(* RELOCATION N-STREAM ******************************************************)
corec definition compose: rtmap → rtmap → rtmap.
-#f2 * #n1 #f1 @(seq â\80¦ (f2@â\9d´n1â\9dµ)) @(compose ? f1) -compose -f1
-@(â\86\93*[⫯n1] f2)
+#f2 * #n1 #f1 @(seq â\80¦ (f2@â\9d¨n1â\9d©)) @(compose ? f1) -compose -f1
+@(â«°*[â\86\91n1] f2)
defined.
interpretation "functional composition (nstream)"
(* Basic properies on compose ***********************************************)
-lemma compose_rew: â\88\80f2,f1,n1. f2@â\9d´n1â\9dµ@(â\86\93*[⫯n1]f2)â\88\98f1 = f2â\88\98(n1@f1).
-#f2 #f1 #n1 <(stream_rew … (f2∘(n1@f1))) normalize //
+lemma compose_rew: â\88\80f2,f1,n1. f2@â\9d¨n1â\9d©â¨®(â«°*[â\86\91n1]f2)â\88\98f1 = f2â\88\98(n1⨮f1).
+#f2 #f1 #n1 <(stream_rew … (f2∘(n1⨮f1))) normalize //
qed.
-lemma compose_next: â\88\80f2,f1,f. f2â\88\98f1 = f â\86\92 (⫯f2)â\88\98f1 = ⫯f.
+lemma compose_next: â\88\80f2,f1,f. f2â\88\98f1 = f â\86\92 (â\86\91f2)â\88\98f1 = â\86\91f.
#f2 * #n1 #f1 #f <compose_rew <compose_rew
* -f <tls_S1 /2 width=1 by eq_f2/
qed.
(* Basic inversion lemmas on compose ****************************************)
-lemma compose_inv_rew: ∀f2,f1,f,n1,n. f2∘(n1@f1) = n@f →
- f2@â\9d´n1â\9dµ = n â\88§ (â\86\93*[⫯n1]f2)∘f1 = f.
-#f2 #f1 #f #n1 #n <(stream_rew … (f2∘(n1@f1))) normalize
+lemma compose_inv_rew: ∀f2,f1,f,n1,n. f2∘(n1⨮f1) = n⨮f →
+ f2@â\9d¨n1â\9d© = n â\88§ (â«°*[â\86\91n1]f2)∘f1 = f.
+#f2 #f1 #f #n1 #n <(stream_rew … (f2∘(n1⨮f1))) normalize
#H destruct /2 width=1 by conj/
qed-.
-lemma compose_inv_O2: ∀f2,f1,f,n2,n. (n2@f2)∘(↑f1) = n@f →
+lemma compose_inv_O2: ∀f2,f1,f,n2,n. (n2⨮f2)∘(⫯f1) = n⨮f →
n2 = n ∧ f2∘f1 = f.
#f2 #f1 #f #n2 #n <compose_rew
#H destruct /2 width=1 by conj/
qed-.
-lemma compose_inv_S2: ∀f2,f1,f,n2,n1,n. (n2@f2)∘(⫯n1@f1) = n@f →
- ⫯(n2+f2@â\9d´n1â\9dµ) = n â\88§ f2â\88\98(n1@f1) = f2@â\9d´n1â\9dµ@f.
+lemma compose_inv_S2: ∀f2,f1,f,n2,n1,n. (n2⨮f2)∘(↑n1⨮f1) = n⨮f →
+ â\86\91(n2+f2@â\9d¨n1â\9d©) = n â\88§ f2â\88\98(n1⨮f1) = f2@â\9d¨n1â\9d©â¨®f.
#f2 #f1 #f #n2 #n1 #n <compose_rew
#H destruct <tls_S1 /2 width=1 by conj/
qed-.
-lemma compose_inv_S1: â\88\80f2,f1,f,n1,n. (⫯f2)â\88\98(n1@f1) = n@f →
- ⫯(f2@â\9d´n1â\9dµ) = n â\88§ f2â\88\98(n1@f1) = f2@â\9d´n1â\9dµ@f.
+lemma compose_inv_S1: â\88\80f2,f1,f,n1,n. (â\86\91f2)â\88\98(n1⨮f1) = n⨮f →
+ â\86\91(f2@â\9d¨n1â\9d©) = n â\88§ f2â\88\98(n1⨮f1) = f2@â\9d¨n1â\9d©â¨®f.
#f2 #f1 #f #n1 #n <compose_rew
#H destruct <tls_S1 /2 width=1 by conj/
qed-.
(* Specific properties on after *********************************************)
-lemma after_O2: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f →
- ∀n. n@f2 ⊚ ↑f1 ≡ n@f.
+lemma after_O2: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f →
+ ∀n. n⨮f2 ⊚ ⫯f1 ≘ n⨮f.
#f2 #f1 #f #Hf #n elim n -n /2 width=7 by after_refl, after_next/
qed.
-lemma after_S2: ∀f2,f1,f,n1,n. f2 ⊚ n1@f1 ≡ n@f →
- ∀n2. n2@f2 ⊚ ⫯n1@f1 ≡ ⫯(n2+n)@f.
+lemma after_S2: ∀f2,f1,f,n1,n. f2 ⊚ n1⨮f1 ≘ n⨮f →
+ ∀n2. n2⨮f2 ⊚ ↑n1⨮f1 ≘ ↑(n2+n)⨮f.
#f2 #f1 #f #n1 #n #Hf #n2 elim n2 -n2 /2 width=7 by after_next, after_push/
qed.
-lemma after_apply: â\88\80n1,f2,f1,f. (â\86\93*[⫯n1] f2) â\8a\9a f1 â\89¡ f â\86\92 f2 â\8a\9a n1@f1 â\89¡ f2@â\9d´n1â\9dµ@f.
+lemma after_apply: â\88\80n1,f2,f1,f. (â«°*[â\86\91n1] f2) â\8a\9a f1 â\89\98 f â\86\92 f2 â\8a\9a n1⨮f1 â\89\98 f2@â\9d¨n1â\9d©â¨®f.
#n1 elim n1 -n1
[ * /2 width=1 by after_O2/
| #n1 #IH * /3 width=1 by after_S2/
]
qed-.
-corec lemma after_total_aux: â\88\80f2,f1,f. f2 â\88\98 f1 = f â\86\92 f2 â\8a\9a f1 â\89¡ f.
+corec lemma after_total_aux: â\88\80f2,f1,f. f2 â\88\98 f1 = f â\86\92 f2 â\8a\9a f1 â\89\98 f.
* #n2 #f2 * #n1 #f1 * #n #f cases n2 -n2
[ cases n1 -n1
[ #H cases (compose_inv_O2 … H) -H /3 width=7 by after_refl, eq_f2/
]
qed-.
-theorem after_total: â\88\80f1,f2. f2 â\8a\9a f1 â\89¡ f2 ∘ f1.
+theorem after_total: â\88\80f1,f2. f2 â\8a\9a f1 â\89\98 f2 ∘ f1.
/2 width=1 by after_total_aux/ qed.
(* Specific inversion lemmas on after ***************************************)
-lemma after_inv_xpx: ∀f2,g2,f,n2,n. n2@f2 ⊚ g2 ≡ n@f → ∀f1. ↑f1 = g2 →
- f2 â\8a\9a f1 â\89¡ f ∧ n2 = n.
+lemma after_inv_xpx: ∀f2,g2,f,n2,n. n2⨮f2 ⊚ g2 ≘ n⨮f → ∀f1. ⫯f1 = g2 →
+ f2 â\8a\9a f1 â\89\98 f ∧ n2 = n.
#f2 #g2 #f #n2 elim n2 -n2
[ #n #Hf #f1 #H2 elim (after_inv_ppx … Hf … H2) -g2 [2,3: // ]
#g #Hf #H elim (push_inv_seq_dx … H) -H destruct /2 width=1 by conj/
]
qed-.
-lemma after_inv_xnx: ∀f2,g2,f,n2,n. n2@f2 ⊚ g2 ≡ n@f → ∀f1. ⫯f1 = g2 →
- â\88\83â\88\83m. f2 â\8a\9a f1 â\89¡ m@f & ⫯(n2+m) = n.
+lemma after_inv_xnx: ∀f2,g2,f,n2,n. n2⨮f2 ⊚ g2 ≘ n⨮f → ∀f1. ↑f1 = g2 →
+ â\88\83â\88\83m. f2 â\8a\9a f1 â\89\98 m⨮f & â\86\91(n2+m) = n.
#f2 #g2 #f #n2 elim n2 -n2
[ #n #Hf #f1 #H2 elim (after_inv_pnx … Hf … H2) -g2 [2,3: // ]
#g #Hf #H elim (next_inv_seq_dx … H) -H
]
qed-.
-lemma after_inv_const: ∀f2,f1,f,n1,n. n@f2 ⊚ n1@f1 ≡ n@f → f2 ⊚ f1 ≡ f ∧ 0 = n1.
+lemma after_inv_const: ∀f2,f1,f,n1,n. n⨮f2 ⊚ n1⨮f1 ≘ n⨮f → f2 ⊚ f1 ≘ f ∧ 0 = n1.
#f2 #f1 #f #n1 #n elim n -n
[ #H elim (after_inv_pxp … H) -H [ |*: // ]
#g2 #Hf #H elim (push_inv_seq_dx … H) -H /2 width=1 by conj/
]
qed-.
-lemma after_inv_total: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89¡ f â\86\92 f2 â\88\98 f1 â\89\97 f.
+lemma after_inv_total: â\88\80f2,f1,f. f2 â\8a\9a f1 â\89\98 f â\86\92 f2 â\88\98 f1 â\89¡ f.
/2 width=4 by after_mono/ qed-.
(* Specific forward lemmas on after *****************************************)
-lemma after_fwd_hd: ∀f2,f1,f,n1,n. f2 ⊚ n1@f1 ≡ n@f → f2@❴n1❵ = n.
+lemma after_fwd_hd: ∀f2,f1,f,n1,n. f2 ⊚ n1⨮f1 ≘ n⨮f → f2@❨n1❩ = n.
#f2 #f1 #f #n1 #n #H lapply (after_fwd_at ? n1 0 … H) -H [1,2,3: // ]
/3 width=2 by at_inv_O1, sym_eq/
qed-.
-lemma after_fwd_tls: ∀f,f1,n1,f2,n2,n. n2@f2 ⊚ n1@f1 ≡ n@f →
- (â\86\93*[n1]f2) â\8a\9a f1 â\89¡ f.
+lemma after_fwd_tls: ∀f,f1,n1,f2,n2,n. n2⨮f2 ⊚ n1⨮f1 ≘ n⨮f →
+ (â«°*[n1]f2) â\8a\9a f1 â\89\98 f.
#f #f1 #n1 elim n1 -n1
[ #f2 #n2 #n #H elim (after_inv_xpx … H) -H //
| #n1 #IH * #m1 #f2 #n2 #n #H elim (after_inv_xnx … H) -H [2,3: // ]
]
qed-.
-lemma after_inv_apply: ∀f2,f1,f,n2,n1,n. n2@f2 ⊚ n1@f1 ≡ n@f →
- (n2@f2)@❴n1❵ = n ∧ (↓*[n1]f2) ⊚ f1 ≡ f.
+lemma after_inv_apply: ∀f2,f1,f,n2,n1,n. n2⨮f2 ⊚ n1⨮f1 ≘ n⨮f →
+ (n2⨮f2)@❨n1❩ = n ∧ (⫰*[n1]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.