1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground_2/notation/functions/upspoon_1.ma".
16 include "ground_2/lib/streams_tls.ma".
18 (* RELOCATION N-STREAM ******************************************************)
20 definition rtmap: Type[0] ≝ stream nat.
22 definition push: rtmap → rtmap ≝ λf. 0⨮f.
24 interpretation "push (nstream)" 'UpSpoon f = (push f).
26 definition next: rtmap → rtmap.
30 interpretation "next (nstream)" 'UpArrow f = (next f).
32 (* Basic properties *********************************************************)
34 lemma push_rew: ∀f. 0⨮f = ⫯f.
37 lemma next_rew: ∀f,n. (↑n)⨮f = ↑(n⨮f).
40 (* Basic inversion lemmas ***************************************************)
42 lemma injective_push: injective ? ? push.
43 #f1 #f2 normalize #H destruct //
46 lemma discr_push_next: ∀f1,f2. ⫯f1 = ↑f2 → ⊥.
47 #f1 * #n2 #f2 normalize #H destruct
50 lemma discr_next_push: ∀f1,f2. ↑f1 = ⫯f2 → ⊥.
51 * #n1 #f1 #f2 normalize #H destruct
54 lemma injective_next: injective ? ? next.
55 * #n1 #f1 * #n2 #f2 normalize #H destruct //
58 lemma push_inv_seq_sn: ∀f,g,n. n⨮g = ⫯f → 0 = n ∧ g = f.
59 #f #g #n <push_rew #H destruct /2 width=1 by conj/
62 lemma push_inv_seq_dx: ∀f,g,n. ⫯f = n⨮g → 0 = n ∧ g = f.
63 #f #g #n <push_rew #H destruct /2 width=1 by conj/
66 lemma next_inv_seq_sn: ∀f,g,n. n⨮g = ↑f → ∃∃m. m⨮g = f & ↑m = n.
67 * #m #f #g #n <next_rew #H destruct /2 width=3 by ex2_intro/
70 lemma next_inv_seq_dx: ∀f,g,n. ↑f = n⨮g → ∃∃m. m⨮g = f & ↑m = n.
71 * #m #f #g #n <next_rew #H destruct /2 width=3 by ex2_intro/
74 lemma case_prop: ∀R:predicate rtmap.
75 (∀f. R (⫯f)) → (∀f. R (↑f)) → ∀f. R f.
79 lemma case_type0: ∀R:rtmap→Type[0].
80 (∀f. R (⫯f)) → (∀f. R (↑f)) → ∀f. R f.
84 lemma iota_push: ∀R,a,b,f. a f = case_type0 R a b (⫯f).
87 lemma iota_next: ∀R,a,b,f. b f = case_type0 R a b (↑f).
91 (* Specific properties ******************************************************)
93 lemma tl_push: ∀f. f = ⫰⫯f.
96 lemma tl_next: ∀f. ⫰f = ⫰↑f.