X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Flfxs.ma;h=c9d55b73ff582d2c06748b8ad5a0b819d4393fb0;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=b3c7a7fe06e8d24c8164bf582743c0ad46004757;hpb=e9da8e091898b6e67a2f270581bdc5cdbe80e9b0;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index b3c7a7fe0..c9d55b73f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -14,161 +14,307 @@ include "ground_2/relocation/rtmap_id.ma". include "basic_2/notation/relations/relationstar_4.ma". -include "basic_2/grammar/ceq.ma". +include "basic_2/syntax/cext2.ma". include "basic_2/relocation/lexs.ma". 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 ⪤*[cext2 R, cfull, f] L2. interpretation "generic extension on referred entries (local environment)" 'RelationStar R T L1 L2 = (lfxs R T L1 L2). -(* Basic properties ***********************************************************) +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 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. -lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆. -/3 width=3 by lexs_atom, frees_atom, ex2_intro/ -qed. - -lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s. - 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. -#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/ -qed. +definition lfxs_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. -lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i. - 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. +definition lfxs_transitive: relation3 ? (relation3 ?? term) … ≝ + λR1,R2,R3. + ∀K1,K,V1. K1 ⪤*[R1, V1] K → + ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2. -lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l. - 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. +(* Basic inversion lemmas ***************************************************) -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. -#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/ +lemma lfxs_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤*[R, T] Y2 → Y2 = ⋆. +#R #Y2 #T * /2 width=4 by lexs_inv_atom1/ qed-. -(* Basic inversion lemmas ***************************************************) +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_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆. -#R #I #Y2 * /2 width=4 by lexs_inv_atom1/ +lemma 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 #I1 ] #Y2 #s * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| lapply (frees_inv_sort … H1) -H1 #Hf + elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct + /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/ +] qed-. -lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆. -#R #I #Y1 * /2 width=4 by lexs_inv_atom2/ +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 & + Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 + | ∃∃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 (lexs_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/ +| elim (frees_inv_unit … H1) -H1 #g #HX #H destruct + elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct + >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/ +| elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct + elim (lexs_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H + elim (ext2_inv_pair_sn … H) -H + /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/ +] qed-. -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 & - 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/ -| #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/ +lemma lfxs_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤*[R, #↑i] Y2 → + ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ + | ∃∃I1,I2,L1,L2. L1 ⪤*[R, #i] L2 & + Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. +#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2 +[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ +| elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct + elim (lexs_inv_push1 … H2) -H2 + /4 width=7 by ex3_4_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 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 #I1 ] #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 #I2 #L2 #H12 #_ #H destruct + /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/ ] 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. +(* 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. #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/ +/6 width=6 by sle_lexs_trans, lexs_inv_tl, ext2_pair, 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. +(* 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. #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_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 lfxs_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 (lfxs_inv_sort … H) -H * +[ #H destruct +| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_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 (lfxs_inv_sort … H) -H * +[ #_ #H destruct +| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_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 (lfxs_inv_zero … H) -H * [ #H destruct -| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct +| #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/ +| #f #Z #Y1 #Y2 #_ #_ #H destruct ] 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 lfxs_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 (lfxs_inv_zero … H) -H * [ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct +| #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/ +| #f #Z #Y1 #Y2 #_ #_ #_ #H destruct +] +qed-. + +lemma lfxs_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 (lfxs_inv_zero … H) -H * +[ #H destruct +| #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct +| #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/ +] +qed-. + +lemma lfxs_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 (lfxs_inv_zero … H) -H * +[ #_ #H destruct +| #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct +| #f #Z #K1 #Y2 #Hf #HK12 #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 * +lemma lfxs_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 (lfxs_inv_lref … H) -H * [ #H destruct -| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +| #Z1 #I2 #Y1 #K2 #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 * +lemma lfxs_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 (lfxs_inv_lref … H) -H * [ #_ #H destruct -| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_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 (lfxs_inv_gref … H) -H * +[ #H destruct +| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ +] +qed-. + +lemma lfxs_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 (lfxs_inv_gref … H) -H * +[ #_ #H destruct +| #I1 #Z2 #K1 #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 +lemma lfxs_fwd_zero_pair (R): ∀I,K1,K2,V1,V2. + K1.ⓑ{I}V1 ⪤*[R, #0] K2.ⓑ{I}V2 → K1 ⪤*[R, V1] K2. +#R #I #K1 #K2 #V1 #V2 #H +elim (lfxs_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 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-. -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. +(* 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. #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV // qed-. -lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2. +(* 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. #R #I #L1 #L2 #V #T #H elim (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 // +lemma lfxs_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 (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct +/2 width=3 by ex1_2_intro/ +qed-. + +(* Basic properties *********************************************************) + +lemma lfxs_atom (R): ∀I. ⋆ ⪤*[R, ⓪{I}] ⋆. +#R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/ +qed. + +lemma lfxs_sort (R): ∀I1,I2,L1,L2,s. + 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, lexs_push, isid_push, ex2_intro/ +qed. + +lemma lfxs_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, lexs_next, ex2_intro/ +qed. + +lemma lfxs_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, lexs_next, ext2_unit, ex2_intro/ qed. + +lemma lfxs_lref (R): ∀I1,I2,L1,L2,i. + L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #↑i] L2.ⓘ{I2}. +#R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/ +qed. + +lemma lfxs_gref (R): ∀I1,I2,L1,L2,l. + 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, lexs_push, isid_push, ex2_intro/ +qed. + +lemma lfxs_bind_repl_dx (R): ∀I,I1,L1,L2,T. + L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I1} → + ∀I2. cext2 R L1 I I2 → + L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I2}. +#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR +/3 width=5 by lexs_pair_repl, ex2_intro/ +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. +#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by lexs_co, cext2_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. +#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 * +/4 width=7 by lexs_co_isid, ex2_intro/ 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/ +lemma lfxs_unit_sn (R1) (R2): + ∀I,K1,L2. K1.ⓤ{I} ⪤*[R1, #0] L2 → K1.ⓤ{I} ⪤*[R2, #0] L2. +#R1 #R2 #I #K1 #L2 #H +elim (lfxs_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct +/3 width=7 by lfxs_unit, lexs_co_isid/ 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 9: + llpx_sn_skip llpx_sn_lref llpx_sn_free + llpx_sn_fwd_lref + llpx_sn_Y llpx_sn_ge_up llpx_sn_ge + llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx *)