X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Fsex.ma;h=2a6946fcab7ac6441c2109b34fbdfa422e9b625c;hb=dc605ae41c39773f55381f241b1ed3db4acf5edd;hp=df09463c8085877bc9bdd7fc0c7ae95dcc99a1d9;hpb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma index df09463c8..2a6946fca 100644 --- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma +++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex.ma @@ -12,8 +12,8 @@ (* *) (**************************************************************************) -include "ground_2/relocation/rtmap_sle.ma". -include "ground_2/relocation/rtmap_sdj.ma". +include "ground/relocation/rtmap_sle.ma". +include "ground/relocation/rtmap_sdj.ma". include "static_2/notation/relations/relation_5.ma". include "static_2/syntax/lenv.ma". @@ -23,45 +23,68 @@ inductive sex (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝ | sex_atom: ∀f. sex RN RP f (⋆) (⋆) | sex_next: ∀f,I1,I2,L1,L2. sex RN RP f L1 L2 → RN L1 I1 I2 → - sex RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) + sex RN RP (↑f) (L1.ⓘ[I1]) (L2.ⓘ[I2]) | sex_push: ∀f,I1,I2,L1,L2. sex RN RP f L1 L2 → RP L1 I1 I2 → - sex RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2}) + sex RN RP (⫯f) (L1.ⓘ[I1]) (L2.ⓘ[I2]) . -interpretation "generic entrywise extension (local environment)" - 'Relation RN RP f L1 L2 = (sex RN RP f L1 L2). - -definition R_pw_confluent2_sex: relation3 lenv bind bind → relation3 lenv bind bind → - relation3 lenv bind bind → relation3 lenv bind bind → - relation3 lenv bind bind → relation3 lenv bind bind → - relation3 rtmap lenv bind ≝ - λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0. - ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 → - ∀L1. L0 ⪤[RN1, RP1, f] L1 → ∀L2. L0 ⪤[RN2, RP2, f] L2 → - ∃∃I. R2 L1 I1 I & R1 L2 I2 I. - -definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind → - relation3 lenv bind bind → - relation3 lenv bind bind → relation3 lenv bind bind → - relation3 rtmap lenv bind ≝ - λR1,R2,R3,RN,RP,f,L1,I1. - ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN, RP, f] L2 → - ∀I2. R2 L2 I I2 → R3 L1 I1 I2. +interpretation + "generic entrywise extension (local environment)" + 'Relation RN RP f L1 L2 = (sex RN RP f L1 L2). + +definition R_pw_transitive_sex: + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 rtmap lenv bind ≝ + λR1,R2,R3,RN,RP,f,L1,I1. + ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 → + ∀I2. R2 L2 I I2 → R3 L1 I1 I2. + +definition R_pw_confluent1_sex: + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 rtmap lenv bind ≝ + λR1,R2,RN,RP,f,L1,I1. + ∀I2. R1 L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 → R2 L2 I1 I2. + +definition R_pw_confluent2_sex: + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 rtmap lenv bind ≝ + λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0. + ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 → + ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 → + ∃∃I. R2 L1 I1 I & R1 L2 I2 I. + +definition R_pw_replace3_sex: + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 lenv bind bind → relation3 lenv bind bind → + relation3 rtmap lenv bind ≝ + λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0. + ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 → + ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 → + ∀I. R2 L1 I1 I → R1 L2 I2 I. (* Basic inversion lemmas ***************************************************) -fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆. +fact sex_inv_atom1_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I1 #I2 #L1 #L2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom1 *) -lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆. +lemma sex_inv_atom1 (RN) (RP): + ∀f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆. /2 width=6 by sex_inv_atom1_aux/ qed-. -fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. +fact sex_inv_next1_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ↑g → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2]. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct | #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct @@ -71,12 +94,14 @@ fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K qed-. (* Basic_2A1: includes lpx_sn_inv_pair1 *) -lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ↑g] Y → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}. +lemma sex_inv_next1 (RN) (RP): + ∀g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,↑g] Y → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ[J2]. /2 width=7 by sex_inv_next1_aux/ qed-. -fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. +fact sex_inv_push1_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ[J1] → f = ⫯g → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2]. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J1 #K1 #H destruct | #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H) @@ -85,21 +110,25 @@ fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K ] qed-. -lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ⫯g] Y → - ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}. +lemma sex_inv_push1 (RN) (RP): + ∀g,J1,K1,Y. K1.ⓘ[J1] ⪤[RN,RP,⫯g] Y → + ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ[J2]. /2 width=7 by sex_inv_push1_aux/ qed-. -fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆. +fact sex_inv_atom2_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆. #RN #RP #f #X #Y * -f -X -Y // #f #I1 #I2 #L1 #L2 #_ #_ #H destruct qed-. (* Basic_2A1: includes lpx_sn_inv_atom2 *) -lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆. +lemma sex_inv_atom2 (RN) (RP): + ∀f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆. /2 width=6 by sex_inv_atom2_aux/ qed-. -fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. +fact sex_inv_next2_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ↑g → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1]. #RN #RP #f #X #Y * -f -X -Y [ #f #g #J2 #K2 #H destruct | #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct @@ -109,12 +138,14 @@ fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K qed-. (* Basic_2A1: includes lpx_sn_inv_pair2 *) -lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ↑g] K2.ⓘ{J2} → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}. +lemma sex_inv_next2 (RN) (RP): + ∀g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ[J2] → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ[J1]. /2 width=7 by sex_inv_next2_aux/ qed-. -fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. +fact sex_inv_push2_aux (RN) (RP): + ∀f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ[J2] → f = ⫯g → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1]. #RN #RP #f #X #Y * -f -X -Y [ #f #J2 #K2 #g #H destruct | #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H) @@ -123,37 +154,41 @@ fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K ] qed-. -lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ⫯g] K2.ⓘ{J2} → - ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}. +lemma sex_inv_push2 (RN) (RP): + ∀g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ[J2] → + ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ[J1]. /2 width=7 by sex_inv_push2_aux/ qed-. (* Basic_2A1: includes lpx_sn_inv_pair *) -lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2. +lemma sex_inv_next (RN) (RP): + ∀f,I1,I2,L1,L2. + L1.ⓘ[I1] ⪤[RN,RP,↑f] L2.ⓘ[I2] → + L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. -lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2. +lemma sex_inv_push (RN) (RP): + ∀f,I1,I2,L1,L2. + L1.ⓘ[I1] ⪤[RN,RP,⫯f] L2.ⓘ[I2] → + L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2. #RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H #I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/ qed-. -lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 → - RN L1 I1 I2 → RP L1 I1 I2 → - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}. +lemma sex_inv_tl (RN) (RP): + ∀f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫰f] L2 → + RN L1 I1 I2 → RP L1 I1 I2 → + L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2]. #RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) * /2 width=1 by sex_next, sex_push/ qed-. (* Basic forward lemmas *****************************************************) -lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → - L1 ⪤[RN, RP, ⫱f] L2. +lemma sex_fwd_bind (RN) (RP): + ∀f,I1,I2,L1,L2. + L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] → L1 ⪤[RN,RP,⫰f] L2. #RN #RP #f #I1 #I2 #L1 #L2 #Hf elim (pn_split f) * #g #H destruct [ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf // @@ -161,7 +196,8 @@ qed-. (* Basic properties *********************************************************) -lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2). +lemma sex_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 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H [ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/ @@ -169,40 +205,45 @@ lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] ] qed-. -lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2). +lemma sex_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 sex_eq_repl_back/ (**) (* full auto fails *) qed-. -lemma sex_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP → - ∀f.reflexive … (sex RN RP f). +lemma sex_refl (RN) (RP): + c_reflexive … RN → c_reflexive … RP → + ∀f.reflexive … (sex RN RP f). #RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L // #L #I #IH #f elim (pn_split f) * #g #H destruct /2 width=1 by sex_next, sex_push/ qed. -lemma sex_sym: ∀RN,RP. - (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) → - (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) → - ∀f. symmetric … (sex RN RP f). +lemma sex_sym (RN) (RP): + (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) → + (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) → + ∀f. symmetric … (sex RN RP f). #RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f /3 width=2 by sex_next, sex_push/ qed-. -lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2. - L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} → - ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 → - L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}. +lemma sex_pair_repl (RN) (RP): + ∀f,I1,I2,L1,L2. + L1.ⓘ[I1] ⪤[RN,RP,f] L2.ⓘ[I2] → + ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 → + L1.ⓘ[J1] ⪤[RN,RP,f] L2.ⓘ[J2]. /3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-. -lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 → - ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2. +lemma sex_co (RN1) (RP1) (RN2) (RP2): + RN1 ⊆ RN2 → RP1 ⊆ RP2 → + ∀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 sex_atom, sex_next, sex_push/ qed-. -lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 → - ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ → - L1 ⪤[RN2, RP2, f] L2. +lemma sex_co_isid (RN1) (RP1) (RN2) (RP2): + RP1 ⊆ RP2 → + ∀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 #I1 #I2 #K1 #K2 #_ #HI12 #IH #H [ elim (isid_inv_next … H) -H // @@ -210,9 +251,10 @@ lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 → ] qed-. -lemma sex_sdj: ∀RN,RP. RP ⊆ RN → - ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 → - ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2. +lemma sex_sdj (RN) (RP): + RP ⊆ RN → + ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 → + ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2. #RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 // #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 [ elim (sdj_inv_nx … H12) -H12 [2,3: // ] @@ -222,9 +264,10 @@ lemma sex_sdj: ∀RN,RP. RP ⊆ RN → ] qed-. -lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP → - ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 → - ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2. +lemma sle_sex_trans (RN) (RP): + RN ⊆ RP → + ∀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 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12 [ elim (pn_split f1) * ] @@ -235,9 +278,10 @@ lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP → ] qed-. -lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN → - ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 → - ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2. +lemma sle_sex_conf (RN) (RP): + RP ⊆ RN → + ∀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 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12 [2: elim (pn_split f2) * ] @@ -248,9 +292,10 @@ lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN → ] qed-. -lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → - ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ⊆ g → - ∃∃L. L1 ⪤[R1, RP, g] L & L ⪤[R2, cfull, f] L2. +lemma sex_sle_split_sn (R1) (R2) (RP): + c_reflexive … R1 → c_reflexive … R2 → + ∀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 sex_atom, ex2_intro/ ] #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H @@ -261,9 +306,10 @@ lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → ] qed-. -lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → - ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ∥ g → - ∃∃L. L1 ⪤[RP, R1, g] L & L ⪤[R2, cfull, f] L2. +lemma sex_sdj_split_sn (R1) (R2) (RP): + c_reflexive … R1 → c_reflexive … R2 → + ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g → + ∃∃L. L1 ⪤[RP,R1,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 sex_atom, ex2_intro/ ] #f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H @@ -274,17 +320,17 @@ lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 → ] qed-. -lemma sex_dec: ∀RN,RP. - (∀L,I1,I2. Decidable (RN L I1 I2)) → - (∀L,I1,I2. Decidable (RP L I1 I2)) → - ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2). +lemma sex_dec (RN) (RP): + (∀L,I1,I2. Decidable (RN L I1 I2)) → + (∀L,I1,I2. Decidable (RP L I1 I2)) → + ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2). #RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ] [ /2 width=1 by sex_atom, or_introl/ | #L2 #I2 #f @or_intror #H lapply (sex_inv_atom1 … H) -H #H destruct | #f @or_intror #H lapply (sex_inv_atom2 … H) -H #H destruct -| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12 +| #L2 #I2 #f elim (IH L2 (⫰f)) -IH #HL12 [2: /4 width=3 by sex_fwd_bind, or_intror/ ] elim (pn_split f) * #g #H destruct [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12