+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground/relocation/pstream_tls.ma".
-include "ground/relocation/pstream_istot.ma".
-include "ground/relocation/rtmap_after.ma".
-
-(* RELOCATION N-STREAM ******************************************************)
-
-corec definition compose: rtmap → rtmap → rtmap.
-#f2 * #p1 #f1 @(stream_cons … (f2@❨p1❩)) @(compose ? f1) -compose -f1
-@(⫰*[p1]f2)
-defined.
-
-interpretation "functional composition (nstream)"
- 'compose f2 f1 = (compose f2 f1).
-
-(* Basic properies on compose ***********************************************)
-
-lemma compose_rew: ∀f2,f1,p1. f2@❨p1❩⨮(⫰*[p1]f2)∘f1 = f2∘(p1⨮f1).
-#f2 #f1 #p1 <(stream_rew … (f2∘(p1⨮f1))) normalize //
-qed.
-
-lemma compose_next: ∀f2,f1,f. f2∘f1 = f → (↑f2)∘f1 = ↑f.
-#f2 * #p1 #f1 #f <compose_rew <compose_rew
-* -f /2 width=1 by eq_f2/
-qed.
-
-(* Basic inversion lemmas on compose ****************************************)
-
-lemma compose_inv_rew: ∀f2,f1,f,p1,p. f2∘(p1⨮f1) = p⨮f →
- f2@❨p1❩ = p ∧ (⫰*[p1]f2)∘f1 = f.
-#f2 #f1 #f #p1 #p <compose_rew
-#H destruct /2 width=1 by conj/
-qed-.
-
-lemma compose_inv_O2: ∀f2,f1,f,p2,p. (p2⨮f2)∘(⫯f1) = p⨮f →
- p2 = p ∧ f2∘f1 = f.
-#f2 #f1 #f #p2 #p <compose_rew
-#H destruct /2 width=1 by conj/
-qed-.
-
-lemma compose_inv_S2: ∀f2,f1,f,p2,p1,p. (p2⨮f2)∘(↑p1⨮f1) = p⨮f →
- f2@❨p1❩+p2 = p ∧ f2∘(p1⨮f1) = f2@❨p1❩⨮f.
-#f2 #f1 #f #p2 #p1 #p <compose_rew
-#H destruct >nsucc_inj <stream_tls_swap
-/2 width=1 by conj/
-qed-.
-
-lemma compose_inv_S1: ∀f2,f1,f,p1,p. (↑f2)∘(p1⨮f1) = p⨮f →
- ↑(f2@❨p1❩) = p ∧ f2∘(p1⨮f1) = f2@❨p1❩⨮f.
-#f2 #f1 #f #p1 #p <compose_rew
-#H destruct /2 width=1 by conj/
-qed-.
-
-(* Specific properties on after *********************************************)
-
-lemma after_O2: ∀f2,f1,f. f2 ⊚ f1 ≘ f →
- ∀p. p⨮f2 ⊚ ⫯f1 ≘ p⨮f.
-#f2 #f1 #f #Hf #p elim p -p
-/2 width=7 by after_refl, after_next/
-qed.
-
-lemma after_S2: ∀f2,f1,f,p1,p. f2 ⊚ p1⨮f1 ≘ p⨮f →
- ∀p2. p2⨮f2 ⊚ ↑p1⨮f1 ≘ (p+p2)⨮f.
-#f2 #f1 #f #p1 #p #Hf #p2 elim p2 -p2
-/2 width=7 by after_next, after_push/
-qed.
-
-lemma after_apply: ∀p1,f2,f1,f.
- (⫰*[ninj p1] f2) ⊚ f1 ≘ f → f2 ⊚ p1⨮f1 ≘ f2@❨p1❩⨮f.
-#p1 elim p1 -p1
-[ * /2 width=1 by after_O2/
-| #p1 #IH * #p2 #f2 >nsucc_inj <stream_tls_swap
- /3 width=1 by after_S2/
-]
-qed-.
-
-corec lemma after_total_aux: ∀f2,f1,f. f2 ∘ f1 = f → f2 ⊚ f1 ≘ f.
-* #p2 #f2 * #p1 #f1 * #p #f cases p2 -p2
-[ cases p1 -p1
- [ #H cases (compose_inv_O2 … H) -H /3 width=7 by after_refl, eq_f2/
- | #p1 #H cases (compose_inv_S2 … H) -H * -p /3 width=7 by after_push/
- ]
-| #p2 >next_rew #H cases (compose_inv_S1 … H) -H * -p >next_rew
- /3 width=5 by after_next/
-]
-qed-.
-
-theorem after_total: ∀f1,f2. f2 ⊚ f1 ≘ f2 ∘ f1.
-/2 width=1 by after_total_aux/ qed.
-
-(* Specific inversion lemmas on after ***************************************)
-
-lemma after_inv_xpx: ∀f2,g2,f,p2,p. p2⨮f2 ⊚ g2 ≘ p⨮f → ∀f1. ⫯f1 = g2 →
- f2 ⊚ f1 ≘ f ∧ p2 = p.
-#f2 #g2 #f #p2 elim p2 -p2
-[ #p #Hf #f1 #H2 elim (after_inv_ppx … Hf … H2) -g2 [|*: // ]
- #g #Hf #H elim (push_inv_seq_dx … H) -H destruct /2 width=1 by conj/
-| #p2 #IH #p #Hf #f1 #H2 elim (after_inv_nxx … Hf) -Hf [|*: // ]
- #g1 #Hg #H1 elim (next_inv_seq_dx … H1) -H1
- #x #Hx #H destruct elim (IH … Hg) [|*: // ] -IH -Hg
- #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma after_inv_xnx: ∀f2,g2,f,p2,p. p2⨮f2 ⊚ g2 ≘ p⨮f → ∀f1. ↑f1 = g2 →
- ∃∃q. f2 ⊚ f1 ≘ q⨮f & q+p2 = p.
-#f2 #g2 #f #p2 elim p2 -p2
-[ #p #Hf #f1 #H2 elim (after_inv_pnx … Hf … H2) -g2 [|*: // ]
- #g #Hf #H elim (next_inv_seq_dx … H) -H
- #x #Hx #Hg destruct /2 width=3 by ex2_intro/
-| #p2 #IH #p #Hf #f1 #H2 elim (after_inv_nxx … Hf) -Hf [|*: // ]
- #g #Hg #H elim (next_inv_seq_dx … H) -H
- #x #Hx #H destruct elim (IH … Hg) -IH -Hg [|*: // ]
- #m #Hf #Hm destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma after_inv_const: ∀f2,f1,f,p1,p.
- p⨮f2 ⊚ p1⨮f1 ≘ p⨮f → f2 ⊚ f1 ≘ f ∧ 𝟏 = p1.
-#f2 #f1 #f #p1 #p elim p -p
-[ #H elim (after_inv_pxp … H) -H [|*: // ]
- #g2 #Hf #H elim (push_inv_seq_dx … H) -H /2 width=1 by conj/
-| #p #IH #H lapply (after_inv_nxn … H ????) -H /2 width=5 by/
-]
-qed-.
-
-lemma after_inv_total: ∀f2,f1,f. f2 ⊚ f1 ≘ f → f2 ∘ f1 ≡ f.
-/2 width=4 by after_mono/ qed-.
-
-(* Specific forward lemmas on after *****************************************)
-
-lemma after_fwd_hd: ∀f2,f1,f,p1,p. f2 ⊚ p1⨮f1 ≘ p⨮f → f2@❨p1❩ = p.
-#f2 #f1 #f #p1 #p #H lapply (after_fwd_at ? p1 (𝟏) … H) -H [4:|*: // ]
-/3 width=2 by at_inv_O1, sym_eq/
-qed-.
-
-lemma after_fwd_tls: ∀f,f1,p1,f2,p2,p. p2⨮f2 ⊚ p1⨮f1 ≘ p⨮f →
- (⫰*[↓p1]f2) ⊚ f1 ≘ f.
-#f #f1 #p1 elim p1 -p1
-[ #f2 #p2 #p #H elim (after_inv_xpx … H) -H //
-| #p1 #IH * #q2 #f2 #p2 #p #H elim (after_inv_xnx … H) -H [|*: // ]
- #x #Hx #H destruct /2 width=3 by/
-]
-qed-.
-
-lemma after_inv_apply: ∀f2,f1,f,p2,p1,p. p2⨮f2 ⊚ p1⨮f1 ≘ p⨮f →
- (p2⨮f2)@❨p1❩ = p ∧ (⫰*[↓p1]f2) ⊚ f1 ≘ f.
-/3 width=3 by after_fwd_tls, after_fwd_hd, conj/ qed-.
-
-(* Properties on apply ******************************************************)
-
-lemma compose_apply (f2) (f1) (i): f2@❨f1@❨i❩❩ = (f2∘f1)@❨i❩.
-/4 width=6 by after_fwd_at, at_inv_total, sym_eq/ qed.
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