notation "\rationals" non associative with precedence 99 for @{'q}.
interpretation "Q" 'q = Q.
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
+definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
+record q_f : Type ≝ { start : ℚ; bars: list bar }.
axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-
axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
axiom qlt : ℚ → ℚ → CProp.
+interpretation "Q plus" 'plus x y = (qp x y).
+interpretation "Q minus" 'minus x y = (qm x y).
interpretation "Q less than" 'lt x y = (qlt x y).
inductive q_comparison (a,b:ℚ) : CProp ≝
definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-interpretation "Q less or equal than" 'le x y = (qle x y).
+interpretation "Q less or equal than" 'leq x y = (qle x y).
+axiom q_le_minus: ∀a,b,c:ℚ. a ≤ c - b → a + b ≤ c.
+axiom q_lt_plus: ∀a,b,c:ℚ. a - b < c → a < c + b.
+
+axiom dist : ℚ → ℚ → ℚ.
+
+
+interpretation "list nth" 'nth = (nth _).
+interpretation "list nth" 'nth_appl l d i = (nth _ l d i).
notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71
-for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
+notation < "'nth' \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i"
+with precedence 69 for @{'nth_appl $l $d $i}.
notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
interpretation "Q x Q" 'q2 = (Prod Q Q).
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ nil ?
- | S m ⇒ def m :: make_list A def m].
+definition make_list ≝
+ λA:Type.λdef:nat→A.
+ let rec make_list (n:nat) on n ≝
+ match n with [ O ⇒ nil ? | S m ⇒ def m :: make_list m]
+ in make_list.
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).
-notation < "'mk_list' \nbsp f \nbsp n"
-with precedence 71 for @{'mk_list_appl $f $n}.
interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
+interpretation "'mk_list'" 'mk_list = (make_list _).
+notation "'mk_list'" with precedence 90 for @{'mk_list}.
+notation < "'mk_list' \nbsp term 90 f \nbsp term 90 n"
+with precedence 69 for @{'mk_list_appl $f $n}.
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
+definition empty_bar : bar ≝ 〈one,OQ〉.
+notation "\rect" with precedence 90 for @{'empty_bar}.
+interpretation "q0" 'empty_bar = empty_bar.
+notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
+interpretation "lq2" 'lq2 = (list bar).
notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
+interpretation "len" 'len = (length _).
+notation < "'len' \nbsp term 90 l" with precedence 69 for @{'len_appl $l}.
interpretation "len appl" 'len_appl l = (length _ l).
-definition eject ≝
- λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
- λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
-
(* a local letin makes russell fail *)
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).
+definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l)).
+
+lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.len (mk_list f n) = n.
+intros; elim n; [reflexivity] simplify; rewrite > H; reflexivity;
+qed.
+
+let rec sum_bases (l:list bar) (i:nat)on i ≝
+ match i with
+ [ O ⇒ OQ
+ | S m ⇒
+ match l with
+ [ nil ⇒ sum_bases l m + Qpos one
+ | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
+definition eject1 ≝
+ λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
+coercion eject1.
+definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
+coercion inject1 with 0 1 nocomposites.
+
+definition value :
+ ∀f:q_f.∀i:ℚ.∃p:nat × ℚ.
+ match q_cmp i (start f) with
+ [ q_lt _ ⇒ \snd p = OQ
+ | _ ⇒
+ And3
+ (sum_bases (bars f) (\fst p) ≤ i - start f)
+ (i - start f < sum_bases (bars f) (S (\fst p)))
+ (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
+intros;
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
-definition rebase:
- q_f → q_f →
- ∃p:q_f × q_f.∀i.
- \fst (nth (bars (\fst p)) q0 i) =
- \fst (nth (bars (\snd p)) q0 i).
-intros (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
-letin spec ≝ (λl1,l2:list (ℚ × ℚ).λm:nat.λz:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).True);
+letin value ≝ (
+ let rec value (p: ℚ) (l : list bar) on l ≝
+ match l with
+ [ nil ⇒ 〈O,OQ〉
+ | cons x tl ⇒
+ match q_cmp p (Qpos (\fst x)) with
+ [ q_lt _ ⇒ 〈O, \snd x〉
+ | _ ⇒
+ let rc ≝ value (p - Qpos (\fst x)) tl in
+ 〈S (\fst rc),\snd rc〉]]
+ in value :
+ ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
+ And3
+ (sum_bases l (\fst p) ≤ acc)
+ (acc < sum_bases l (S (\fst p)))
+ (\snd p = \snd (nth l ▭ (\fst p))));
+[5: clearbody value;
+ cases (q_cmp i (start f));
+ [2: exists [apply 〈O,OQ〉] simplify; reflexivity;
+ |*: cases (value (i - start f) (bars f)) (p Hp);
+ cases Hp; clear Hp value;
+ exists[1,3:apply p]; simplify; split; assumption;]
+|1,3: intros; split;
+ [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1); cases H2; clear H2;
+ simplify; apply q_le_minus; assumption;
+ |2,5: cases (value (q-Qpos (\fst b)) l1); cases H3; clear H3 H2 value;
+ change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
+ apply q_lt_plus; assumption;
+ |*: cases (value (q-Qpos (\fst b)) l1); simplify; cases H3; clear H3 value H2;
+ assumption;]
+|2: clear value H2; simplify; split;
+ [1:
+
+
+definition same_shape ≝
+ λl1,l2:q_f.
+ ∀input.∃col.
+
+ And3
+ (sum_bases (bars l2) j ≤ offset - start l2)
+ (offset - start l2 ≤ sum_bases (bars l2) (S j))
+ (\snd (nth (bars l2)) q0 j) = \snd (nth (bars l1) q0 i).
+
+…┐─┌┐…
+\ldots\boxdl\boxh\boxdr\boxdl\ldots
+
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+definition rebase_spec ≝
+ ∀l1,l2:q_f.∃p:q_f × q_f.
+ And4
+ (len (bars (\fst p)) = len (bars (\snd p)))
+ (start (\fst p) = start (\snd p))
+ (∀i.\fst (nth (bars (\fst p)) q0 i) = \fst (nth (bars (\snd p)) q0 i))
+ (∀i,offset.
+ sum_bases (bars l1) i ≤ offset - start l1 →
+ offset - start l1 ≤ sum_bases (bars l1) (S i) →
+ ∃j.
+ And3
+ (sum_bases (bars (\fst p)) j ≤ offset - start (\fst p))
+ (offset - start (\fst p) ≤ sum_bases (bars (\fst p)) (S j))
+ (\snd (nth (bars (\fst p)) q0 j) = \snd (nth (bars l1) q0 i)) ∧
+ And3
+ (sum_bases (bars (\snd p)) j ≤ offset - start (\snd p))
+ (offset - start (\snd p) ≤ sum_bases (bars (\snd p)) (S j))
+ (\snd (nth (bars (\snd p)) q0 j) = \snd (nth (bars l2) q0 i))).
+
+definition rebase_spec_simpl ≝
+ λl1,l2:list (ℚ × ℚ).λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).
+ len ( (\fst p)) = len ( (\snd p)) ∧
+ (∀i.
+ \fst (nth ( (\fst p)) q0 i) =
+ \fst (nth ( (\snd p)) q0 i)) ∧
+ ∀i,offset.
+ sum_bases ( l1) i ≤ offset ∧
+ offset ≤ sum_bases ( l1) (S i)
+ →
+ ∃j.
+ sum_bases ( (\fst p)) j ≤ offset ∧
+ offset ≤ sum_bases ((\fst p)) (S j) ∧
+ \snd (nth ( (\fst p)) q0 j) =
+ \snd (nth ( l1) q0 i).
+
+definition eject ≝
+ λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
+coercion eject.
+definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
+coercion inject with 0 1 nocomposites.
+
+definition rebase: rebase_spec.
+intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
+letin spec ≝ (
+ λl1,l2:list (ℚ × ℚ).λm:nat.λz:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).
+ len l1 + len l2 < m → rebase_spec_simpl l1 l2 z);
letin aux ≝ (
let rec aux (l1,l2:list (ℚ × ℚ)) (n:nat) on n : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
match n with
let height1 ≝ (\snd he1) in
let height2 ≝ (\snd he2) in
match q_cmp base1 base2 with
- [ q_eq _ ⇒ 〈[],[]〉 (*
+ [ q_eq _ ⇒
let rc ≝ aux tl1 tl2 m in
- 〈he1 :: \fst rc,he2 :: \snd rc〉 *)
- | q_lt _ ⇒ 〈[],[]〉 (*
+ 〈he1 :: \fst rc,he2 :: \snd rc〉
+ | q_lt _ ⇒
let rest ≝ base2 - base1 in
let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
- 〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉 *)
- | q_gt _ ⇒ 〈[],[]〉 (*
+ 〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉
+ | q_gt _ ⇒
let rest ≝ base1 - base2 in
let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
- 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉 *)
+ 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
]]]]
-in aux : ∀l1,l2,m.∃z.spec l1 l2 m z);
+in aux : ∀l1,l2,m.∃z.spec l1 l2 m z); unfold spec;
+[7: clearbody aux; unfold spec in aux; clear spec;
+ cases (q_cmp s1 s2);
+ [1: cases (aux l1 l2 (S (len l1 + len l2)));
+ cases (H1 (le_n ?)); clear H1;
+ exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] repeat split;
+ [1: cases H2; assumption;
+ |2: assumption;
+ |3: cases H2; assumption;
+ |4: intros; cases (H3 i (offset - s1));
+ [2:
+
+
+|1,2: simplify; generalize in ⊢ (? ? (? (? ? (? ? ? (? ? %)))) (? (? ? (? ? ? (? ? %))))); intro X;
+ cases X (rc OK); clear X; simplify; apply eq_f; assumption;
+|3: cases (aux l4 l5 n1) (rc OK); simplify; apply eq_f; assumption;
+|4,5: simplify; unfold cb0h; rewrite > len_mk_list; reflexivity;
+|6: reflexivity]
+clearbody aux; unfold spec in aux; clear spec;
+
+
+
qed.
\ No newline at end of file
projects / helm.git / commitdiff