partial commit in static_2

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

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma
index cc71886a23b085d078cd8cd52fad721eeecc9033..19078cfbe6a5ad84435abbd6ff9e3da79768d65b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/sex_sex.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/relocation/rtmap_sand.ma".
+include "ground/relocation/rtmap_sand.ma".
 include "static_2/relocation/drops.ma".
 
 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
@@ -20,18 +20,18 @@ include "static_2/relocation/drops.ma".
 (* Main properties **********************************************************)
 
 theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
-                      ∀L1,f.
-                      (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
-                      (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) →
-                      ∀L0. L1 ⪤[RN1,RP1,f] L0 →
-                      ∀L2. L0 ⪤[RN2,RP2,f] L2 →
-                      L1 ⪤[RN,RP,f] L2.
+        ∀L1,f.
+        (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_transitive_sex RN1 RN2 RN RN1 RP1 g K I) →
+        (∀g,I,K,n. ⇩[n] L1 ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_transitive_sex RP1 RP2 RP RN1 RP1 g K I) →
+        ∀L0. L1 ⪤[RN1,RP1,f] L0 →
+        ∀L2. L0 ⪤[RN2,RP2,f] L2 →
+        L1 ⪤[RN,RP,f] L2.
 #RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
 [ #f #_ #_ #L0 #H1 #L2 #H2
   lapply (sex_inv_atom1 … H1) -H1 #H destruct
   lapply (sex_inv_atom1 … H2) -H2 #H destruct
   /2 width=1 by sex_atom/
-| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
+| #K1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
   #HN #HP #L0 #H1 #L2 #H2
   [ elim (sex_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
     elim (sex_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
@@ -45,30 +45,32 @@ theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
 ]
 qed-.
 
-theorem sex_trans (RN) (RP) (f): (∀g,I,K. sex_transitive RN RN RN RN RP g K I) →
-                                 (∀g,I,K. sex_transitive RP RP RP RN RP g K I) →
-                                 Transitive … (sex RN RP f).
+theorem sex_trans (RN) (RP) (f):
+        (∀g,I,K. R_pw_transitive_sex RN RN RN RN RP g K I) →
+        (∀g,I,K. R_pw_transitive_sex RP RP RP RN RP g K I) →
+        Transitive … (sex RN RP f).
 /2 width=9 by sex_trans_gen/ qed-.
 
-theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈❪f❫ →
-                            ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2.
+theorem sex_trans_id_cfull (R1) (R2) (R3):
+        ∀L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈❪f❫ →
+        ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2.
 #R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
 [ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ]
 #f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
-[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
+[ elim (pr_isi_inv_next … Hf) | lapply (pr_isi_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
 elim (sex_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
 /3 width=1 by sex_push/
 qed-.
 
 theorem sex_conf (RN1) (RP1) (RN2) (RP2):
-                 ∀L,f.
-                 (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ↑g = ⫱*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
-                 (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ⫯g = ⫱*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
-                 pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L.
+        ∀L,f.
+        (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+        (∀g,I,K,n. ⇩[n] L ≘ K.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
+        pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L.
 #RN1 #RP1 #RN2 #RP2 #L elim L -L
 [ #f #_ #_ #L1 #H1 #L2 #H2 >(sex_inv_atom1 … H1) >(sex_inv_atom1 … H2) -H2 -H1
   /2 width=3 by sex_atom, ex2_intro/
-| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
+| #L #I0 #IH #f elim (pr_map_split_tl f) * #g #H destruct
   #HN #HP #Y1 #H1 #Y2 #H2
   [ elim (sex_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
     elim (sex_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
@@ -82,38 +84,62 @@ theorem sex_conf (RN1) (RP1) (RN2) (RP2):
 ]
 qed-.
 
-theorem sex_canc_sn: ∀RN,RP,f. Transitive … (sex RN RP f) →
-                               symmetric … (sex RN RP f) →
-                               left_cancellable … (sex RN RP f).
+lemma sex_repl (RN) (RP) (SN) (SP) (L1) (f):
+      (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ↑g = ⫰*[n] f → R_pw_replace3_sex … RN SN RN RP SN SP g K1 I) →
+      (∀g,I,K1,n. ⇩[n] L1 ≘ K1.ⓘ[I] → ⫯g = ⫰*[n] f → R_pw_replace3_sex … RP SP RN RP SN SP g K1 I) →
+      ∀L2. L1 ⪤[RN,RP,f] L2 → ∀K1. L1 ⪤[SN,SP,f] K1 →
+      ∀K2. L2 ⪤[SN,SP,f] K2 → K1 ⪤[RN,RP,f] K2.
+#RN #RP #SN #SP #L1 elim L1 -L1
+[ #f #_ #_ #Y #HY #Y1 #HY1 #Y2 #HY2
+  lapply (sex_inv_atom1 … HY) -HY #H destruct
+  lapply (sex_inv_atom1 … HY1) -HY1 #H destruct
+  lapply (sex_inv_atom1 … HY2) -HY2 #H destruct //
+| #L1 #I1 #IH #f elim (pr_map_split_tl f) * #g #H destruct
+  #HN #HP #Y #HY #Y1 #HY1 #Y2 #HY2
+  [ elim (sex_inv_push1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct
+    elim (sex_inv_push1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct
+    elim (sex_inv_push1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct
+    /5 width=13 by sex_push, drops_refl, drops_drop/
+  | elim (sex_inv_next1 … HY) -HY #I2 #L2 #HL12 #HI12 #H destruct
+    elim (sex_inv_next1 … HY1) -HY1 #J1 #K1 #HLK1 #HIJ1 #H destruct
+    elim (sex_inv_next1 … HY2) -HY2 #J2 #K2 #HLK2 #HIJ2 #H destruct
+    /5 width=13 by sex_next, drops_refl, drops_drop/
+  ]
+]
+qed-.
+
+theorem sex_canc_sn (RN) (RP):
+        ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) →
+        left_cancellable … (sex RN RP f).
 /3 width=3 by/ qed-.
 
-theorem sex_canc_dx: ∀RN,RP,f. Transitive … (sex RN RP f) →
-                               symmetric … (sex RN RP f) →
-                               right_cancellable … (sex RN RP f).
+theorem sex_canc_dx (RN) (RP):
+        ∀f. Transitive … (sex RN RP f) → symmetric … (sex RN RP f) →
+        right_cancellable … (sex RN RP f).
 /3 width=3 by/ qed-.
 
-lemma sex_meet: ∀RN,RP,L1,L2.
-                ∀f1. L1 ⪤[RN,RP,f1] L2 →
-                ∀f2. L1 ⪤[RN,RP,f2] L2 →
-                ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
+lemma sex_meet (RN) (RP) (L1) (L2):
+      ∀f1. L1 ⪤[RN,RP,f1] L2 →
+      ∀f2. L1 ⪤[RN,RP,f2] L2 →
+      ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
-elim (pn_split f2) * #g2 #H2 destruct
+elim (pr_map_split_tl f2) * #g2 #H2 destruct
 try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
-[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
-| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
+[ elim (pr_sand_inv_next_push … Hf) | elim (pr_sand_inv_next_bi … Hf)
+| elim (pr_sand_inv_push_bi … Hf) | elim (pr_sand_inv_push_next … Hf)
 ] -Hf /3 width=5 by sex_next, sex_push/
 qed-.
 
-lemma sex_join: ∀RN,RP,L1,L2.
-                ∀f1. L1 ⪤[RN,RP,f1] L2 →
-                ∀f2. L1 ⪤[RN,RP,f2] L2 →
-                ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
+lemma sex_join (RN) (RP) (L1) (L2):
+      ∀f1. L1 ⪤[RN,RP,f1] L2 →
+      ∀f2. L1 ⪤[RN,RP,f2] L2 →
+      ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
 #RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
 #f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
-elim (pn_split f2) * #g2 #H2 destruct
+elim (pr_map_split_tl f2) * #g2 #H2 destruct
 try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
-[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
-| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+[ elim (pr_sor_inv_next_push … Hf) | elim (pr_sor_inv_next_bi … Hf)
+| elim (pr_sor_inv_push_bi … Hf) | elim (pr_sor_inv_push_next … Hf)
 ] -Hf /3 width=5 by sex_next, sex_push/
 qed-.