(* *)
(**************************************************************************)
-include "ground/notation/functions/apply_2.ma".
+include "ground/notation/functions/atsharp_2.ma".
include "ground/arith/pnat_plus.ma".
include "ground/relocation/tr_map.ma".
interpretation
"functional positive application (total relocation maps)"
- 'Apply f i = (tr_pap i f).
+ 'AtSharp f i = (tr_pap i f).
(* Basic constructions ******************************************************)
(*** apply_O1 *)
-lemma tr_pap_unit (f):
- ∀p. p = (p⨮f)@❨𝟏❩.
+lemma tr_cons_pap_unit (f):
+ ∀p. p = (p⨮f)@⧣❨𝟏❩.
// qed.
(*** apply_S1 *)
-lemma tr_pap_succ (f):
- ∀p,i. f@❨i❩+p = (p⨮f)@❨↑i❩.
+lemma tr_cons_pap_succ (f):
+ ∀p,i. f@⧣❨i❩+p = (p⨮f)@⧣❨↑i❩.
// qed.
-(*
-(*** apply_S2 *)
-lemma tr_pap_next (f):
- ∀i. ↑(f@❨i❩) = (↑f)@❨i❩.
-* #p #f * //
-qed.
-
-
-
-(*** apply_eq_repl *)
-lemma apply_eq_repl (i):
- ∀f1,f2. f1 ≗ f2 → f1@❨i❩ = f2@❨i❩.
-
-
-(i): pr_eq_repl … (λf1,f2. f1@❨i❩ = f2@❨i❩).
-#i elim i -i [2: #i #IH ] * #p1 #f1 * #p2 #f2 #H
-elim (eq_inv_seq_aux … H) -H #Hp #Hf //
->apply_S1 >apply_S1 /3 width=1 by eq_f2/
-qed.
-
-
-(* Main inversion lemmas ****************************************************)
-
-theorem apply_inj: ∀f,i1,i2,j. f@❨i1❩ = j → f@❨i2❩ = j → i1 = i2.
-/2 width=4 by gr_pat_inj/ qed-.
-
-corec theorem nstream_eq_inv_ext: ∀f1,f2. (∀i. f1@❨i❩ = f2@❨i❩) → f1 ≗ f2.
-* #p1 #f1 * #p2 #f2 #Hf @stream_eq_cons
-[ @(Hf (𝟏))
-| @nstream_eq_inv_ext -nstream_eq_inv_ext #i
- lapply (Hf (𝟏)) >apply_O1 >apply_O1 #H destruct
- lapply (Hf (↑i)) >apply_S1 >apply_S1 #H
- /3 width=2 by eq_inv_pplus_bi_dx, eq_inv_psucc_bi/
-]
-qed-.
-
-(*
-include "ground/relocation/pstream_eq.ma".
-*)
-
-(*
-include "ground/relocation/rtmap_istot.ma".
-
-lemma at_istot: ∀f. 𝐓❨f❩.
-/2 width=2 by ex_intro/ qed.
-*)
-*)
\ No newline at end of file