(* RELOCATION N-STREAM ******************************************************)
let corec compose: rtmap → rtmap → rtmap ≝ ?.
-#f1 * #b2 #f2 @(seq … (f1@❴b2❵)) @(compose ? f2) -compose -f2
-@(tln … (⫯b2) f1)
-qed.
+#f1 * #n2 #f2 @(seq … (f1@❴n2❵)) @(compose ? f2) -compose -f2
+@(tln … (⫯n2) f1)
+defined.
interpretation "functional composition (nstream)"
'compose f1 f2 = (compose f1 f2).
coinductive after: relation3 rtmap rtmap rtmap ≝
-| after_zero: ∀f1,f2,f,b1,b2,b.
- after f1 f2 f →
- b1 = 0 → b2 = 0 → b = 0 →
- after (b1@f1) (b2@f2) (b@f)
-| after_skip: ∀f1,f2,f,b1,b2,b,a2,a.
- after f1 (a2@f2) (a@f) →
- b1 = 0 → b2 = ⫯a2 → b = ⫯a →
- after (b1@f1) (b2@f2) (b@f)
-| after_drop: ∀f1,f2,f,b1,b,a1,a.
- after (a1@f1) f2 (a@f) →
- b1 = ⫯a1 → b = ⫯a →
- after (b1@f1) f2 (b@f)
+| after_refl: ∀f1,f2,f,g1,g2,g.
+ after f1 f2 f → g1 = ↑f1 → g2 = ↑f2 → g = ↑f → after g1 g2 g
+| after_push: ∀f1,f2,f,g1,g2,g.
+ after f1 f2 f → g1 = ↑f1 → g2 = ⫯f2 → g = ⫯f → after g1 g2 g
+| after_next: ∀f1,f2,f,g1,g.
+ after f1 f2 f → g1 = ⫯f1 → g = ⫯f → after g1 f2 g
.
interpretation "relational composition (nstream)"
(* Basic properies on compose ***********************************************)
-lemma compose_unfold: ∀f1,f2,a2. f1∘(a2@f2) = f1@❴a2❵@tln … (⫯a2) f1∘f2.
-#f1 #f2 #a2 >(stream_expand … (f1∘(a2@f2))) normalize //
+lemma compose_unfold: ∀f1,f2,n2. f1∘(n2@f2) = f1@❴n2❵@tln … (⫯n2) f1∘f2.
+#f1 #f2 #n2 >(stream_expand … (f1∘(n2@f2))) normalize //
qed.
-lemma compose_drop: ∀f1,f2,f,a1,a. (a1@f1)∘f2 = a@f → (⫯a1@f1)∘f2 = ⫯a@f.
-#f1 * #a2 #f2 #f #a1 #a >compose_unfold >compose_unfold
+lemma compose_next: ∀f1,f2,f. f1∘f2 = f → (⫯f1)∘f2 = ⫯f.
+* #n1 #f1 * #n2 #f2 #f >compose_unfold >compose_unfold
#H destruct normalize //
qed.
(* Basic inversion lemmas on compose ****************************************)
-lemma compose_inv_unfold: ∀f1,f2,f,a2,a. f1∘(a2@f2) = a@f →
- f1@❴a2❵ = a ∧ tln … (⫯a2) f1∘f2 = f.
-#f1 #f2 #f #a2 #a >(stream_expand … (f1∘(a2@f2))) normalize
+lemma compose_inv_unfold: ∀f1,f2,f,n2,n. f1∘(n2@f2) = n@f →
+ f1@❴n2❵ = n ∧ tln … (⫯n2) f1∘f2 = f.
+#f1 #f2 #f #n2 #n >(stream_expand … (f1∘(n2@f2))) normalize
#H destruct /2 width=1 by conj/
qed-.
-lemma compose_inv_O2: ∀f1,f2,f,a1,a. (a1@f1)∘(O@f2) = a@f →
- a = a1 ∧ f1∘f2 = f.
-#f1 #f2 #f #a1 #a >compose_unfold
+lemma compose_inv_O2: ∀f1,f2,f,n1,n. (n1@f1)∘(↑f2) = n@f →
+ n = n1 ∧ f1∘f2 = f.
+#f1 #f2 #f #n1 #n >compose_unfold
#H destruct /2 width=1 by conj/
qed-.
-lemma compose_inv_S2: ∀f1,f2,f,a1,a2,a. (a1@f1)∘(⫯a2@f2) = a@f →
- a = ⫯(a1+f1@❴a2❵) ∧ f1∘(a2@f2) = f1@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
+lemma compose_inv_S2: ∀f1,f2,f,n1,n2,n. (n1@f1)∘(⫯n2@f2) = n@f →
+ n = ⫯(n1+f1@❴n2❵) ∧ f1∘(n2@f2) = f1@❴n2❵@f.
+#f1 #f2 #f #n1 #n2 #n >compose_unfold
#H destruct /2 width=1 by conj/
qed-.
-lemma compose_inv_S1: ∀f1,f2,f,a1,a2,a. (⫯a1@f1)∘(a2@f2) = a@f →
- a = ⫯((a1@f1)@❴a2❵) ∧ (a1@f1)∘(a2@f2) = (a1@f1)@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
+lemma compose_inv_S1: ∀f1,f2,f,n1,n2,n. (⫯n1@f1)∘(n2@f2) = n@f →
+ n = ⫯((n1@f1)@❴n2❵) ∧ (n1@f1)∘(n2@f2) = (n1@f1)@❴n2❵@f.
+#f1 #f2 #f #n1 #n2 #n >compose_unfold
#H destruct /2 width=1 by conj/
qed-.
(* Basic properties on after ************************************************)
lemma after_O2: ∀f1,f2,f. f1 ⊚ f2 ≡ f →
- ∀b. b@f1 ⊚ O@f2 ≡ b@f.
-#f1 #f2 #f #Ht #b elim b -b /2 width=5 by after_drop, after_zero/
+ ∀n. n@f1 ⊚ ↑f2 ≡ n@f.
+#f1 #f2 #f #Ht #n elim n -n /2 width=7 by after_refl, after_next/
qed.
-lemma after_S2: ∀f1,f2,f,b2,b. f1 ⊚ b2@f2 ≡ b@f →
- ∀b1. b1@f1 ⊚ ⫯b2@f2 ≡ ⫯(b1+b)@f.
-#f1 #f2 #f #b2 #b #Ht #b1 elim b1 -b1 /2 width=5 by after_drop, after_skip/
+lemma after_S2: ∀f1,f2,f,n2,n. f1 ⊚ n2@f2 ≡ n@f →
+ ∀n1. n1@f1 ⊚ ⫯n2@f2 ≡ ⫯(n1+n)@f.
+#f1 #f2 #f #n2 #n #Ht #n1 elim n1 -n1 /2 width=7 by after_next, after_push/
qed.
-lemma after_apply: ∀b2,f1,f2,f. (tln … (⫯b2) f1) ⊚ f2 ≡ f → f1 ⊚ b2@f2 ≡ f1@❴b2❵@f.
-#b2 elim b2 -b2
+lemma after_apply: ∀n2,f1,f2,f. (tln … (⫯n2) f1) ⊚ f2 ≡ f → f1 ⊚ n2@f2 ≡ f1@❴n2❵@f.
+#n2 elim n2 -n2
[ * /2 width=1 by after_O2/
-| #b2 #IH * /3 width=1 by after_S2/
+| #n2 #IH * /3 width=1 by after_S2/
]
qed-.
let corec after_total_aux: ∀f1,f2,f. f1 ∘ f2 = f → f1 ⊚ f2 ≡ f ≝ ?.
-* #a1 #f1 * #a2 #f2 * #a #f cases a1 -a1
-[ cases a2 -a2
+* #n1 #f1 * #n2 #f2 * #n #f cases n1 -n1
+[ cases n2 -n2
[ #H cases (compose_inv_O2 … H) -H
- /3 width=1 by after_zero/
- | #a2 #H cases (compose_inv_S2 … H) -H
- /3 width=5 by after_skip, eq_f/
+ /3 width=7 by after_refl, eq_f2/
+ | #n2 #H cases (compose_inv_S2 … H) -H
+ /3 width=7 by after_push/
]
-| #a1 #H cases (compose_inv_S1 … H) -H
- /3 width=5 by after_drop, eq_f/
+| #n1 #H cases (compose_inv_S1 … H) -H
+ /4 width=7 by after_next, next_rew_sn/
]
qed-.
(* Basic inversion lemmas on after ******************************************)
-fact after_inv_O1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1. f1 = 0@g1 →
- (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
- ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
-| #b2 #b #a2 #a #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=7 by ex3_4_intro, or_intror/
-| #b #a1 #a #_ #H1 #H3 #g1 #H destruct
+fact after_inv_OOx_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. g1 = ↑f1 → g2 = ↑f2 →
+ ∃∃f. f1 ⊚ f2 ≡ f & g = ↑f.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
+ <(injective_push … Hx1) <(injective_push … Hx2) -x2 -x1
+ /2 width=3 by ex2_intro/
+| #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+ elim (discr_next_push … Hx2)
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+ elim (discr_next_push … Hx1)
]
qed-.
-fact after_inv_O1_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 →
- (∧∧ f1 ⊚ f2 ≡ f & b2 = 0 & b = 0) ∨
- ∃∃a2,a. f1 ⊚ a2@f2 ≡ a@f & b2 = ⫯a2 & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H elim (after_inv_O1_aux … Ht) -Ht [4: // |2: skip ] *
-[ #g2 #g #Hu #H1 #H2 destruct /3 width=1 by and3_intro, or_introl/
-| #g2 #g #a2 #a #Hu #H1 #H2 destruct /3 width=5 by ex3_2_intro, or_intror/
+lemma after_inv_OOx: ∀f1,f2,g. ↑f1 ⊚ ↑f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ↑f.
+/2 width=5 by after_inv_OOx_aux/ qed-.
+
+fact after_inv_OSx_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. g1 = ↑f1 → g2 = ⫯f2 →
+ ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+ elim (discr_push_next … Hx2)
+| #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
+ <(injective_push … Hx1) <(injective_next … Hx2) -x2 -x1
+ /2 width=3 by ex2_intro/
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+ elim (discr_next_push … Hx1)
]
qed-.
-lemma after_inv_O1: ∀g1,f2,f. 0@g1 ⊚ f2 ≡ f →
- (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
- ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-/2 width=3 by after_inv_O1_aux/ qed-.
+lemma after_inv_OSx: ∀f1,f2,g. ↑f1 ⊚ ⫯f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+/2 width=5 by after_inv_OSx_aux/ qed-.
-fact after_inv_zero_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → b2 = 0 →
- f1 ⊚ f2 ≡ f ∧ b = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ /2 width=1 by conj/
-| #a1 #a2 #_ #H0 destruct
+fact after_inv_Sxx_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g → ∀f1. g1 = ⫯f1 →
+ ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+#g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+ elim (discr_push_next … Hx1)
+| #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+ elim (discr_push_next … Hx1)
+| #g #Hf #H1 #H #x1 #Hx1 destruct
+ <(injective_next … Hx1) -x1
+ /2 width=3 by ex2_intro/
]
qed-.
-lemma after_inv_zero: ∀g1,g2,f. 0@g1 ⊚ 0@g2 ≡ f →
- ∃∃g. g1 ⊚ g2 ≡ g & f = 0@g.
-#g1 #g2 #f #H elim (after_inv_O1 … H) -H *
-[ #x2 #g #Hu #H1 #H2 destruct /2 width=3 by ex2_intro/
-| #x2 #g #a2 #a #Hu #H destruct
-]
-qed-.
+lemma after_inv_Sxx: ∀f1,f2,g. ⫯f1 ⊚ f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+/2 width=5 by after_inv_Sxx_aux/ qed-.
-fact after_inv_skip_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → ∀a2. b2 = ⫯a2 →
- ∃∃a. f1 ⊚ a2@f2 ≡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #a2 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ #_ #H0 destruct
-| #x2 #x #H #H0 #H1 destruct /2 width=3 by ex2_intro/
-]
-qed-.
+(* Advanced inversion lemmas on after ***************************************)
-lemma after_inv_skip: ∀g1,g2,f,b2. 0@g1 ⊚ ⫯b2@g2 ≡ f →
- ∃∃g,b. g1 ⊚ b2@g2 ≡ b@g & f = ⫯b@g.
-#g1 #g2 * #b #f #b2 #Ht elim (after_inv_skip_aux2 … Ht) [2,4: // |3: skip ] -Ht
-#a #Ht #H destruct /2 width=4 by ex2_2_intro/
+fact after_inv_OOO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ ∀f1,f2,f. g1 = ↑f1 → g2 = ↑f2 → g = ↑f → f1 ⊚ f2 ≡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct >(injective_push … Hx) -f //
qed-.
-fact after_inv_S1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1,b1. f1 = ⫯b1@g1 →
- ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b #a1 #a #Ht #H1 #H3 #g1 #x1 #H destruct /2 width=4 by ex2_2_intro/
-]
+fact after_inv_OOS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ ∀f1,f2,f. g1 = ↑f1 → g2 = ↑f2 → g = ⫯f → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct elim (discr_next_push … Hx)
qed-.
-fact after_inv_S1_aux2: ∀f1,f2,f,b1,b. b1@f1 ⊚ f2 ≡ b@f → ∀a1. b1 = ⫯a1 →
- â\88\83â\88\83a. a1@f1 â\8a\9a f2 â\89¡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b #Ht #a #H elim (after_inv_S1_aux … Ht) -Ht [4: // |2,3: skip ]
-#g #x #Hu #H0 destruct /2 width=3 by ex2_intro/
+fact after_inv_OSS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ â\88\80f1,f2,f. g1 = â\86\91f1 â\86\92 g2 = ⫯f2 â\86\92 g = ⫯f â\86\92 f1 â\8a\9a f2 â\89¡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct >(injective_next … Hx) -f //
qed-.
-lemma after_inv_S1: ∀g1,f2,f,b1. ⫯b1@g1 ⊚ f2 ≡ f →
- ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-/2 width=3 by after_inv_S1_aux/ qed-.
-
-fact after_inv_drop_aux2: ∀f1,f2,f,a1,a. a1@f1 ⊚ f2 ≡ a@f → ∀b1,b. a1 = ⫯b1 → a = ⫯b →
- b1@f1 ⊚ f2 ≡ b@f.
-#f1 #f2 #f #a1 #a #Ht #b1 #b #H1 #H elim (after_inv_S1_aux2 … Ht … H1) -a1
-#x #Ht #Hx destruct //
+fact after_inv_OSO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ ∀f1,f2,f. g1 = ↑f1 → g2 = ⫯f2 → g = ↑f → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct elim (discr_push_next … Hx)
qed-.
-lemma after_inv_drop: ∀f1,f2,f,b1,b. ⫯b1@f1 ⊚ f2 ≡ ⫯b@f → b1@f1 ⊚ f2 ≡ b@f.
-/2 width=5 by after_inv_drop_aux2/ qed-.
+fact after_inv_SxS_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
+ ∀f1,f. g1 = ⫯f1 → g = ⫯f → f1 ⊚ f2 ≡ f.
+#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_Sxx_aux … Hg … H1) -g1
+#x #Hf #Hx destruct >(injective_next … Hx) -f //
+qed-.
-fact after_inv_O3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g. f = 0@g →
- ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g #H destruct /2 width=5 by ex3_2_intro/
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g #H destruct
-| #b #a1 #a #_ #H1 #H3 #g #H destruct
-]
+fact after_inv_SxO_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
+ ∀f1,f. g1 = ⫯f1 → g = ↑f → ⊥.
+#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_Sxx_aux … Hg … H1) -g1
+#x #Hf #Hx destruct elim (discr_push_next … Hx)
qed-.
-fact after_inv_O3_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b = 0 →
- ∧∧ f1 ⊚ f2 ≡ f & b1 = 0 & b2 = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 elim (after_inv_O3_aux1 … Ht) [2: // |3: skip ] -b
-#g1 #g2 #Ht #H1 #H2 destruct /2 width=1 by and3_intro/
+fact after_inv_OxO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ ∀f1,f. g1 = ↑f1 → g = ↑f →
+ ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ↑f2.
+#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
+[ elim (after_inv_OSO_aux … Hg … H1 … H) -g1 -g -f1 -f //
+| lapply (after_inv_OOO_aux … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
+]
qed-.
-lemma after_inv_O3: ∀f1,f2,g. f1 ⊚ f2 ≡ 0@g →
- ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-/2 width=3 by after_inv_O3_aux1/ qed-.
+lemma after_inv_OxO: ∀f1,g2,f. ↑f1 ⊚ g2 ≡ ↑f →
+ ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ↑f2.
+/2 width=5 by after_inv_OxO_aux/ qed-.
-fact after_inv_S3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g,b. f = ⫯b@g →
- (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
- ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g #a #H destruct
-| #b2 #b #a2 #a #HT #H1 #H2 #H3 #g #x #H destruct /3 width=6 by ex3_3_intro, or_introl/
-| #b #a1 #a #HT #H1 #H3 #g #x #H destruct /3 width=4 by ex2_2_intro, or_intror/
+fact after_inv_OxS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+ ∀f1,f. g1 = ↑f1 → g = ⫯f →
+ ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ⫯f2.
+#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
+[ lapply (after_inv_OSS_aux … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
+| elim (after_inv_OOS_aux … Hg … H1 … H) -g1 -g -f1 -f //
]
qed-.
-fact after_inv_S3_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → ∀b. a = ⫯b →
- (∃∃b2. f1 ⊚ b2@f2 ≡ b@f & a1 = 0 & a2 = ⫯b2) ∨
- ∃∃b1. b1@f1 ⊚ a2@f2 ≡ b@f & a1 = ⫯b1.
-#f1 #f2 #f #a1 #a2 #a #Ht #b #H elim (after_inv_S3_aux1 … Ht) [3: // |4,5: skip ] -a *
-[ #g1 #g2 #b2 #Ht #H1 #H2 destruct /3 width=3 by ex3_intro, or_introl/
-| #g1 #b1 #Ht #H1 destruct /3 width=3 by ex2_intro, or_intror/
+fact after_inv_xxO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. g = ↑f →
+ ∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ↑f2.
+* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
+[ elim (after_inv_SxO_aux … Hg … H) -g2 -g -f //
+| elim (after_inv_OxO_aux … Hg … H) -g /2 width=5 by ex3_2_intro/
]
qed-.
-lemma after_inv_S3: ∀f1,f2,g,b. f1 ⊚ f2 ≡ ⫯b@g →
- (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
- ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-/2 width=3 by after_inv_S3_aux1/ qed-.
-
-(* Advanced inversion lemmas on after ***************************************)
+lemma after_inv_xxO: ∀g1,g2,f. g1 ⊚ g2 ≡ ↑f →
+ ∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ↑f2.
+/2 width=3 by after_inv_xxO_aux/ qed-.
-fact after_inv_O2_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → a2 = 0 →
- a1 = a ∧ f1 ⊚ f2 ≡ f.
-#f1 #f2 #f #a1 #a2 elim a1 -a1
-[ #a #H #H2 elim (after_inv_zero_aux2 … H … H2) -a2 /2 width=1 by conj/
-| #a1 #IH #a #H #H2 elim (after_inv_S1_aux2 … H) -H [3: // |2: skip ]
- #b #H #H1 elim (IH … H) // -a2
- #H2 destruct /2 width=1 by conj/
+fact after_inv_xxS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. g = ⫯f →
+ (∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ⫯f2) ∨
+ ∃∃f1. f1 ⊚ g2 ≡ f & g1 = ⫯f1.
+* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
+[ /4 width=5 by after_inv_SxS_aux, or_intror, ex2_intro/
+| elim (after_inv_OxS_aux … Hg … H) -g
+ /3 width=5 by or_introl, ex3_2_intro/
]
qed-.
-lemma after_inv_O2: ∀f1,g2,f. f1 ⊚ 0@g2 ≡ f →
- ∃∃g1,g,a. f1 = a@g1 & f = a@g & g1 ⊚ g2 ≡ g.
-* #a1 #f1 #f2 * #a #f #H elim (after_inv_O2_aux2 … H) -H //
-/2 width=6 by ex3_3_intro/
+lemma after_inv_xxS: ∀g1,g2,f. g1 ⊚ g2 ≡ ⫯f →
+ (∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ⫯f2) ∨
+ ∃∃f1. f1 ⊚ g2 ≡ f & g1 = ⫯f1.
+/2 width=3 by after_inv_xxS_aux/ qed-.
+
+fact after_inv_xOx_aux: ∀f1,g2,f,n1,n. n1@f1 ⊚ g2 ≡ n@f → ∀f2. g2 = ↑f2 →
+ f1 ⊚ f2 ≡ f ∧ n1 = n.
+#f1 #g2 #f #n1 elim n1 -n1
+[ #n #Hf #f2 #H2 elim (after_inv_OOx_aux … Hf … H2) -g2 [3: // |2: skip ]
+ #g #Hf #H elim (push_inv_seq_sn … H) -H destruct /2 width=1 by conj/
+| #n1 #IH #n #Hf #f2 #H2 elim (after_inv_Sxx_aux … Hf) -Hf [3: // |2: skip ]
+ #g1 #Hg #H1 elim (next_inv_seq_sn … H1) -H1
+ #x #Hx #H destruct elim (IH … Hg) [2: // |3: skip ] -IH -Hg
+ #H destruct /2 width=1 by conj/
+]
qed-.
-lemma after_inv_const: ∀a,f1,b2,g2,f. a@f1 ⊚ b2@g2 ≡ a@f → b2 = 0.
-#a elim a -a
-[ #f1 #b2 #g2 #f #H elim (after_inv_O3 … H) -H
- #g1 #x2 #_ #_ #H destruct //
-| #a #IH #f1 #b2 #g2 #f #H elim (after_inv_S1 … H) -H
- #x #b #Hx #H destruct >(IH … Hx) -f1 -g2 -x -b2 -b //
+lemma after_inv_xOx: ∀f1,f2,f,n1,n. n1@f1 ⊚ ↑f2 ≡ n@f →
+ f1 ⊚ f2 ≡ f ∧ n1 = n.
+/2 width=3 by after_inv_xOx_aux/ qed-.
+
+fact after_inv_xSx_aux: ∀f1,g2,f,n1,n. n1@f1 ⊚ g2 ≡ n@f → ∀f2. g2 = ⫯f2 →
+ ∃∃m. f1 ⊚ f2 ≡ m@f & n = ⫯(n1+m).
+#f1 #g2 #f #n1 elim n1 -n1
+[ #n #Hf #f2 #H2 elim (after_inv_OSx_aux … Hf … H2) -g2 [3: // |2: skip ]
+ #g #Hf #H elim (next_inv_seq_sn … H) -H
+ #x #Hx #Hg destruct /2 width=3 by ex2_intro/
+| #n1 #IH #n #Hf #f2 #H2 elim (after_inv_Sxx_aux … Hf) -Hf [3: // |2: skip ]
+ #g #Hg #H elim (next_inv_seq_sn … H) -H
+ #x #Hx #H destruct elim (IH … Hg) -IH -Hg [3: // |2: skip ]
+ #m #Hf #Hm destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma after_inv_S2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ ⫯a2@f2 ≡ a@f → ∀b. a = ⫯(a1+b) →
- f1 ⊚ a2@f2 ≡ b@f.
-#f1 #f2 #f #a1 elim a1 -a1
-[ #a2 #a #Ht #b #Hb
- elim (after_inv_skip_aux2 … Ht) -Ht [3,4: // |2: skip ]
- #c #Ht #Hc destruct //
-| #a1 #IH #a2 #a #Ht #b #Hb
- lapply (after_inv_drop_aux2 … Ht … Hb) -a [ // | skip ]
- /2 width=3 by/
+lemma after_inv_xSx: ∀f1,f2,f,n1,n. n1@f1 ⊚ ⫯f2 ≡ n@f →
+ ∃∃m. f1 ⊚ f2 ≡ m@f & n = ⫯(n1+m).
+/2 width=3 by after_inv_xSx_aux/ qed-.
+
+lemma after_inv_const: ∀f1,f2,f,n2,n. n@f1 ⊚ n2@f2 ≡ n@f → f1 ⊚ f2 ≡ f ∧ n2 = 0.
+#f1 #f2 #f #n2 #n elim n -n
+[ #H elim (after_inv_OxO … H) -H
+ #g2 #Hf #H elim (push_inv_seq_sn … H) -H /2 width=1 by conj/
+| #n #IH #H lapply (after_inv_SxS_aux … H ????) -H /2 width=5 by/
]
qed-.
lemma after_at_fwd: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
#f #i1 #i #H elim H -f -i1 -i
-[ #f #f2 #f1 #H elim (after_inv_O3 … H) -H
- /2 width=3 by at_zero, ex2_intro/
-| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_O3 … H) -H
- #g2 #g1 #Hu #H1 #H2 destruct elim (IH … Hu) -f
+[ #f #f2 #f1 #H elim (after_inv_xxO … H) -H
+ /2 width=3 by at_refl, ex2_intro/
+| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_xxO … H) -H
+ #g2 #g1 #Hg #H1 #H2 destruct elim (IH … Hg) -f
/3 width=3 by at_S1, ex2_intro/
-| #f #b #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_S3 … H) -H *
- [ #g2 #g1 #b2 #Hu #H1 #H2 destruct elim (IH … Hu) -f -b
- /3 width=3 by at_S1, at_lift, ex2_intro/
- | #g1 #b1 #Hu #H destruct elim (IH … Hu) -f -b
- /3 width=3 by at_lift, ex2_intro/
+| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_xxS … H) -H *
+ [ #g2 #g1 #Hg #H2 #H1 destruct elim (IH … Hg) -f
+ /3 width=3 by at_S1, at_next, ex2_intro/
+ | #g1 #Hg #H destruct elim (IH … Hg) -f
+ /3 width=3 by at_next, ex2_intro/
]
]
qed-.
lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2,f. f2 ⊚ f1 ≡ f →
∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
#f1 #i1 #i2 #H elim H -f1 -i1 -i2
-[ #f1 #f2 #f #H elim (after_inv_O2 … H) -H /2 width=3 by ex2_intro/
-| #f1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
- [ #f2 #f #H elim (after_inv_zero … H) -H
- #g #Hu #H destruct elim (IH … Hu) -f1
- /3 width=3 by at_S1, at_skip, ex2_intro/
- | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
- #g #b #Hu #H destruct elim (IH … Hu) -f1
- /3 width=3 by at_lift, ex2_intro/
- ]
-| #f1 #b1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
- [ #f2 #f #H elim (after_inv_skip … H) -H
- #g #a #Hu #H destruct elim (IH … Hu) -f1 -b1
- /3 width=3 by at_S1, at_lift, ex2_intro/
- | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
- #g #b #Hu #H destruct elim (IH … Hu) -f1 -b1
- /3 width=3 by at_lift, ex2_intro/
- ]
+[ #f1 * #n2 #f2 * #n #f #H elim (after_inv_xOx … H) -H /2 width=3 by ex2_intro/
+| #f1 #i1 #i2 #_ #IH * #n2 #f2 * #n #f #H elim (after_inv_xOx … H) -H
+ #Hf #H destruct elim (IH … Hf) -f1 /3 width=3 by at_S1, ex2_intro/
+| #f1 #i1 #i2 #_ #IH * #n2 #f2 * #n #f #H elim (after_inv_xSx … H) -H
+ #m #Hf #Hm destruct elim (IH … Hf) -f1
+ /4 width=3 by at_plus2, at_S1, at_next, ex2_intro/
]
qed-.
lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
∀f2. f2 ⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
#f #i1 #i #H elim H -f -i1 -i
-[ #f #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
+[ #f #f1 #i2 #Ht1 #f2 #H elim (after_inv_xxO … H) -H
#g2 #g1 #_ #H1 #H2 destruct >(at_inv_OOx … Ht1) -f -g1 -i2 //
-| #f #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
+| #f #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_xxO … H) -H
#g2 #g1 #Hu #H1 #H2 destruct elim (at_inv_SOx … Ht1) -Ht1
- /3 width=3 by at_skip/
-| #f #b #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_S3 … H) -H *
- [ #g2 #g1 #a1 #Hu #H1 #H2 destruct elim (at_inv_xSx … Ht1) -Ht1
- /3 width=3 by at_skip/
- | #g2 #a2 #Hu #H destruct /3 width=3 by at_lift/
+ /3 width=3 by at_push/
+| #f #i1 #i #_ #IH #f1 #i2 #Hf1 #f2 #H elim (after_inv_xxS … H) -H *
+ [ #g2 #g1 #Hg #H2 #H1 destruct elim (at_inv_xSx … Hf1) -Hf1
+ /3 width=3 by at_push/
+ | #g2 #Hg #H destruct /3 width=3 by at_next/
]
]
qed-.
(* Advanced forward lemmas on after *****************************************)
-lemma after_fwd_hd: ∀f1,f2,f,a2,a. f1 ⊚ a2@f2 ≡ a@f → a = f1@❴a2❵.
-#f1 #f2 #f #a2 #a #Ht lapply (after_fwd_at … 0 … Ht) -Ht [4: // | // |2,3: skip ]
+lemma after_fwd_hd: ∀f1,f2,f,n2,n. f1 ⊚ n2@f2 ≡ n@f → n = f1@❴n2❵.
+#f1 #f2 #f #n2 #n #H lapply (after_fwd_at … 0 … H) -H [1,4: // |2,3: skip ]
/3 width=2 by at_inv_O1, sym_eq/
qed-.
-lemma after_fwd_tl: ∀f,f2,a2,f1,a1,a. a1@f1 ⊚ a2@f2 ≡ a@f →
- tln … a2 f1 ⊚ f2 ≡ f.
-#f #f2 #a2 elim a2 -a2
-[ #f1 #a1 #a #Ht elim (after_inv_O2_aux2 … Ht) -Ht //
-| #a2 #IH * #b1 #f1 #a1 #a #Ht
- lapply (after_fwd_hd … Ht) #Ha
- lapply (after_inv_S2 … Ht … Ha) -a
- /2 width=3 by/
+lemma after_fwd_tl: ∀f,f2,n2,f1,n1,n. n1@f1 ⊚ n2@f2 ≡ n@f →
+ tln … n2 f1 ⊚ f2 ≡ f.
+#f #f2 #n2 elim n2 -n2
+[ #f1 #n1 #n #H elim (after_inv_xOx … H) -H //
+| #n2 #IH * #m1 #f1 #n1 #n #H elim (after_inv_xSx_aux … H ??) -H [3: // |2: skip ]
+ #m #Hm #H destruct /2 width=3 by/
]
qed-.
lemma after_inv_apply: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f →
a = (a1@f1)@❴a2❵ ∧ tln … a2 f1 ⊚ f2 ≡ f.
-/3 width=3 by after_fwd_tl, after_fwd_hd, conj/ qed-.
+/3 width=3 by after_fwd_tl, after_fwd_hd, conj/ qed-.
(* Main properties on after *************************************************)
-let corec after_trans1: ∀f1,f2,f0. f1 ⊚ f2 ≡ f0 →
- ∀f3,f4. f0 ⊚ f3 ≡ f4 →
+let corec after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
+ ∀f1,f2. f1 ⊚ f2 ≡ f0 →
∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4 ≝ ?.
-#f1 #f2 #f0 * -f1 -f2 -f0 #f1 #f2 #f0 #b1 [1,2: #b2 ] #b0
-[ #Ht0 #H1 #H2 #H0 * #b3 #f3 * #b4 #f4 #Ht4 * #b #f #Ht
- cases (after_inv_O1_aux2 … Ht4 H0) -Ht4 -H0 *
- [ #Ht4 #H3 #H4 cases (after_inv_zero_aux2 … Ht H2 H3) -Ht -H2 -H3
- #Ht #H /3 width=6 by after_zero/
- | #a0 #a4 #Ht4 #H3 #H4 cases (after_inv_skip_aux2 … Ht H2 … H3) -Ht -H2 -H3
- #a #Ht3 #H /3 width=6 by after_skip/
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (after_inv_xxO_aux … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (after_inv_OOx_aux … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by after_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (after_inv_xxO_aux … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (after_inv_OSx_aux … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by after_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (after_inv_xxS_aux … Hg0 … H0) -g0 *
+ [ #f1 #f2 #Hf0 #H1 #H2
+ cases (after_inv_Sxx_aux … Hg … H2) -g2
+ #f #Hf #H /3 width=7 by after_push/
+ | #f1 #Hf0 #H1 /3 width=6 by after_next/
]
-| #a2 #a0 #Ht0 #H1 #H2 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
- #a4 #Ht4 #H4 * #b #f #H cases (after_inv_S1_aux2 … H … H2) -H -H2
- #a #Ht3 #H /3 width=6 by after_skip/
-| #a1 #a0 #Ht0 #H1 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
- #a4 #Ht4 #H4 * #b #f #Ht /3 width=6 by after_drop/
]
qed-.
let corec after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≡ f4 →
∀f2, f3. f2 ⊚ f3 ≡ f0 →
∀f. f1 ⊚ f2 ≡ f → f ⊚ f3 ≡ f4 ≝ ?.
-#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #b1 [1,2: #b0 ] #b4
-[ #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
- cases (after_inv_O3_aux2 … Ht0 H0) -b0
- #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
- #Ht #H /3 width=6 by after_zero/
-| #a0 #a4 #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
- cases (after_inv_S3_aux2 … Ht0 … H0) -b0 *
- [ #a3 #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
- #Ht #H /3 width=6 by after_skip/
- | #a2 #Ht0 #H2 cases (after_inv_skip_aux2 … Ht H1 … H2) -b1 -b2
- #a #Ht #H /3 width=6 by after_drop/
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (after_inv_xxO_aux … Hg0 … H0) -g0
+ #f2 #f3 #Hf0 #H2 #H3
+ cases (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by after_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (after_inv_xxS_aux … Hg0 … H0) -g0 *
+ [ #f2 #f3 #Hf0 #H2 #H3
+ cases (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by after_push/
+ | #f2 #Hf0 #H2
+ cases (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=6 by after_next/
]
-| #a1 #a4 #Ht4 #H1 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
- cases (after_inv_S1_aux2 … Ht … H1) -b1
- #a #Ht #H /3 width=6 by after_drop/
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+ cases (after_inv_Sxx_aux … Hg … H1) -g1
+ #f #Hg #H /3 width=6 by after_next/
]
qed-.
+(* Main inversion lemmas on after *******************************************)
+
let corec after_mono: ∀f1,f2,x. f1 ⊚ f2 ≡ x → ∀y. f1 ⊚ f2 ≡ y → x ≐ y ≝ ?.
-* #a1 #f1 * #a2 #f2 * #c #x #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd #Hy
+* #n1 #f1 * #n2 #f2 * #n #x #Hx * #m #y #Hy
+cases (after_inv_apply … Hx) -Hx #Hn #Hx
+cases (after_inv_apply … Hy) -Hy #Hm #Hy
/3 width=4 by eq_seq/
qed-.
let corec after_inj: ∀f1,x,f. f1 ⊚ x ≡ f → ∀y. f1 ⊚ y ≡ f → x ≐ y ≝ ?.
-* #a1 #f1 * #c #x * #a #f #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd
-cases (apply_inj_aux … Hc Hd) //
-#Hy -a -d /3 width=4 by eq_seq/
+* #n1 #f1 * #n2 #x * #n #f #Hx * #m2 #y #Hy
+cases (after_inv_apply … Hx) -Hx #Hn2 #Hx
+cases (after_inv_apply … Hy) -Hy #Hm2
+cases (apply_inj_aux … Hn2 Hm2) -n -m2 /3 width=4 by eq_seq/
qed-.
-(* Main inversion lemmas on after *******************************************)
-
theorem after_inv_total: ∀f1,f2,f. f1 ⊚ f2 ≡ f → f1 ∘ f2 ≐ f.
/2 width=4 by after_mono/ qed-.
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