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=0848e850358c8878053c135b7aa41bb1e85af4c2;hp=515e24bef730445b73a2579592925d7808501a60;hb=f308429a0fde273605a2330efc63268b4ac36c99;hpb=87f57ddc367303c33e19c83cd8989cd561f3185b diff --git a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma index 515e24bef..0848e8503 100644 --- a/matita/matita/contribs/lambdadelta/static_2/static/rex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/static/rex.ma @@ -21,7 +21,7 @@ include "static_2/static/frees.ma". (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****) definition rex (R) (T): relation lenv ≝ - λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R, cfull, f] L2. + λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R,cfull,f] L2. interpretation "generic extension on referred entries (local environment)" 'Relation R T L1 L2 = (rex R T L1 L2). @@ -30,32 +30,32 @@ definition R_confluent2_rex: relation4 (relation3 lenv term term) (relation3 lenv term term) … ≝ λR1,R2,RP1,RP2. ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 → - ∀L1. L0 ⪤[RP1, T0] L1 → ∀L2. L0 ⪤[RP2, T0] L2 → + ∀L1. L0 ⪤[RP1,T0] L1 → ∀L2. L0 ⪤[RP2,T0] L2 → ∃∃T. R2 L1 T1 T & R1 L2 T2 T. definition rex_confluent: relation … ≝ λR1,R2. - ∀K1,K,V1. K1 ⪤[R1, V1] K → ∀V. R1 K1 V1 V → - ∀K2. K ⪤[R2, V] K2 → K ⪤[R2, V1] K2. + ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V → + ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2. definition rex_transitive: relation3 ? (relation3 ?? term) … ≝ λR1,R2,R3. - ∀K1,K,V1. K1 ⪤[R1, V1] K → + ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2. (* Basic inversion lemmas ***************************************************) -lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R, T] Y2 → Y2 = ⋆. +lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R,T] Y2 → Y2 = ⋆. #R #Y2 #T * /2 width=4 by sex_inv_atom1/ qed-. -lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R, T] ⋆ → Y1 = ⋆. +lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R,T] ⋆ → Y1 = ⋆. #R #I #Y1 * /2 width=4 by sex_inv_atom2/ qed-. -lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 → +lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R,⋆s] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, ⋆s] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -66,11 +66,11 @@ lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 → ] qed-. -lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 → +lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R,#0] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 & + | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 & Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2 - | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R, cfull, f] L2 & + | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R,cfull,f] L2 & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}. #R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/ @@ -84,9 +84,9 @@ lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 → ] qed-. -lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 → +lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R,#↑i] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, #i] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,#i] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -96,9 +96,9 @@ lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 → ] qed-. -lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 → +lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R,§l] Y2 → ∨∨ Y1 = ⋆ ∧ Y2 = ⋆ - | ∃∃I1,I2,L1,L2. L1 ⪤[R, §l] L2 & + | ∃∃I1,I2,L1,L2. L1 ⪤[R,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}. #R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/ @@ -110,39 +110,39 @@ lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 → qed-. (* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *) -lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → - ∧∧ L1 ⪤[R, V1] L2 & L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2. +lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → R L1 V1 V2 → + ∧∧ L1 ⪤[R,V1] L2 & L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2. #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf /6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Basic_2A1: uses: llpx_sn_inv_flat *) -lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → - ∧∧ L1 ⪤[R, V] L2 & L1 ⪤[R, T] L2. +lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → + ∧∧ L1 ⪤[R,V] L2 & L1 ⪤[R,T] L2. #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf /5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/ qed-. (* Advanced inversion lemmas ************************************************) -lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R, ⋆s] L2 → - ∃∃I2,K2. K1 ⪤[R, ⋆s] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R,⋆s] L2 → + ∃∃I2,K2. K1 ⪤[R,⋆s] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R, ⋆s] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, ⋆s] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R,⋆s] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,⋆s] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 → - ∃∃K2,V2. K1 ⪤[R, V1] K2 & R K1 V1 V2 & +lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R,#0] L2 → + ∃∃K2,V2. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2. #R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H * [ #H destruct @@ -152,8 +152,8 @@ lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 → ] qed-. -lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 → - ∃∃K1,V1. K1 ⪤[R, V1] K2 & R K1 V1 V2 & +lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R,#0] K2.ⓑ{I}V2 → + ∃∃K1,V1. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1. #R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H * [ #_ #H destruct @@ -163,8 +163,8 @@ lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 → ] qed-. -lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 → - ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 & +lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R,#0] L2 → + ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L2 = K2.ⓤ{I}. #R #I #K1 #L2 #H elim (rex_inv_zero … H) -H * [ #H destruct @@ -173,8 +173,8 @@ lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 → ] qed-. -lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} → - ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 & +lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R,#0] K2.ⓤ{I} → + ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L1 = K1.ⓤ{I}. #R #I #L1 #K2 #H elim (rex_inv_zero … H) -H * [ #_ #H destruct @@ -183,32 +183,32 @@ lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} → ] qed-. -lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R, #↑i] L2 → - ∃∃I2,K2. K1 ⪤[R, #i] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R,#↑i] L2 → + ∃∃I2,K2. K1 ⪤[R,#i] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R, #↑i] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, #i] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R,#↑i] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,#i] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R, §l] L2 → - ∃∃I2,K2. K1 ⪤[R, §l] K2 & L2 = K2.ⓘ{I2}. +lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R,§l] L2 → + ∃∃I2,K2. K1 ⪤[R,§l] K2 & L2 = K2.ⓘ{I2}. #R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H * [ #H destruct | #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ ] qed-. -lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R, §l] K2.ⓘ{I2} → - ∃∃I1,K1. K1 ⪤[R, §l] K2 & L1 = K1.ⓘ{I1}. +lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R,§l] K2.ⓘ{I2} → + ∃∃I1,K1. K1 ⪤[R,§l] K2 & L1 = K1.ⓘ{I1}. #R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H * [ #_ #H destruct | #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/ @@ -218,30 +218,30 @@ qed-. (* Basic forward lemmas *****************************************************) lemma rex_fwd_zero_pair (R): ∀I,K1,K2,V1,V2. - K1.ⓑ{I}V1 ⪤[R, #0] K2.ⓑ{I}V2 → K1 ⪤[R, V1] K2. + K1.ⓑ{I}V1 ⪤[R,#0] K2.ⓑ{I}V2 → K1 ⪤[R,V1] K2. #R #I #K1 #K2 #V1 #V2 #H elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct // qed-. (* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *) -lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R, ②{I}V.T] L2 → L1 ⪤[R, V] L2. +lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R,②{I}V.T] L2 → L1 ⪤[R,V] L2. #R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL [ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf /4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/ qed-. (* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *) -lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2. +lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2. #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV // qed-. (* Basic_2A1: uses: llpx_sn_fwd_flat_dx *) -lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → L1 ⪤[R, T] L2. +lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → L1 ⪤[R,T] L2. #R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H // qed-. -lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R, T] K2.ⓘ{I2} → +lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ{I2} → ∃∃I1,K1. L1 = K1.ⓘ{I1}. #R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct [ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct @@ -250,63 +250,63 @@ qed-. (* Basic properties *********************************************************) -lemma rex_atom (R): ∀I. ⋆ ⪤[R, ⓪{I}] ⋆. +lemma rex_atom (R): ∀I. ⋆ ⪤[R,⓪{I}] ⋆. #R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/ qed. lemma rex_sort (R): ∀I1,I2,L1,L2,s. - L1 ⪤[R, ⋆s] L2 → L1.ⓘ{I1} ⪤[R, ⋆s] L2.ⓘ{I2}. + L1 ⪤[R,⋆s] L2 → L1.ⓘ{I1} ⪤[R,⋆s] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #s * #f #Hf #H12 lapply (frees_inv_sort … Hf) -Hf /4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/ qed. -lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 → - R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, #0] L2.ⓑ{I}V2. +lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 → + R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,#0] L2.ⓑ{I}V2. #R #I1 #I2 #L1 #L2 #V1 * /4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/ qed. -lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 → - L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}. +lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 → + L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}. /4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed. lemma rex_lref (R): ∀I1,I2,L1,L2,i. - L1 ⪤[R, #i] L2 → L1.ⓘ{I1} ⪤[R, #↑i] L2.ⓘ{I2}. + L1 ⪤[R,#i] L2 → L1.ⓘ{I1} ⪤[R,#↑i] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/ qed. lemma rex_gref (R): ∀I1,I2,L1,L2,l. - L1 ⪤[R, §l] L2 → L1.ⓘ{I1} ⪤[R, §l] L2.ⓘ{I2}. + L1 ⪤[R,§l] L2 → L1.ⓘ{I1} ⪤[R,§l] L2.ⓘ{I2}. #R #I1 #I2 #L1 #L2 #l * #f #Hf #H12 lapply (frees_inv_gref … Hf) -Hf /4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/ qed. lemma rex_bind_repl_dx (R): ∀I,I1,L1,L2,T. - L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I1} → + L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I1} → ∀I2. cext2 R L1 I I2 → - L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I2}. + L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I2}. #R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR /3 width=5 by sex_pair_repl, ex2_intro/ qed-. (* Basic_2A1: uses: llpx_sn_co *) lemma rex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) → - ∀L1,L2,T. L1 ⪤[R1, T] L2 → L1 ⪤[R2, T] L2. + ∀L1,L2,T. L1 ⪤[R1,T] L2 → L1 ⪤[R2,T] L2. #R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/ qed-. lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2. (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) → (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) → - L1 ⪤[R1, T1] L2 → L1 ⪤[R2, T2] L2. + L1 ⪤[R1,T1] L2 → L1 ⪤[R2,T2] L2. #R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 * /4 width=7 by sex_co_isid, ex2_intro/ qed-. lemma rex_unit_sn (R1) (R2): - ∀I,K1,L2. K1.ⓤ{I} ⪤[R1, #0] L2 → K1.ⓤ{I} ⪤[R2, #0] L2. + ∀I,K1,L2. K1.ⓤ{I} ⪤[R1,#0] L2 → K1.ⓤ{I} ⪤[R2,#0] L2. #R1 #R2 #I #K1 #L2 #H elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct /3 width=7 by rex_unit, sex_co_isid/