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 dist : ℚ → ℚ → ℚ.
+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 < "'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.
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].
[ 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)))
+ (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".
letin value ≝ (
let rec value (p: ℚ) (l : list bar) on l ≝
match l with
- [ nil ⇒ 〈O,OQ〉
+ [ nil ⇒ 〈nat_of_q p,OQ〉
| cons x tl ⇒
match q_cmp p (Qpos (\fst x)) with
[ q_lt _ ⇒ 〈O, \snd x〉
[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;
+ |*: 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; clear H2;
+ [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 H3; clear H3 H2 value;
+ |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 H3; clear H3 value H2;
+ |*: 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; split;
- [1:
+|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_shape ≝
+definition same_values ≝
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).
+ ∀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)]
+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.
-…┐─┌┐…
-\ldots\boxdl\boxh\boxdr\boxdl\ldots
+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;
+
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;
-
-
-
+ |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 / commitdiff