update in staic_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / relocation / drops_sex.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma
index edec3dcc5f88841327da1a83856e5675629cfa05..d9a9588a346a996138c2c8edcda6c6a8f9123284 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops_sex.ma
@@ -20,7 +20,8 @@ include "static_2/relocation/drops.ma".
 (* Properties with entrywise extension of context-sensitive relations *******)
 
 (**) (* changed after commit 13218 *)
-lemma sex_co_dropable_sn: ∀RN,RP. co_dropable_sn (sex RN RP).
+lemma sex_co_dropable_sn (RN) (RP):
+      co_dropable_sn (sex RN RP).
 #RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(sex_inv_atom1 … H) -X
   /4 width=3 by sex_atom, drops_atom, ex2_intro/
@@ -41,10 +42,11 @@ lemma sex_co_dropable_sn: ∀RN,RP. co_dropable_sn (sex RN RP).
 ]
 qed-.
 
-lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
-                                     ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
-                                     ∀b,f. ⇩*[b,f] L1 ≘ K1 → ⇩*[b,f] L2 ≘ K2 →
-                                     f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2.
+lemma sex_liftable_co_dedropable_bi (RN) (RP):
+      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+      ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+      ∀b,f. ⇩*[b,f] L1 ≘ K1 → ⇩*[b,f] L2 ≘ K2 →
+      f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2.
 #RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
 #g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
 [ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
@@ -68,9 +70,10 @@ lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN →
 ]
 qed-.
 
-lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
-                                     d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
-                                     co_dedropable_sn (sex RN RP).
+lemma sex_liftable_co_dedropable_sn (RN) (RP):
+      (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+      co_dedropable_sn (sex RN RP).
 #RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
 [ #f #Hf #X #f1 #H #f2 #Hf2 >(sex_inv_atom1 … H) -X
   /4 width=4 by drops_atom, sex_atom, ex3_intro/
@@ -87,9 +90,10 @@ lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) →
 ]
 qed-.
 
-fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f❫ →
-                          ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
-                          ∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
+fact sex_dropable_dx_aux (RN) (RP):
+     ∀b,f,L2,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f❫ →
+     ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+     ∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
 #RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
 [ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H
   #H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/
@@ -109,79 +113,112 @@ fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f
 ]
 qed-.
 
-lemma sex_co_dropable_dx: ∀RN,RP. co_dropable_dx (sex RN RP).
+lemma sex_co_dropable_dx (RN) (RP):
+      co_dropable_dx (sex RN RP).
 /2 width=5 by sex_dropable_dx_aux/ qed-.
 
-lemma sex_drops_conf_next: ∀RN,RP.
-                           ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
-                           ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
-                           ∀f1. f ~⊚ ↑f1 ≘ f2 →
-                           ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
+lemma sex_drops_conf_next (RN) (RP):
+      ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+      ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
+      ∀f1. f ~⊚ ↑f1 ≘ f2 →
+      ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
 #X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
 #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma sex_drops_conf_push: ∀RN,RP.
-                           ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
-                           ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
-                           ∀f1. f ~⊚ ⫯f1 ≘ f2 →
-                           ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
+lemma sex_drops_conf_push (RN) (RP):
+      ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+      ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
+      ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+      ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
 elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
 #X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
 #I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
-                            ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
-                            ∀f1. f ~⊚ ↑f1 ≘ f2 →
-                            ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
+lemma sex_drops_trans_next (RN) (RP):
+      ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+      ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
+      ∀f1. f ~⊚ ↑f1 ≘ f2 →
+      ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
 #I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
-                            ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
-                            ∀f1. f ~⊚ ⫯f1 ≘ f2 →
-                            ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
+lemma sex_drops_trans_push (RN) (RP): ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+      ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
+      ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+      ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
 #RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
 elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
 #X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
 #I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
-                            d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
-                            ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
-                            ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
-                            ∀f2. f ~⊚ f1 ≘ ↑f2 →
-                            ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
+lemma drops_sex_trans_next (RN) (RP):
+      (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+      ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+      ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
+      ∀f2. f ~⊚ f1 ≘ ↑f2 →
+      ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
 #X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
 #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
 qed-.
 
-lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
-                            d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
-                            ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
-                            ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
-                            ∀f2. f ~⊚ f1 ≘ ⫯f2 →
-                            ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
+lemma drops_sex_trans_push (RN) (RP):
+      (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+      d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+      ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+      ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
+      ∀f2. f ~⊚ f1 ≘ ⫯f2 →
+      ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
 #RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
 elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
 #X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
 #I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
 qed-.
 
-lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⇩*[b,f1] L1 ≘ ⋆ → 𝐔❪f1❫ →
-                            ∀f,L2. L1 ⪤[RN,RP,f] L2 →
-                            ∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆.
+lemma drops_atom2_sex_conf (RN) (RP):
+      ∀b,f1,L1. ⇩*[b,f1] L1 ≘ ⋆ → 𝐔❪f1❫ →
+      ∀f,L2. L1 ⪤[RN,RP,f] L2 →
+      ∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆.
 #RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
 elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
 #L #H #HL2 lapply (sex_inv_atom1 … H) -H //
 qed-.
+
+lemma sex_sdj_split_dx (R1) (R2) (RP):
+      c_reflexive … R1 → c_reflexive … R2 → c_reflexive … RP →
+      ∀L1,f1.
+      (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫱*[n] f1 → R_pw_confluent1_sex R1 R1 R2 cfull g K I) →
+      ∀L2. L1 ⪤[R1,RP,f1] L2 → ∀f2. f1 ∥ f2 →
+      ∃∃L. L1 ⪤[R2,cfull,f1] L & L ⪤[RP,R1,f2] L2.
+#R1 #R2 #RP #HR1 #HR2 #HRP #L1 elim L1 -L1
+[ #f1 #_ #L2 #H #f2 #_
+  lapply (sex_inv_atom1 … H) -H #H destruct
+  /2 width=3 by sex_atom, ex2_intro/
+| #K1 #I1 #IH #f1 elim (pn_split f1) * #g1 #H destruct
+  #HR #L2 #H #f2 #Hf
+  [ elim (sex_inv_push1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct
+    elim (sdj_inv_px … Hf ??) -Hf [1,3: * |*: // ] #g2 #Hg #H destruct
+    elim (IH … HK12 … Hg) -IH -HK12 -Hg
+    [1,3: /3 width=5 by sex_next, sex_push, ex2_intro/
+    |2,4: /3 width=3 by drops_drop/
+    ]
+  | elim (sex_inv_next1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct
+    elim (sdj_inv_nx … Hf ??) -Hf [|*: // ] #g2 #Hg #H destruct
+    elim (IH … HK12 … Hg) -IH -HK12 -Hg
+    [ /5 width=11 by sex_next, sex_push, drops_refl, ex2_intro/
+    | /3 width=3 by drops_drop/
+    ]
+  ]
+]
+qed-.