(* *)
(**************************************************************************)
-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma".
-
-
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q.
-
-definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
-record q_f : Type ≝ { start : ℚ; bars: list bar }.
-
-axiom qp : ℚ → ℚ → ℚ.
-axiom qm : ℚ → ℚ → ℚ.
-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 ≝
- | q_eq : a = b → q_comparison a b
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-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 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).
-
-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.
-
-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 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 term 90 l" with precedence 69 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-(* a local letin makes russell fail *)
-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".
-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
+include "nat_ordered_set.ma".
+include "models/q_shift.ma".
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)))
+ (*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))).
+ (same_bases (\fst p) (\snd p))
+ (same_values l1 (\fst p))
+ (same_values l2 (\snd p)).
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).
+ λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
+ And3
+ (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
+ (same_values (mk_q_f start l1) (mk_q_f start (\fst p)))
+ (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
+
+(* a local letin makes russell fail *)
+definition cb0h : list bar → list bar ≝
+ λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
definition eject ≝
λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
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);
+ λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
+alias symbol "pi1" (instance 34) = "exT \fst".
+alias symbol "pi1" (instance 21) = "exT \fst".
letin aux ≝ (
-let rec aux (l1,l2:list (ℚ × ℚ)) (n:nat) on n : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
+let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
match n with
[ O ⇒ 〈 nil ? , nil ? 〉
| S m ⇒
match l2 with
[ nil ⇒ 〈l1, cb0h l1〉
| cons he2 tl2 ⇒
- let base1 ≝ (\fst he1) in
- let base2 ≝ (\fst he2) in
+ let base1 ≝ Qpos (\fst he1) in
+ let base2 ≝ Qpos (\fst he2) in
let height1 ≝ (\snd he1) in
let height2 ≝ (\snd he2) in
match q_cmp base1 base2 with
[ q_eq _ ⇒
let rc ≝ aux tl1 tl2 m in
〈he1 :: \fst rc,he2 :: \snd rc〉
- | q_lt _ ⇒
+ | q_lt Hp ⇒
let rest ≝ base2 - base1 in
- let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
- 〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉
- | q_gt _ ⇒
+ let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
+ 〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉
+ | q_gt Hp ⇒
let rest ≝ base1 - base2 in
- let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
- 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
+ let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
+ 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
]]]]
-in aux : ∀l1,l2,m.∃z.spec l1 l2 m z); unfold spec;
-[7: clearbody aux; unfold spec in aux; clear spec;
+in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
+[9: 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;
+ cases (H1 s1 (le_n ?)); clear H1;
+ exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
+ [1,2: assumption;
+ |3: intro; apply (H3 input);
+ |4: intro; rewrite > H in H4;
+ rewrite > (H4 input); reflexivity;]
+ |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
+ apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption]
+ cases (aux l1 l2' (S (len l1 + len l2')));
+ cases (H1 s1 (le_n ?)); clear H1 aux;
+ exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
+ [1: reflexivity
|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
+ |3: assumption;
+ |4: intro; rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input);
+ rewrite < (H4 input); reflexivity;]
+ |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
+ apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption]
+ cases (aux l1' l2 (S (len l1' + len l2)));
+ cases (H1 s2 (le_n ?)); clear H1 aux;
+ exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
+ [1: reflexivity
+ |2: assumption;
+ |4: assumption;
+ |3: intro; rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input);
+ rewrite < (H3 input); reflexivity;]]
+|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption;
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
+|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ simplify in ⊢ (???? (??? (??? (??? (?? (? (?? (??? % ?) ?) ??)))) ?));
+ simplify in ⊢ (???? (???? (??? (??? (?? (? (?? (??? % ?) ?) ??))))));
+ clear H4; cases (aux (〈w,\snd b〉::l4) l5 n1); clear aux;
+ cut (len (〈w,\snd b〉::l4) + len l5 < n1) as K;[2:
+ simplify in H5; simplify; rewrite > sym_plus in H5; simplify in H5;
+ rewrite > sym_plus in H5; apply le_S_S_to_le; apply H5;]
+ split;
+ [1: simplify in ⊢ (? % ?); simplify in ⊢ (? ? %);
+ cases (H4 s K); clear K H4; intro input; cases input; [reflexivity]
+ simplify; apply H7;
+ |2: simplify in ⊢ (? ? %); cases (H4 s K); clear H4 K H5 spec;
+ intro;
+ (* input < s + b1 || input >= s + b1 *)
+ |3: simplify in ⊢ (? ? %);]
+|4: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ (* duale del 3 *)
+|5: intros; (* triviale, caso in cui non fa nulla *)
+|6,7: (* casi base in cui allunga la lista più corta *)
+]
+qed.
projects / helm.git / blobdiff