ{I}V1. #R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H * [ #_ #H destruct | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ @@ -245,24 +245,24 @@ qed-. (* Basic forward lemmas *****************************************************) (* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *) -lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2. +lemma lfxs_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_lexs_trans, sor_inv_sle_sn, ex2_intro/ qed-. (* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *) -lemma lfxs_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 lfxs_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 (lfxs_inv_bind … H HV) -H -HV // qed-. (* Basic_2A1: uses: llpx_sn_fwd_flat_dx *) -lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2. +lemma lfxs_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 (lfxs_inv_flat … H) -H // qed-. -lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 → +lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⪤*[R, T] K2.ⓑ{I}V2 → ∃∃K1,V1. L1 = K1.ⓑ{I}V1. #R #I #L1 #K2 #T #V2 * #f elim (pn_split f) * #g #Hg #_ #Hf destruct [ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #K1 #V1 #_ #_ #H destruct diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma index b39c11f95..1cb6621b6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma @@ -21,18 +21,18 @@ include "basic_2/static/lfxs.ma". definition 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. + ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≡ U → + ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2. definition 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. + ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≡ U → + ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≡ K2. definition dropable_dx: predicate (relation3 lenv term term) ≝ - λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 → + 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. + ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⪤*[R, T] K2. (* Properties with generic slicing for local environments *******************) @@ -70,23 +70,23 @@ elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2 qed-. (* Basic_2A1: was: llpx_sn_inv_lift_O *) -lemma lfxs_inv_lifts_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 → +lemma lfxs_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤*[R, U] L2 → ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 → - ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2. + ∀T. ⬆*[i] T ≡ U → K1 ⪤*[R, T] K2. #R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY lapply (drops_mono … HY … HLK2) -L2 -i #H destruct // qed-. -lemma lfxs_inv_lref_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 lfxs_inv_lref_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 (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 // #Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ qed-. -lemma lfxs_inv_lref_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 lfxs_inv_lref_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 (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 // #Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/ diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fqup.ma index e0fc99355..c54aaa406 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fqup.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fqup.ma @@ -20,12 +20,12 @@ include "basic_2/static/lfxs.ma". (* Advanced properties ******************************************************) (* Basic_2A1: uses: llpx_sn_refl *) -lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L. +lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤*[R, T] L. #R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/ qed. lemma lfxs_pair: ∀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 diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_length.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_length.ma index 386e2ba88..3b8208e4a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_length.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_length.ma @@ -20,6 +20,6 @@ include "basic_2/static/lfxs.ma". (* Forward lemmas with length for local environments ************************) (* Basic_2A1: uses: llpx_sn_fwd_length *) -lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|. +lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|. #R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma index 4cf5505cc..8f92b0f34 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -21,14 +21,14 @@ include "basic_2/static/lfxs.ma". (* Advanced properties ******************************************************) -lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → - ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⦻*[R, cfull, f] L2. +lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → + ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⪤*[R, cfull, f] L2. #R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/ qed-. (* Basic_2A1: uses: llpx_sn_dec *) lemma lfxs_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 (lexs_dec R cfull HR … L1 L2 f) @@ -37,8 +37,8 @@ qed-. lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → lexs_frees_confluent … R1 cfull → - ∀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 @@ -56,8 +56,8 @@ qed-. lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → lexs_frees_confluent … R1 cfull → - ∀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 @@ -72,8 +72,8 @@ qed-. lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → lexs_frees_confluent … R1 cfull → - ∀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 @@ -93,8 +93,8 @@ qed-. (* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *) theorem lfxs_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 elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2)) /3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/ @@ -102,8 +102,8 @@ qed. (* Basic_2A1: llpx_sn_flat *) theorem lfxs_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, lexs_join, ex2_intro/ qed. @@ -135,16 +135,16 @@ qed-. (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *) lemma lfnxs_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 (lfxs_dec … HR L1 L2 V) /4 width=2 by lfxs_bind, or_intror, or_introl/ qed-. (* Basic_2A1: uses: nllpx_sn_inv_flat *) lemma lfnxs_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 (lfxs_dec … HR L1 L2 V) /4 width=1 by lfxs_flat, or_intror, or_introl/ qed-. diff --git a/matita/matita/predefined_virtuals.ml b/matita/matita/predefined_virtuals.ml index 12a7c9442..36507a42d 100644 --- a/matita/matita/predefined_virtuals.ml +++ b/matita/matita/predefined_virtuals.ml @@ -1580,8 +1580,8 @@ let predefined_classes = [ ["V"; "𝕍"; "𝐕"; "Ⓥ"; ] ; ["w"; "ω"; "𝕨"; "𝐰"; "𝛚"; "ⓦ"; ] ; ["W"; "Ω"; "𝕎"; "𝐖"; "𝛀"; "Ⓦ"; ] ; - ["x"; "ξ"; "χ"; "ϰ"; "𝕩"; "𝐱"; "𝛏"; "𝛘"; "𝛞"; "ⓧ"; ] ; - ["X"; "Ξ"; "𝕏";"𝐗"; "𝚵"; "Ⓧ"; "⦻"; ] ; + ["x"; "ξ"; "χ"; "ϰ"; "𝕩"; "𝐱"; "𝛏"; "𝛘"; "𝛞"; "ⓧ"; "⨴"; "⨵"; ] ; + ["X"; "Ξ"; "𝕏";"𝐗"; "𝚵"; "Ⓧ"; "⦻"; "⪤" ] ; ["y"; "υ"; "𝕪"; "𝐲"; "ⓨ"; ] ; ["Y"; "ϒ"; "𝕐"; "𝐘"; "𝚼"; "Ⓨ"; ] ; ["z"; "ζ"; "𝕫"; "𝐳"; "𝛇"; "ⓩ"; ] ; -- 2.39.2