- ground_2: support for lifts_div4

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_after.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
index 80401a3d14ab99b9e016c4cbcfb5225212db6e71..a00a10d1696d858896f005f9800cbc240d97b9f1 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
@@ -14,7 +14,6 @@
 
 include "ground_2/notation/relations/rafter_3.ma".
 include "ground_2/relocation/rtmap_istot.ma".
-include "ground_2/relocation/rtmap_uni.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
@@ -321,6 +320,24 @@ lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ⫯f2
 /5 width=7 by isid_after_dx, after_eq_repl_back_2, after_next, after_push, eq_push_inv_isid/
 qed.
 
+lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ⊚ f1 ≡ f → f2 ⊚ ⫯f1 ≡ f.
+#f2 #H elim H -f2
+[ #f2 #Hf2 #f1 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+  /4 width=7 by after_isid_inv_sn, isid_after_sn, after_eq_repl_back_0, eq_next/
+| #f2 #_ #g2 #H2 #IH #f1 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+  /3 width=5 by after_next/
+]
+qed.
+
+(* Properties on uni ********************************************************)
+
+lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+@nat_elim2
+/4 width=5 by after_uni_next2, isid_after_sn, isid_after_dx, after_next/
+qed.
+
 (* Forward lemmas on at *****************************************************)
 
 lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
@@ -384,8 +401,8 @@ qed-.
 
 (* Properties with at *******************************************************)
 
-lemma after_isuni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                      ∀f. f2 ⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f.
+lemma after_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+                    ∀f. f2 ⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -403,8 +420,8 @@ lemma after_isuni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed.
 
-lemma after_isuni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                      ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f → f2 ⊚ 𝐔❴i1❵ ≡ f.
+lemma after_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+                    ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f → f2 ⊚ 𝐔❴i1❵ ≡ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -419,6 +436,50 @@ lemma after_isuni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed-.
 
+lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+                         ∀f. f2 ⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+  elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+  elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
+  lapply (after_isid_inv_dx … Hg ?) -Hg
+  /4 width=5 by isid_after_sn, after_eq_repl_back_0, after_next/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #j1 #Hg2 #H1 #H2 destruct
+    elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+    /3 width=5 by after_next/
+  | #g2 #Hg2 #H2 destruct
+    elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+    /3 width=5 by after_next/
+  ]
+]
+qed.
+
+lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+                         ∀f. 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f → f2 ⊚ 𝐔❴⫯i1❵ ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+  elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+  elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+  lapply (after_isid_inv_sn … Hg ?) -Hg
+  /4 width=7 by isid_after_dx, after_eq_repl_back_0, after_push/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
+  | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
+  ]
+]
+qed-.
+
+lemma after_uni_one_dx: ∀f2,f. ↑f2 ⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ⊚ f2 ≡ f.
+#f2 #f #H @(after_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
+qed.
+
+lemma after_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ⊚ f1 ≡ f → ↑f1 ⊚ 𝐔❴⫯O❵ ≡ f.
+/3 width=3 by after_uni_succ_sn, at_refl/ qed-.
+
 (* Forward lemmas on istot **************************************************)
 
 lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.