include "list/list.ma".
include "cprop_connectives.ma".
+
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_le_minus_r: ∀a,b,c:ℚ. a - b ≤ c → a ≤ c + b.
+axiom q_lt_plus: ∀a,b,c:ℚ. a - b < c → a < c + b.
+axiom q_lt_minus: ∀a,b,c:ℚ. a + b < c → a < c - b.
+
+axiom q_dist : ℚ → ℚ → ℚ.
+
+notation "hbox(\dd [term 19 x, break term 19 y])" with precedence 90
+for @{'distance $x $y}.
+interpretation "ℚ distance" 'distance x y = (q_dist x y).
+
+axiom q_dist_ge_OQ : ∀x,y:ℚ. OQ ≤ ⅆ[x,y].
+
+axiom q_lt_to_le: ∀a,b:ℚ.a < b → a ≤ b.
+axiom q_le_to_diff_ge_OQ : ∀a,b.a ≤ b → OQ ≤ b-a.
+axiom q_plus_OQ: ∀x:ℚ.x + OQ = x.
+axiom q_plus_sym: ∀x,y:ℚ.x + y = y + x.
+axiom nat_of_q: ℚ → nat.
+
+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).
-notation "'nth'" left associative with precedence 70 for @{'nth}.
-notation < "\nth \nbsp l \nbsp d \nbsp i" left associative with precedence 70 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).
+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 < "\rationals \sup 2" non associative with precedence 40 for @{'q2}.
-interpretation "Q x Q" 'q2 = (product Q Q).
+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}.
-let rec mk_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ []
- | S m ⇒ def m :: mk_list A def m].
-interpretation "mk_list appl" 'mk_list f n = (mk_list f n).
-interpretation "mk_list" 'mk_list = mk_list.
-notation < "\mk_list \nbsp f \nbsp n" left associative with precedence 70 for @{'mk_list_appl $f $n}.
-notation "'mk_list'" left associative with precedence 70 for @{'mk_list}.
+definition empty_bar : bar ≝ 〈one,OQ〉.
+notation "\rect" with precedence 90 for @{'empty_bar}.
+interpretation "q0" 'empty_bar = empty_bar.
-alias symbol "pair" = "pair".
-definition q00 : ℚ × ℚ ≝ 〈OQ,OQ〉.
+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).
+
+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)]].
+
+axiom sum_bases_empty_nat_of_q_ge_OQ:
+ ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q).
+axiom sum_bases_empty_nat_of_q_le_q:
+ ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
+axiom sum_bases_empty_nat_of_q_le_q_one:
+ ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
+
+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) ≤ ⅆ[i,start f])
+ (ⅆ[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 ⇒ 〈nat_of_q p,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 ⅆ[i,start f] (bars f)) (p Hp);
+ cases (Hp (q_dist_ge_OQ ? ?)); 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 (q_le_to_diff_ge_OQ ?? (? H1)));
+ [1,3: intros; [right|left;symmetry] assumption]
+ simplify; apply q_le_minus; assumption;
+ |2,5: cases (value (q-Qpos (\fst b)) l1);
+ cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
+ [1,3: intros; [right|left;symmetry] assumption]
+ 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 (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
+ [1,3: intros; [right|left;symmetry] assumption]
+ assumption;]
+|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
+ rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
+|4: simplify; intros; split;
+ [1: apply sum_bases_empty_nat_of_q_le_q;
+ |2: apply sum_bases_empty_nat_of_q_le_q_one;
+ |3: elim (nat_of_q q); [reflexivity] simplify; assumption]]
+qed.
+
+
+definition same_values ≝
+ λl1,l2:q_f.
+ ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)).
+
+definition same_bases ≝
+ λl1,l2:q_f.
+ (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)).
+
+axiom q_lt_corefl: ∀x:Q.x < x → False.
+axiom q_lt_antisym: ∀x,y:Q.x < y → y < x → False.
+axiom q_neg_gt: ∀r:ratio.OQ < Qneg r → False.
+axiom q_d_x_x: ∀x:Q.ⅆ[x,x] = OQ.
+axiom q_pos_OQ: ∀x.Qpos x ≤ OQ → False.
+axiom q_lt_plus_trans:
+ ∀x,y:Q.OQ ≤ x → OQ < y → OQ < x + y.
+axiom q_pos_lt_OQ: ∀x.OQ < Qpos x.
+axiom q_le_plus_trans:
+ ∀x,y:Q. OQ ≤ x → OQ ≤ y → OQ ≤ x + y.
+axiom q_lt_trans: ∀x,y,z:Q. x < y → y < z → x < z.
+axiom q_le_trans: ∀x,y,z:Q. x ≤ y → y ≤ z → x ≤ z.
+axiom q_d_noabs: ∀x,y. x ≤ y → ⅆ[x,y] = y - x.
+axiom q_d_sym: ∀x,y. ⅆ[x,y] = ⅆ[y,x].
+axiom q_le_S: ∀x,y,z.OQ ≤ x → x + y ≤ z → y ≤ z.
+axiom q_plus_minus: ∀x.Qpos x + Qneg x = OQ.
+axiom q_minus: ∀x,y. y - Qpos x = y + Qneg x.
+axiom q_minus_r: ∀x,y. y + Qpos x = y - Qneg x.
+axiom q_plus_assoc: ∀x,y,z.x + (y + z) = x + y + z.
+
+lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
+intro; cases x; intros; [2:exists [apply r] reflexivity]
+cases (?:False);
+[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
+qed.
+
+notation < "\blacksquare" non associative with precedence 90 for @{'hide}.
+definition hide ≝ λT:Type.λx:T.x.
+interpretation "hide" 'hide = (hide _ _).
+
+lemma sum_bases_ge_OQ:
+ ∀l,n. OQ ≤ sum_bases (bars l) n.
+intro; elim (bars l); simplify; intros;
+[1: elim n; [left;reflexivity] simplify;
+ apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
+|2: cases n; [left;reflexivity] simplify;
+ apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
+qed.
+
+lemma sum_bases_O:
+ ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O.
+intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
+cases H;
+[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %);
+|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l);
+simplify; apply q_lt_plus_trans;
+try apply q_pos_lt_OQ;
+try apply (sum_bases_ge_OQ (mk_q_f OQ []));
+apply (sum_bases_ge_OQ (mk_q_f OQ l1));
+qed.
+
+lemma initial_shift_same_values:
+ ∀l1:q_f.∀init.init < start l1 →
+ same_values l1
+ (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
+[apply hide; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
+intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
+cases (unpos (start l1-init) H1); intro input;
+simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
+cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input);
+simplify in ⊢ (? ? ? (? ? ? %));
+cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
+whd in ⊢ (% → ?); simplify in H3;
+[1: intro; cases H4; clear H4; rewrite > H3;
+ cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
+ [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
+ |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
+ |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
+ rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
+ symmetry; apply le_n_O_to_eq;
+ rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]
+ clear H6 w2; simplify in H5:(? ? (? ? %));
+ destruct H3; rewrite > q_d_x_x in H5; assumption;]
+|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
+ cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
+ [1: cases (?:False); clear w2 H4 w1 H2 w H1;
+ apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
+ |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
+ |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
+ apply (q_lt_trans ??? H3 H);]
+|3: intro; cases H4; clear H4;
+ cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
+ [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+ simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+ cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+ cut (\fst w2 = O); [2: clear H10;
+ symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
+ apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x;
+ left; reflexivity;]
+ rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
+ cut (ⅆ[input,init] = Qpos w) as E; [2:
+ rewrite > H2; rewrite < H4; rewrite > q_d_sym;
+ rewrite > q_d_noabs; [reflexivity] right; assumption;]
+ cases (\fst w1) in H5 H6; intros;
+ [1: cases (?:False); clear H5; simplify in H6;
+ apply (q_lt_corefl ⅆ[input,init]);
+ rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
+ rewrite > q_plus_sym; assumption;
+ |2: cases n in H5 H6; [intros; reflexivity] intros;
+ cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
+ [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));]
+ apply (q_le_S ??? (sum_bases_ge_OQ (mk_q_f init ?) n1));[apply [];|3:apply l]
+ simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus w);
+ apply q_le_minus_r; rewrite < q_minus_r;
+ rewrite < E in ⊢ (??%); assumption;]
+ |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+ simplify in H5 H6 ⊢ %;
+ simplify in H5:(? ? (? ? %));
+
+
+
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
-alias symbol "pair" = "pair".
-definition rebase:
- q_f → q_f →
- ∃p:q_f × q_f.∀i.
- fst (nth (bars (fst p)) q00 i) =
- fst (nth (bars (snd p)) q00 i).
-intros (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
-letin aux ≝ (
-let rec aux (l1,l2:list (ℚ × ℚ)) on l1 : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
+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))
+ (same_bases (\fst p) (\snd p))
+ (same_values l1 (\fst p))
+ (same_values l2 (\snd p)).
+
+definition rebase_spec_simpl ≝
+ 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].
+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 ≝ (
+ 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 bar) (n:nat) on n : (list bar) × (list bar) ≝
+match n with
+[ O ⇒ 〈 nil ? , nil ? 〉
+| S m ⇒
match l1 with
- [ nil ⇒ 〈mk_list (λi.〈fst (nth l2 q00 i),OQ〉) (length ? l2),l2〉
- | cons he tl ⇒ 〈[],[]〉] in aux);
-
-cases (q_cmp s1 s2);
-[1: apply (mk_q_f s1);
-|2: apply (mk_q_f s1); cases l2;
- [1: letin l2' ≝ (
-[1: (* offset: the smallest one *)
- cases
+ [ nil ⇒ 〈cb0h l2, l2〉
+ | cons he1 tl1 ⇒
+ match l2 with
+ [ nil ⇒ 〈l1, cb0h l1〉
+ | cons he2 tl2 ⇒
+ 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 Hp ⇒
+ let rest ≝ base2 - base1 in
+ 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 (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
+ 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
+]]]]
+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 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: assumption;
+ |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
+ cases (value (mk_q_f s1 l2') input);
+ cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
+ whd in ⊢ (% → ?);
+ [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
+ cases (value (mk_q_f s2 l2) input);
+ cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
+ whd in ⊢ (% → ?);
+ [1: intros; cases H6; clear H6; change with (w1 = w);
+
+ (* TODO *) ]]
+|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption;
+|3:(* TODO *)
+|4:(* TODO *)
+|5:(* TODO *)
+|6:(* TODO *)
+|7:(* TODO *)
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff