X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fi_static%2Ftc_lfxs.ma;h=da4959b88dbe4c7f9dedcafa27445409f4bc79bb;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=4c30dc0a9b999f679d4da476fa6fba8332158c3f;hpb=cbc645186c8836c88c559c787a4deea63b7a12b0;p=helm.git 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 4c30dc0a9..da4959b88 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 @@ -12,249 +12,184 @@ (* *) (**************************************************************************) +include "ground_2/lib/star.ma". include "basic_2/notation/relations/relationstarstar_4.ma". include "basic_2/static/lfxs.ma". (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***) -definition tc_lfxs (R) (T): relation lenv ≝ TC … (lfxs R T). +definition tc_lfxs (R): term → relation lenv ≝ CTC … (lfxs R). interpretation "iterated extension on referred entries (local environment)" 'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2). (* Basic properties *********************************************************) -lemma tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆. +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. +#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *) +qed-. + +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, step, inj/ +/3 width=4 by lfxs_sort, tc_lfxs_step_dx, inj/ +qed. + +lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) → + ∀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_pair, tc_lfxs_step_dx, inj/ qed. +lemma tc_lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 → + L1.ⓤ{I} ⪤**[R, #0] L2.ⓤ{I}. +/3 width=3 by lfxs_unit, 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, step, inj/ +/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, step, inj/ +/3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/ qed. -lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull → - (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) → - ∀T. symmetric … (tc_lfxs R T). -#R #H1R #H2R #T #L1 #L2 #H elim H -L2 -/4 width=3 by lfxs_sym, TC_strap, inj/ -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, step, inj/ +/4 width=5 by lfxs_co, tc_lfxs_step_dx, inj/ qed-. (* Basic inversion lemmas ***************************************************) -lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆. +(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *) +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-. -lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆. +(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *) +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 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. +lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤**[R, ⋆s] Y2 → + ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ + | ∃∃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 (lfxs_inv_sort … H) -H * - /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/ + /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/ | #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H * - [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/ - | #I #L #L2 #V #V2 #HL2 #H #H2 * * + [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ + | #I #I2 #L #L2 #HL2 #H #H2 * * [ #H1 #H0 destruct - | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct - /4 width=8 by ex3_5_intro, step, or_intror/ + | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct + /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/ ] ] ] qed-. -(* -lemma tc_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 & - Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2. -#R #Y1 #Y2 #H elim H -Y2 -[ #Y2 #H elim (lfxs_inv_zero … H) -H * - /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/ -| #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H * - [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/ - | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * * + +lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤**[R, §l] Y2 → + ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ + | ∃∃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 (lfxs_inv_gref … H) -H * + /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/ +| #Y #Y2 #_ #H elim (lfxs_inv_gref … H) -H * + [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/ + | #I #I2 #L #L2 #HL2 #H #H2 * * [ #H1 #H0 destruct - | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct - @or_intror @ex4_5_intro [6,7: |*: /width=7/ ] - - /4 width=8 by ex3_5_intro, step, or_intror/ + | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct + /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/ ] ] ] qed-. - - - - -#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/ -| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg - /4 width=9 by ex4_5_intro, ex2_intro, or_intror/ -] -qed-. - -lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃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/ -| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg - /4 width=8 by ex3_5_intro, ex2_intro, or_intror/ +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. +#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 lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 → - (Y1 = ⋆ ∧ Y2 = ⋆) ∨ - ∃∃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/ -| lapply (frees_inv_gref … H1) -H1 #Hf - elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct - elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct - /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/ +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/ ] qed-. -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-. - -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. -#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H * +lemma tc_lfxs_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 (tc_lfxs_inv_sort … H) -H * [ #H destruct -| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +| #Z #I2 #Y1 #L2 #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. -#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H * +lemma tc_lfxs_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 (tc_lfxs_inv_sort … H) -H * [ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +| #I1 #Z #L1 #Y2 #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 & - Y2 = L2.ⓑ{I}V2. -#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H * +lemma tc_lfxs_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 (tc_lfxs_inv_gref … H) -H * [ #H destruct -| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct - /2 width=5 by ex3_2_intro/ +| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] 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 & - Y1 = L1.ⓑ{I}V1. -#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H * +lemma tc_lfxs_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 (tc_lfxs_inv_gref … H) -H * [ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct - /2 width=5 by ex3_2_intro/ -] -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. -#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. -#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. -#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. -#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/ +| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. (* Basic forward lemmas *****************************************************) -lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2. -#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf -/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/ +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 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 // +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. +#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind … H) -H // qed-. -lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2. -#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H // +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-. -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_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2. -#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/ -qed-. - -(* Basic_2A1: removed theorems 24: - llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref - llpx_sn_bind llpx_sn_flat - llpx_sn_inv_bind llpx_sn_inv_flat - llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length - llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx - llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co - llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx -*) +(* Basic_2A1: removed theorems 2: + TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2 *)