(* *)
(**************************************************************************)
-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 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).
-
-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).
-
-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]
+include "nat_ordered_set.ma".
+include "models/q_bars.ma".
+
+lemma sum_bars_increasing:
+ ∀l,x.sum_bases l x < sum_bases l (S x).
+intro; elim l;
+[1: elim x;
+ [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ apply q_pos_lt_OQ;
+ |2: simplify in H ⊢ %;
+ apply q_lt_plus; rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus; rewrite > q_plus_OQ;
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]]
+|2: elim x;
+ [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ apply q_pos_lt_OQ;
+ |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;
+ apply q_lt_plus; rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
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.
-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)]
+lemma q_lt_canc_plus_r:
+ ∀x,y,z:Q.x + z < y + z → x < y.
+intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
+rewrite > q_elim_minus; rewrite > q_plus_assoc;
+apply q_lt_plus; rewrite > q_elim_opp; assumption;
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;]]
+lemma q_lt_inj_plus_r:
+ ∀x,y,z:Q.x < y → x + z < y + z.
+intros; apply (q_lt_canc_plus_r ?? (Qopp z));
+do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
+rewrite > q_plus_minus;
+do 2 rewrite > q_plus_OQ; assumption;
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));
+lemma sum_bases_lt_canc:
+ ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
+intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
+generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
+intros 2;
+[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
+ apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
+|2: cases (?:False); simplify in H2;
+ apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
+ apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
+|1: cases n in H2; intro;
+ [1: cases (?:False); apply (q_lt_corefl ? H2);
+ |2: simplify; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_pos_lt_OQ;]]
qed.
+axiom q_minus_distrib:
+ ∀x,y,z:Q.x - (y + z) = x - y - z.
+
+axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
+
lemma initial_shift_same_values:
∀l1:q_f.∀init.init < start l1 →
same_values l1
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:(? ? (? ? %));
+ clear H6 w2; simplify in H5:(? ? (? ? %));
destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|2: intros;
-
+|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;
+ apply q_eq_to_le; 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] apply q_lt_to_le; 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 ? n1));[apply []|3:apply l]
+ simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
+ rewrite > q_elim_minus; apply q_le_minus_r;
+ rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
+ |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+ simplify in H5 H6 ⊢ %;
+ cases (\fst w1) in H5 H6; [intros; reflexivity]
+ cases (bars l1);
+ [1: intros; simplify; elim n [reflexivity] simplify; assumption;
+ |2: simplify; intros; cases (?:False); clear H6;
+ apply (q_lt_le_incompat (input - init) (Qpos w) );
+ [1: rewrite > H2; do 2 rewrite > q_elim_minus;
+ apply q_lt_plus; rewrite > q_elim_minus;
+ rewrite < q_plus_assoc; rewrite < q_elim_minus;
+ rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
+ |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
+ rewrite > q_d_sym; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
+ |3: 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;]
+ simplify in H5 H6;
+ cases (\fst w1) in H5 H6; intros;
+ [1: cases (?:False); clear H5 H9 H10; simplify in H6;
+ apply (q_lt_antisym input (start l1)); [2: assumption]
+ rewrite > q_d_sym in H6;
+ rewrite > q_d_noabs in H6; [2: apply q_lt_to_le; assumption]
+ rewrite > q_plus_sym in H6;
+ rewrite > q_plus_OQ in H6; rewrite > H2 in H6;
+ lapply (q_lt_plus ??? H6) as X; clear H6 H2 H3 H1 H H4 w1 w2 w;
+ rewrite > q_elim_minus in X; rewrite < q_plus_assoc in X;
+ rewrite > (q_plus_sym (Qopp init)) in X;
+ rewrite < q_elim_minus in X; rewrite > q_plus_minus in X;
+ rewrite > q_plus_OQ in X; assumption;
+ |2: simplify in H5; apply eq_f;
+ cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n)+Qpos w);[2:
+ apply (q_le_lt_trans ??? H9);
+ apply (q_lt_trans ??? ? H6);
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+ do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
+ apply q_lt_plus; rewrite > q_plus_sym;
+ rewrite > q_elim_minus; rewrite < q_plus_assoc;
+ rewrite < q_elim_minus; rewrite > q_plus_minus;
+ rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
+ clear H9 H6;
+ cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+ rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+ rewrite > H2; rewrite > (q_elim_minus (start ?));
+ rewrite > q_minus_distrib; rewrite > q_elim_opp;
+ do 2 rewrite > q_elim_minus;
+ do 2 rewrite < q_plus_assoc;
+ rewrite > (q_plus_sym ? init);
+ rewrite > (q_plus_assoc ? init);
+ rewrite > (q_plus_sym ? init);
+ rewrite < (q_elim_minus init); rewrite > q_plus_minus;
+ rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+ rewrite < q_elim_minus; reflexivity;]
+ cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
+ apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
+ rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
+ assumption;] clear Hcut1 H5 H10;
+ generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2;
+ apply (nat_elim2 ???? n (\fst w2));
+ [3: intros (x y); apply eq_f; apply H5; clear H5;
+ [1: clear H7; apply sum_bases_lt_canc; assumption;
+ |2: clear H6; ]
+ |2: intros; cases (?:False); clear H6;
+ cases n1 in H5; intro;
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+ |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
+ elim n2 in H5 H6;
+
+
+ elim (bars l1) 0;
+ [1: intro; elim n1; [reflexivity] cases (?:False);
+
+
+ intros; clear H5;
+ elim n1 in H6; [reflexivity] cases (?:False);
+ [1: apply (q_lt_corefl ? H5);
+ |2: cases (bars l1) in H5; intro;
+ [1: simplify in H5;
+ apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+ apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+ apply q_le_OQ_Qpos;
+ |2: simplify in H5:(??%);
+ lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+ apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+qed.
+
+
+
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
definition rebase_spec ≝
|6:(* TODO *)
|7:(* TODO *)
|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
-qed.
\ No newline at end of file
+qed.
projects / helm.git / blobdiff