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 *********************************************)
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
]
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 *)
]
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