defined.
rec definition fun2 (n1:nat) on n1: rtmap → rtmap.
-* * [ | #n2 #f2 @(n2@f2) ]
+* * [ | #n2 #f2 @(n2⨮f2) ]
#f2 cases n1 -n1 [ @f2 ]
#n1 @(fun2 n1 f2)
defined.
rec definition fun1 (n1:nat) (f1:rtmap) on n1: rtmap → rtmap.
-* * [ | #n2 #f2 @(n1@f1) ]
+* * [ | #n2 #f2 @(n1⨮f1) ]
#f2 cases n1 -n1 [ @f1 ]
#n1 @(fun1 n1 f1 f2)
defined.
* #n2 #f2 * //
qed.
-lemma fun1_xxn: ∀f2,f1,n1. fun1 n1 f1 (↑f2) = n1@f1.
+lemma fun1_xxn: ∀f2,f1,n1. fun1 n1 f1 (↑f2) = n1⨮f1.
* #n2 #f2 #f1 * //
qed.
(* Basic properies on cocompose *********************************************)
-lemma cocompose_rew: ∀f2,f1,n1. (fun0 n1 f2)@(fun2 n1 f2)~∘(fun1 n1 f1 f2) = f2 ~∘ (n1@f1).
-#f2 #f1 #n1 <(stream_rew … (f2~∘(n1@f1))) normalize //
+lemma cocompose_rew: ∀f2,f1,n1. (fun0 n1 f2)⨮(fun2 n1 f2)~∘(fun1 n1 f1 f2) = f2 ~∘ (n1⨮f1).
+#f2 #f1 #n1 <(stream_rew … (f2~∘(n1⨮f1))) normalize //
qed.
(* Basic inversion lemmas on compose ****************************************)
-lemma cocompose_inv_ppx: ∀f2,f1,f,x. (⫯f2) ~∘ (⫯f1) = x@f →
+lemma cocompose_inv_ppx: ∀f2,f1,f,x. (⫯f2) ~∘ (⫯f1) = x⨮f →
0 = x ∧ f2 ~∘ f1 = f.
#f2 #f1 #f #x
<cocompose_rew #H destruct
normalize /2 width=1 by conj/
qed-.
-lemma cocompose_inv_pnx: ∀f2,f1,f,n1,x. (⫯f2) ~∘ ((↑n1)@f1) = x@f →
- ∃∃n. ↑n = x & f2 ~∘ (n1@f1) = n@f.
+lemma cocompose_inv_pnx: ∀f2,f1,f,n1,x. (⫯f2) ~∘ (↑n1⨮f1) = x⨮f →
+ ∃∃n. ↑n = x & f2 ~∘ (n1⨮f1) = n⨮f.
#f2 #f1 #f #n1 #x
<cocompose_rew #H destruct
@(ex2_intro … (fun0 n1 f2)) // <cocompose_rew
/3 width=1 by eq_f2/
qed-.
-lemma cocompose_inv_nxx: ∀f2,f1,f,n1,x. (↑f2) ~∘ (n1@f1) = x@f →
- 0 = x ∧ f2 ~∘ (n1@f1) = f.
+lemma cocompose_inv_nxx: ∀f2,f1,f,n1,x. (↑f2) ~∘ (n1⨮f1) = x⨮f →
+ 0 = x ∧ f2 ~∘ (n1⨮f1) = f.
#f2 #f1 #f #n1 #x
<cocompose_rew #H destruct
/2 width=1 by conj/