include "ground_2/notation/functions/apply_2.ma".
include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/nstream.ma".
+include "ground_2/relocation/nstream_lift.ma".
(* RELOCATION N-STREAM ******************************************************)
let rec apply (i: nat) on i: rtmap → nat ≝ ?.
-* #b #f cases i -i
-[ @b
+* #n #f cases i -i
+[ @n
| #i lapply (apply i f) -apply -i -f
- #i @(⫯(b+i))
+ #i @(⫯(n+i))
]
-qed.
+defined.
interpretation "functional application (nstream)"
'Apply f i = (apply i f).
inductive at: rtmap → relation nat ≝
-| at_zero: ∀f. at (0 @ f) 0 0
-| at_skip: ∀f,i1,i2. at f i1 i2 → at (0 @ f) (⫯i1) (⫯i2)
-| at_lift: ∀f,b,i1,i2. at (b @ f) i1 i2 → at (⫯b @ f) i1 (⫯i2)
+| at_refl: ∀f. at (↑f) 0 0
+| at_push: ∀f,i1,i2. at f i1 i2 → at (↑f) (⫯i1) (⫯i2)
+| at_next: ∀f,i1,i2. at f i1 i2 → at (⫯f) i1 (⫯i2)
.
interpretation "relational application (nstream)"
(* Basic properties on apply ************************************************)
-lemma apply_S1: ∀f,a,i. (⫯a@f)@❴i❵ = ⫯((a@f)@❴i❵).
-#a #f * //
+lemma apply_eq_repl (i): eq_stream_repl … (λf1,f2. f1@❴i❵ = f2@❴i❵).
+#i elim i -i [2: #i #IH ] * #n1 #f1 * #n2 #f2 #H
+elim (eq_stream_inv_seq ????? H) -H normalize //
+#Hn #Hf /4 width=1 by eq_f2, eq_f/
+qed.
+
+lemma apply_S1: ∀f,n,i. (⫯n@f)@❴i❵ = ⫯((n@f)@❴i❵).
+#n #f * //
qed.
(* Basic inversion lemmas on at *********************************************)
-fact at_inv_xOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. f = 0@g →
- (i1 = 0 ∧ i2 = 0) ∨
- ∃∃j1,j2. @⦃j1, g⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
-#f #i1 #i2 * -f -i1 -i2
-[ /3 width=1 by or_introl, conj/
-| #f #i1 #i2 #Hi #g #H destruct /3 width=5 by ex3_2_intro, or_intror/
-| #f #b #i1 #i2 #_ #g #H destruct
+fact at_inv_OOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. i1 = 0 → f = ↑g → i2 = 0.
+#f #i1 #i2 * -f -i1 -i2 //
+[ #f #i1 #i2 #_ #g #H destruct
+| #f #i1 #i2 #_ #g #_ #H elim (discr_next_push … H)
]
qed-.
-lemma at_inv_xOx: ∀f,i1,i2. @⦃i1, 0@f⦄ ≡ i2 →
- (i1 = 0 ∧ i2 = 0) ∨
- ∃∃j1,j2. @⦃j1, f⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
-/2 width=3 by at_inv_xOx_aux/ qed-.
+lemma at_inv_OOx: ∀f,i2. @⦃0, ↑f⦄ ≡ i2 → i2 = 0.
+/2 width=6 by at_inv_OOx_aux/ qed-.
-lemma at_inv_OOx: ∀f,i. @⦃0, 0 @ f⦄ ≡ i → i = 0.
-#f #i #H elim (at_inv_xOx … H) -H * //
-#j1 #j2 #_ #H destruct
+fact at_inv_SOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,j1. i1 = ⫯j1 → f = ↑g →
+ ∃∃j2. @⦃j1, g⦄ ≡ j2 & i2 = ⫯j2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #j1 #H destruct
+| #f #i1 #i2 #Hi #g #j1 #H #Hf <(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/
+| #f #i1 #i2 #_ #g #j1 #_ #H elim (discr_next_push … H)
+]
qed-.
-lemma at_inv_xOO: ∀f,i. @⦃i, 0@f⦄ ≡ 0 → i = 0.
-#f #i #H elim (at_inv_xOx … H) -H * //
-#j1 #j2 #_ #_ #H destruct
+lemma at_inv_SOx: ∀f,i1,i2. @⦃⫯i1, ↑f⦄ ≡ i2 →
+ ∃∃j2. @⦃i1, f⦄ ≡ j2 & i2 = ⫯j2.
+/2 width=5 by at_inv_SOx_aux/ qed-.
+
+fact at_inv_xSx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. f = ⫯g →
+ ∃∃j2. @⦃i1, g⦄ ≡ j2 & i2 = ⫯j2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #H elim (discr_push_next … H)
+| #f #i1 #i2 #_ #g #H elim (discr_push_next … H)
+| #f #i1 #i2 #Hi #g #H <(injective_next … H) -g /2 width=3 by ex2_intro/
+]
qed-.
-lemma at_inv_SOx: ∀f,i1,i2. @⦃⫯i1, 0@f⦄ ≡ i2 →
+lemma at_inv_xSx: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ i2 →
∃∃j2. @⦃i1, f⦄ ≡ j2 & i2 = ⫯j2.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
+/2 width=3 by at_inv_xSx_aux/ qed-.
+
+(* Advanced inversion lemmas on at ******************************************)
+
+lemma at_inv_OOS: ∀f,i2. @⦃0, ↑f⦄ ≡ ⫯i2 → ⊥.
+#f #i2 #H lapply (at_inv_OOx … H) -H
+#H destruct
qed-.
-lemma at_inv_xOS: ∀f,i1,i2. @⦃i1, 0@f⦄ ≡ ⫯i2 →
- ∃∃j1. @⦃j1, f⦄ ≡ i2 & i1 = ⫯j1.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
+lemma at_inv_SOS: ∀f,i1,i2. @⦃⫯i1, ↑f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
+#f #i1 #i2 #H elim (at_inv_SOx … H) -H
+#j2 #H2 #H destruct //
qed-.
-lemma at_inv_SOS: ∀f,i1,i2. @⦃⫯i1, 0@f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct //
-]
+lemma at_inv_SOO: ∀f,i1. @⦃⫯i1, ↑f⦄ ≡ 0 → ⊥.
+#f #i1 #H elim (at_inv_SOx … H) -H
+#j2 #_ #H destruct
qed-.
-lemma at_inv_OOS: ∀f,i. @⦃0, 0@f⦄ ≡ ⫯i → ⊥.
-#f #i #H elim (at_inv_xOx … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #_ #H destruct
-]
+lemma at_inv_xSS: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
+#f #i1 #i2 #H elim (at_inv_xSx … H) -H
+#j2 #H #H2 destruct //
qed-.
-lemma at_inv_SOO: ∀f,i. @⦃⫯i, 0@f⦄ ≡ 0 → ⊥.
-#f #i #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #_ #_ #H destruct
-]
+lemma at_inv_xSO: ∀f,i1. @⦃i1, ⫯f⦄ ≡ 0 → ⊥.
+#f #i1 #H elim (at_inv_xSx … H) -H
+#j2 #_ #H destruct
qed-.
-fact at_inv_xSx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,a. f = ⫯a @ g →
- ∃∃j2. @⦃i1, a@g⦄ ≡ j2 & i2 = ⫯j2.
-#f #i1 #i2 * -f -i1 -i2
-[ #f #g #a #H destruct
-| #f #i1 #i2 #_ #g #a #H destruct
-| #f #b #i1 #i2 #Hi #g #a #H destruct /2 width=3 by ex2_intro/
+lemma at_inv_xOx: ∀f,i1,i2. @⦃i1, ↑f⦄ ≡ i2 →
+ (i1 = 0 ∧ i2 = 0) ∨
+ ∃∃j1,j2. @⦃j1, f⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
+#f * [2: #i1 ] #i2 #H
+[ elim (at_inv_SOx … H) -H
+ #j2 #H2 #H destruct /3 width=5 by or_intror, ex3_2_intro/
+| >(at_inv_OOx … H) -i2 /3 width=1 by conj, or_introl/
]
qed-.
-lemma at_inv_xSx: ∀f,b,i1,i2. @⦃i1, ⫯b@f⦄ ≡ i2 →
- ∃∃j2. @⦃i1, b@f⦄ ≡ j2 & i2 = ⫯j2.
-/2 width=3 by at_inv_xSx_aux/ qed-.
-
-lemma at_inv_xSS: ∀f,b,i1,i2. @⦃i1, ⫯b @ f⦄ ≡ ⫯i2 → @⦃i1, b@f⦄ ≡ i2.
-#f #b #i1 #i2 #H elim (at_inv_xSx … H) -H
-#j2 #Hj #H destruct //
+lemma at_inv_xOO: ∀f,i. @⦃i, ↑f⦄ ≡ 0 → i = 0.
+#f #i #H elim (at_inv_xOx … H) -H * //
+#j1 #j2 #_ #_ #H destruct
qed-.
-lemma at_inv_xSO: ∀f,b,i. @⦃i, ⫯b@f⦄ ≡ 0 → ⊥.
-#f #b #i #H elim (at_inv_xSx … H) -H
-#j2 #_ #H destruct
+lemma at_inv_xOS: ∀f,i1,i2. @⦃i1, ↑f⦄ ≡ ⫯i2 →
+ ∃∃j1. @⦃j1, f⦄ ≡ i2 & i1 = ⫯j1.
+#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
+[ #_ #H destruct
+| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
+]
qed-.
(* alternative definition ***************************************************)
-lemma at_O1: ∀b,f. @⦃0, b@f⦄ ≡ b.
-#b elim b -b /2 width=1 by at_lift/
+lemma at_O1: ∀i2,f. @⦃0, i2@f⦄ ≡ i2.
+#i2 elim i2 -i2 /2 width=1 by at_refl, at_next/
qed.
-lemma at_S1: ∀b,f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃⫯i1, b@f⦄ ≡ ⫯(b+i2).
-#b elim b -b /3 width=1 by at_skip, at_lift/
+lemma at_S1: ∀n,f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃⫯i1, n@f⦄ ≡ ⫯(n+i2).
+#n elim n -n /3 width=1 by at_push, at_next/
qed.
-lemma at_inv_O1: ∀f,b,i2. @⦃0, b@f⦄ ≡ i2 → i2 = b.
-#f #b elim b -b /2 width=2 by at_inv_OOx/
-#b #IH #i2 #H elim (at_inv_xSx … H) -H
+lemma at_inv_O1: ∀f,n,i2. @⦃0, n@f⦄ ≡ i2 → i2 = n.
+#f #n elim n -n /2 width=2 by at_inv_OOx/
+#n #IH #i2 <next_rew #H elim (at_inv_xSx … H) -H
#j2 #Hj #H destruct /3 width=1 by eq_f/
qed-.
-lemma at_inv_S1: ∀f,b,j1,i2. @⦃⫯j1, b@f⦄ ≡ i2 → ∃∃j2. @⦃j1, f⦄ ≡ j2 & i2 =⫯(b+j2).
-#f #b elim b -b /2 width=1 by at_inv_SOx/
-#b #IH #j1 #i2 #H elim (at_inv_xSx … H) -H
+lemma at_inv_S1: ∀f,n,j1,i2. @⦃⫯j1, n@f⦄ ≡ i2 → ∃∃j2. @⦃j1, f⦄ ≡ j2 & i2 =⫯(n+j2).
+#f #n elim n -n /2 width=1 by at_inv_SOx/
+#n #IH #j1 #i2 <next_rew #H elim (at_inv_xSx … H) -H
#j2 #Hj #H destruct elim (IH … Hj) -IH -Hj
#i2 #Hi #H destruct /2 width=3 by ex2_intro/
qed-.
-lemma at_total: ∀i,f. @⦃i, f⦄ ≡ f@❴i❵.
-#i elim i -i
+lemma at_total: ∀i1,f. @⦃i1, f⦄ ≡ f@❴i1❵.
+#i1 elim i1 -i1
[ * // | #i #IH * /3 width=1 by at_S1/ ]
qed.
#f #i1 #i2 #H elim H -f -i1 -i2 /2 width=1 by le_S_S, le_S/
qed-.
-lemma at_increasing_plus: ∀f,b,i1,i2. @⦃i1, b@f⦄ ≡ i2 → i1 + b ≤ i2.
-#f #b *
+lemma at_increasing_plus: ∀f,n,i1,i2. @⦃i1, n@f⦄ ≡ i2 → i1 + n ≤ i2.
+#f #n *
[ #i2 #H >(at_inv_O1 … H) -i2 //
| #i1 #i2 #H elim (at_inv_S1 … H) -H
#j1 #Ht #H destruct
]
qed-.
-lemma at_increasing_strict: ∀f,b,i1,i2. @⦃i1, ⫯b @ f⦄ ≡ i2 →
- i1 < i2 ∧ @⦃i1, b@f⦄ ≡ ⫰i2.
-#f #b #i1 #i2 #H elim (at_inv_xSx … H) -H
+lemma at_increasing_strict: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ i2 →
+ i1 < i2 ∧ @⦃i1, f⦄ ≡ ⫰i2.
+#f #i1 #i2 #H elim (at_inv_xSx … H) -H
#j2 #Hj #H destruct /4 width=2 by conj, at_increasing, le_S_S/
qed-.
-lemma at_fwd_id: ∀f,b,i. @⦃i, b@f⦄ ≡ i → b = 0.
-#f #b *
-[ #H <(at_inv_O1 … H) -f -b //
+lemma at_fwd_id: ∀f,n,i. @⦃i, n@f⦄ ≡ i → n = 0.
+#f #n *
+[ #H <(at_inv_O1 … H) -f -n //
| #i #H elim (at_inv_S1 … H) -H
#j #H #H0 destruct lapply (at_increasing … H) -H
#H lapply (eq_minus_O … H) -H //
]
+qed-.
+
+(* Basic properties on at ***************************************************)
+
+lemma at_plus2: ∀f,i1,i,n,m. @⦃i1, n@f⦄ ≡ i → @⦃i1, (m+n)@f⦄ ≡ m+i.
+#f #i1 #i #n #m #H elim m -m /2 width=1 by at_next/
qed.
-(* Main properties on at ****************************************************)
+(* Advanced properties on at ************************************************)
lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≡ i2 → @⦃i1, f⦄ ≡ i1.
#i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ]
-* #b #f #H lapply (at_fwd_id … H)
+* #n #f #H lapply (at_fwd_id … H)
#H0 destruct /4 width=1 by at_S1, at_inv_SOS/
qed-.
+(* Main properties on at ****************************************************)
+
let corec at_ext: ∀f1,f2. (∀i,i1,i2. @⦃i, f1⦄ ≡ i1 → @⦃i, f2⦄ ≡ i2 → i1 = i2) → f1 ≐ f2 ≝ ?.
-* #b1 #f1 * #b2 #f2 #Hi lapply (Hi 0 b1 b2 ? ?) //
+* #n1 #f1 * #n2 #f2 #Hi lapply (Hi 0 n1 n2 ? ?) //
#H lapply (at_ext f1 f2 ?) /2 width=1 by eq_seq/ -at_ext
-#j #j1 #j2 #H1 #H2 @(injective_plus_r … b2) /4 width=5 by at_S1, injective_S/ (**) (* full auto fails *)
+#j #j1 #j2 #H1 #H2 @(injective_plus_r … n2) /4 width=5 by at_S1, injective_S/ (**) (* full auto fails *)
qed-.
theorem at_monotonic: ∀i1,i2. i1 < i2 → ∀f1,f2. f1 ≐ f2 → ∀j1,j2. @⦃i1, f1⦄ ≡ j1 → @⦃i2, f2⦄ ≡ j2 → j1 < j2.
#i1 #i2 #H @(lt_elim … H) -i1 -i2
-[ #i2 * #b1 #f1 * #b2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
+[ #i2 * #n1 #f1 * #n2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
#H #Ht #j1 #j2 #H1 #H2 destruct
>(at_inv_O1 … H1) elim (at_inv_S1 … H2) -H2 -j1 //
-| #i1 #i2 #IH * #b1 #f1 * #b2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
+| #i1 #i2 #IH * #n1 #f1 * #n2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
#H #Ht #j1 #j2 #H1 #H2 destruct
elim (at_inv_S1 … H2) elim (at_inv_S1 … H1) -H1 -H2
#x1 #Hx1 #H1 #x2 #Hx2 #H2 destruct /4 width=5 by lt_S_S, monotonic_lt_plus_r/
theorem at_inv_monotonic: ∀f1,i1,j1. @⦃i1, f1⦄ ≡ j1 → ∀f2,i2,j2. @⦃i2, f2⦄ ≡ j2 → f1 ≐ f2 → j2 < j1 → i2 < i1.
#f1 #i1 #j1 #H elim H -f1 -i1 -j1
[ #f1 #f2 #i2 #j2 #_ #_ #H elim (lt_le_false … H) //
-| #f1 #i1 #j1 #_ #IH * #b2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
+| #f1 #i1 #j1 #_ #IH * #n2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
#H0 #Ht destruct elim (at_inv_xOx … H) -H *
[ #H1 #H2 destruct //
| #x2 #y2 #Hxy #H1 #H2 destruct /4 width=5 by lt_S_S_to_lt, lt_S_S/
]
-| #f1 #b1 #i1 #j1 #_ #IH * #b2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
- #H0 #Ht destruct elim (at_inv_xSx … H) -H
+| * #n1 #f1 #i1 #j1 #_ #IH * #n2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
+ #H0 #Ht destruct <next_rew in H; #H elim (at_inv_xSx … H) -H
#y2 #Hy #H destruct /3 width=5 by eq_seq, lt_S_S_to_lt/
]
qed-.
lemma at_inv_total: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → i2 = f@❴i1❵.
/2 width=6 by at_mono/ qed-.
-lemma at_repl_back: ∀i1,i2. eq_stream_repl_back ? (λf. @⦃i1, f⦄ ≡ i2).
-#i1 #i2 #f1 #f2 #Ht #H1 lapply (at_total i1 f2)
-#H2 <(at_mono … Ht … H1 … H2) -f1 -i2 //
+lemma at_eq_repl_back: ∀i1,i2. eq_stream_repl_back ? (λf. @⦃i1, f⦄ ≡ i2).
+#i1 #i2 #f1 #H1 #f2 #Hf lapply (at_total i1 f2)
+#H2 <(at_mono … Hf … H1 … H2) -f1 -i2 //
qed-.
-lemma at_repl_fwd: ∀i1,i2. eq_stream_repl_fwd ? (λf. @⦃i1, f⦄ ≡ i2).
-#i1 #i2 @eq_stream_repl_sym /2 width=3 by at_repl_back/
+lemma at_eq_repl_fwd: ∀i1,i2. eq_stream_repl_fwd ? (λf. @⦃i1, f⦄ ≡ i2).
+#i1 #i2 @eq_stream_repl_sym /2 width=3 by at_eq_repl_back/
qed-.
(* Advanced properties on at ************************************************)
(* Note: see also: trace_at/is_at_dec *)
lemma is_at_dec: ∀f,i2. Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
-#f #i2 @(is_at_dec_le ? ? (⫯i2)) /2 width=4 by lt_le_false/
+#f #i2 @(is_at_dec_le ?? (⫯i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on apply *********************************************)
-fact apply_inj_aux: ∀f1,f2. f1 ≐ f2 → ∀i,i1,i2. i = f1@❴i1❵ → i = f2@❴i2❵ → i1 = i2.
+fact apply_inj_aux: ∀f1,f2,j1,j2,i1,i2. j1 = f1@❴i1❵ → j2 = f2@❴i2❵ →
+ j1 = j2 → f1 ≐ f2 → i1 = i2.
/2 width=6 by at_inj/ qed-.
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