+sandwich.ma ordered_uniform.ma
+property_sigma.ma ordered_uniform.ma russell_support.ma
+uniform.ma supremum.ma
bishop_set.ma ordered_set.ma
-ordered_set.ma cprop_connectives.ma
-cprop_connectives.ma logic/equality.ma
-bishop_set_rewrite.ma bishop_set.ma
sequence.ma nat/nat.ma
-uniform.ma supremum.ma
-supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
-nat_ordered_set.ma bishop_set.ma nat/compare.ma
-property_sigma.ma ordered_uniform.ma russell_support.ma
ordered_uniform.ma uniform.ma
+supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
+bishop_set_rewrite.ma bishop_set.ma
+cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
+nat_ordered_set.ma bishop_set.ma nat/compare.ma
lebesgue.ma property_exhaustivity.ma sandwich.ma
-sandwich.ma ordered_uniform.ma
+ordered_set.ma cprop_connectives.ma
russell_support.ma cprop_connectives.ma nat/nat.ma
-models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
+models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/q_function.ma Q/q/q.ma cprop_connectives.ma list/list.ma
+models/q_support.ma Q/q/q.ma
models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
-models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
-models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma
+models/q_function.ma models/q_bars.ma
models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
+models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/list_support.ma list/list.ma
+models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
Q/q/q.ma
datatypes/constructors.ma
list/list.ma
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "list/list.ma".
+
+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}.
+
+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}.
+
+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.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "models/q_support.ma".
+include "models/list_support.ma".
+include "cprop_connectives.ma".
+
+definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
+record q_f : Type ≝ { start : ℚ; bars: list bar }.
+
+notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
+interpretation "Q x Q" 'q2 = (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).
+
+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; [apply q_lt_to_le|apply q_eq_to_le;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; [apply q_lt_to_le|apply q_eq_to_le;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; [apply q_lt_to_le|apply q_eq_to_le;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)).
+
+alias symbol "lt" = "Q less than".
+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; [apply q_eq_to_le;reflexivity] simplify;
+ apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
+|2: cases n; [apply q_eq_to_le;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 (q_le_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.
+
(* *)
(**************************************************************************)
-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_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).
-
-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]
- 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.
+include "models/q_bars.ma".
lemma initial_shift_same_values:
∀l1:q_f.∀init.init < start l1 →
|6:(* TODO *)
|7:(* TODO *)
|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
-qed.
\ No newline at end of file
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Q/q/q.ma".
+include "cprop_connectives.ma".
+
+notation "\rationals" non associative with precedence 99 for @{'q}.
+interpretation "Q" 'q = Q.
+
+(* group over Q *)
+axiom qp : ℚ → ℚ → ℚ.
+axiom qm : ℚ → ℚ → ℚ.
+
+interpretation "Q plus" 'plus x y = (qp x y).
+interpretation "Q minus" 'minus x y = (qm x y).
+
+axiom q_plus_OQ: ∀x:ℚ.x + OQ = x.
+axiom q_plus_sym: ∀x,y:ℚ.x + y = y + x.
+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.
+
+(* order over Q *)
+axiom qlt : ℚ → ℚ → CProp.
+axiom qle : ℚ → ℚ → CProp.
+interpretation "Q less than" 'lt x y = (qlt x y).
+interpretation "Q less or equal than" 'leq x y = (qle 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.
+
+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_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_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_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_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_le_S: ∀x,y,z.OQ ≤ x → x + y ≤ z → y ≤ z.
+axiom q_eq_to_le: ∀x,y. x = y → x ≤ y.
+
+inductive q_le_elimination (a,b:ℚ) : CProp ≝
+| q_le_from_eq : a = b → q_le_elimination a b
+| q_le_from_lt : a < b → q_le_elimination a b.
+
+axiom q_le_cases : ∀x,y:ℚ.x ≤ y → q_le_elimination x y.
+
+(* distance *)
+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_d_x_x: ∀x:Q.ⅆ[x,x] = OQ.
+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].
+
+(* integral part *)
+axiom nat_of_q: ℚ → nat.
+
projects / helm.git / commitdiff
