X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fstatic%2Frex.ma;h=45e9028d70ff7668af576eca059b5e9bb3fdbbd6;hp=bfaf0cbce5d2b0cb831b246a216a7097384b359c;hb=98e786e1a6bd7b621e37ba7cd4098d4a0a6f8278;hpb=d8d00d6f6694155be5be486a8239f5953efe28b7 diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma index bfaf0cbce..45e9028d7 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma @@ -12,11 +12,11 @@ (* *) (**************************************************************************) -include "ground_2/xoa/ex_1_2.ma". -include "ground_2/xoa/ex_3_4.ma". -include "ground_2/xoa/ex_4_4.ma". -include "ground_2/xoa/ex_4_5.ma". -include "ground_2/relocation/rtmap_id.ma". +include "ground/xoa/ex_1_2.ma". +include "ground/xoa/ex_3_4.ma". +include "ground/xoa/ex_4_4.ma". +include "ground/xoa/ex_4_5.ma". +include "ground/relocation/rtmap_id.ma". include "static_2/notation/relations/relation_4.ma". include "static_2/syntax/cext2.ma". include "static_2/relocation/sex.ma". @@ -25,46 +25,62 @@ 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. - -interpretation "generic extension on referred entries (local environment)" - 'Relation R T L1 L2 = (rex R T L1 L2). - -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 → - ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + 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). + +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 → + ∃∃T. R2 L1 T1 T & R1 L2 T2 T. + +definition R_replace3_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 → + ∀T. R2 L1 T1 T → R1 L2 T2 T. + +definition R_transitive_rex: 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. + +definition R_confluent1_rex: relation … ≝ + λR1,R2. + ∀K1,K2,V1. K1 ⪤[R2,V1] K2 → ∀V2. R1 K1 V1 V2 → R1 K2 V1 V2. 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. - -definition rex_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. + λR1,R2. + ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V → + ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2. (* 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 → ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. + | ∃∃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/ | lapply (frees_inv_sort … H1) -H1 #Hf - elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (pr_isi_inv_gen … Hf) -Hf #g #Hg #H destruct elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/ ] @@ -72,11 +88,9 @@ qed-. 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 & - 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}. + ∨∨ ∧∧ 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 (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/ | elim (frees_inv_unit … H1) -H1 #g #HX #H destruct @@ -92,7 +106,7 @@ qed-. lemma rex_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}. + | ∃∃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/ | elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct @@ -104,11 +118,11 @@ qed-. lemma rex_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}. + | ∃∃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/ | lapply (frees_inv_gref … H1) -H1 #Hf - elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct + elim (pr_isi_inv_gen … Hf) -Hf #g #Hg #H destruct elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/ ] @@ -116,25 +130,25 @@ 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. + ∀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/ +/6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, pr_sor_inv_sle_dx, pr_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 → + ∀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/ +/5 width=6 by sle_sex_trans, pr_sor_inv_sle_dx, pr_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}. + ∀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/ @@ -142,8 +156,8 @@ lemma rex_inv_sort_bind_sn (R): 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}. + ∀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/ @@ -151,8 +165,8 @@ lemma rex_inv_sort_bind_dx (R): 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 & L2 = K2.ⓑ{I}V2. + ∀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 | #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct @@ -162,8 +176,8 @@ lemma rex_inv_zero_pair_sn (R): 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 & L1 = K1.ⓑ{I}V1. + ∀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 | #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct @@ -173,8 +187,8 @@ lemma rex_inv_zero_pair_dx (R): 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 & L2 = K2.ⓤ{I}. + ∀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 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct @@ -183,8 +197,8 @@ lemma rex_inv_zero_unit_sn (R): 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 & L1 = K1.ⓤ{I}. + ∀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 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct @@ -193,8 +207,8 @@ lemma rex_inv_zero_unit_dx (R): 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}. + ∀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/ @@ -202,8 +216,8 @@ lemma rex_inv_lref_bind_sn (R): 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}. + ∀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/ @@ -211,8 +225,8 @@ lemma rex_inv_lref_bind_dx (R): 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}. + ∀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/ @@ -220,8 +234,8 @@ lemma rex_inv_gref_bind_sn (R): 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}. + ∀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/ @@ -231,78 +245,81 @@ 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. + ∀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 (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/ +/4 width=6 by sle_sex_trans, pr_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. + ∀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} → - ∃∃I1,K1. L1 = K1.ⓘ{I1}. -#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct + ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ[I2] → + ∃∃I1,K1. L1 = K1.ⓘ[I1]. +#R #I2 #L1 #K2 #T * #f elim (pr_map_split_tl f) * #g #Hg #_ #Hf destruct [ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct /2 width=3 by ex1_2_intro/ 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}. + ∀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, sex_push, isid_push, ex2_intro/ +/4 width=3 by frees_sort, sex_push, pr_isi_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. + 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}. + ∀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}. + ∀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 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}. + ∀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, sex_push, isid_push, ex2_intro/ +/4 width=3 by frees_gref, sex_push, pr_isi_push, ex2_intro/ qed. lemma rex_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}. + ∀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 sex_pair_repl, ex2_intro/ qed-. @@ -316,15 +333,15 @@ qed-. lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2. - (∀f. L1 ⊢ 𝐅+⦃T1⦄ ≘ f → 𝐈⦃f⦄) → - (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅+⦃T2⦄ ≘ f) → + (∀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 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/