- some commutations between the rt-steps and the s-steps proved

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lifts.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
index ceaca58a944a6c4884fd476b000b8f2315bebcd2..afb3e127a6a9b65c02dcd0afe41e0835cb80f1f0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
@@ -15,7 +15,7 @@
 
 include "ground_2/relocation/nstream_after.ma".
 include "basic_2/notation/relations/rliftstar_3.ma".
-include "basic_2/grammar/term.ma".
+include "basic_2/syntax/term.ma".
 
 (* GENERIC RELOCATION FOR TERMS *********************************************)
 
@@ -41,6 +41,13 @@ interpretation "uniform relocation (term)"
 interpretation "generic relocation (term)"
    'RLiftStar f T1 T2 = (lifts f T1 T2).
 
+definition liftable2: predicate (relation term) ≝
+                      λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≡ U1 → 
+                      ∃∃U2. ⬆*[f] T2 ≡ U2 & R U1 U2.
+
+definition deliftable2_sn: predicate (relation term) ≝
+                           λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≡ U1 →
+                           ∃∃T2. ⬆*[f] T2 ≡ U2 & R T1 T2.
 
 (* Basic inversion lemmas ***************************************************)
 
@@ -208,6 +215,30 @@ lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≡ ⓕ{I}V2.T2 →
                                 X = ⓕ{I}V1.T1.
 /2 width=3 by lifts_inv_flat2_aux/ qed-.
 
+(* Advanced inversion lemmas ************************************************)
+
+lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≡ Y →
+                       ∨∨ ∃∃s. I = Sort s & Y = ⋆s
+                        | ∃∃i,j. @⦃i, f⦄ ≡ j & I = LRef i & Y = #j
+                        | ∃∃l. I = GRef l & Y = §l.
+#f * #n #Y #H
+[ lapply (lifts_inv_sort1 … H)
+| elim (lifts_inv_lref1 … H)
+| lapply (lifts_inv_gref1 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
+lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} →
+                       ∨∨ ∃∃s. X = ⋆s & I = Sort s
+                        | ∃∃i,j. @⦃i, f⦄ ≡ j & X = #i & I = LRef j
+                        | ∃∃l. X = §l & I = GRef l.
+#f * #n #X #H
+[ lapply (lifts_inv_sort2 … H)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
 #f #J #V elim V -V
@@ -241,6 +272,28 @@ lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≡ T → ⊥.
 ]
 qed-.
 
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≡ Y → Y = #(l+i).
+#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
+qed-.
+
+lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≡ #i2 →
+                           ∃∃i1. X = #i1 & i2 = l + i1.
+#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
+/3 width=3 by at_inv_uni, ex2_intro/
+qed-.
+
+lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≡ #(l + i) → X = #i.
+#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
+#i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
+qed-.
+
+lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≡ #i → i < l → ⊥.
+#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
+#i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
+qed-.
+
 (* Basic forward lemmas *****************************************************)
 
 (* Basic_2A1: includes: lift_inv_O2 *)
@@ -371,6 +424,11 @@ lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≡ T2).
 ]
 qed-.
 
+(* Properties with uniform relocation ***************************************)
+
+lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≡ U → ⬆*[n1+n2] T ≡ U.
+/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
+
 (* Basic_2A1: removed theorems 14:
               lifts_inv_nil lifts_inv_cons
               lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1