+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
-(* ||A|| E.Tassi, S.Zacchiroli *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU Lesser General Public License Version 2.1 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Q/q".
-
-include "Z/compare.ma".
-include "Z/plus.ma".
-
-(* a fraction is a list of Z-coefficients for primes, in natural
-order. The last coefficient must eventually be different from 0 *)
-
-inductive fraction : Set \def
- pp : nat \to fraction
-| nn: nat \to fraction
-| cons : Z \to fraction \to fraction.
-
-inductive ratio : Set \def
- one : ratio
- | frac : fraction \to ratio.
-
-(* a rational number is either O or a ratio with a sign *)
-inductive Q : Set \def
- OQ : Q
- | Qpos : ratio \to Q
- | Qneg : ratio \to Q.
-
-(* double elimination principles *)
-theorem fraction_elim2:
-\forall R:fraction \to fraction \to Prop.
-(\forall n:nat.\forall g:fraction.R (pp n) g) \to
-(\forall n:nat.\forall g:fraction.R (nn n) g) \to
-(\forall x:Z.\forall f:fraction.\forall m:nat.R (cons x f) (pp m)) \to
-(\forall x:Z.\forall f:fraction.\forall m:nat.R (cons x f) (nn m)) \to
-(\forall x,y:Z.\forall f,g:fraction.R f g \to R (cons x f) (cons y g)) \to
-\forall f,g:fraction. R f g.
-intros 7.elim f.
- apply H.
- apply H1.
- elim g.
- apply H2.
- apply H3.
- apply H4.apply H5.
-qed.
-
-(* boolean equality *)
-let rec eqfb f g \def
-match f with
-[ (pp n) \Rightarrow
- match g with
- [ (pp m) \Rightarrow eqb n m
- | (nn m) \Rightarrow false
- | (cons y g1) \Rightarrow false]
-| (nn n) \Rightarrow
- match g with
- [ (pp m) \Rightarrow false
- | (nn m) \Rightarrow eqb n m
- | (cons y g1) \Rightarrow false]
-| (cons x f1) \Rightarrow
- match g with
- [ (pp m) \Rightarrow false
- | (nn m) \Rightarrow false
- | (cons y g1) \Rightarrow andb (eqZb x y) (eqfb f1 g1)]].
-
-(* discrimination *)
-definition aux \def
- \lambda f. match f with
- [ (pp n) \Rightarrow n
- | (nn n) \Rightarrow n
- | (cons x f) \Rightarrow O].
-
-definition fhd \def
-\lambda f. match f with
- [ (pp n) \Rightarrow (pos n)
- | (nn n) \Rightarrow (neg n)
- | (cons x f) \Rightarrow x].
-
-definition ftl \def
-\lambda f. match f with
- [ (pp n) \Rightarrow (pp n)
- | (nn n) \Rightarrow (nn n)
- | (cons x f) \Rightarrow f].
-
-theorem injective_pp : injective nat fraction pp.
-unfold injective.intros.
-change with ((aux (pp x)) = (aux (pp y))).
-apply eq_f.assumption.
-qed.
-
-theorem injective_nn : injective nat fraction nn.
-unfold injective.intros.
-change with ((aux (nn x)) = (aux (nn y))).
-apply eq_f.assumption.
-qed.
-
-theorem eq_cons_to_eq1: \forall f,g:fraction.\forall x,y:Z.
-(cons x f) = (cons y g) \to x = y.
-intros.
-change with ((fhd (cons x f)) = (fhd (cons y g))).
-apply eq_f.assumption.
-qed.
-
-theorem eq_cons_to_eq2: \forall x,y:Z.\forall f,g:fraction.
-(cons x f) = (cons y g) \to f = g.
-intros.
-change with ((ftl (cons x f)) = (ftl (cons y g))).
-apply eq_f.assumption.
-qed.
-
-theorem not_eq_pp_nn: \forall n,m:nat. pp n \neq nn m.
-intros.unfold Not. intro.
-change with match (pp n) with
-[ (pp n) \Rightarrow False
-| (nn n) \Rightarrow True
-| (cons x f) \Rightarrow True].
-rewrite > H.
-simplify.exact I.
-qed.
-
-theorem not_eq_pp_cons:
-\forall n:nat.\forall x:Z. \forall f:fraction.
-pp n \neq cons x f.
-intros.unfold Not. intro.
-change with match (pp n) with
-[ (pp n) \Rightarrow False
-| (nn n) \Rightarrow True
-| (cons x f) \Rightarrow True].
-rewrite > H.
-simplify.exact I.
-qed.
-
-theorem not_eq_nn_cons:
-\forall n:nat.\forall x:Z. \forall f:fraction.
-nn n \neq cons x f.
-intros.unfold Not. intro.
-change with match (nn n) with
-[ (pp n) \Rightarrow True
-| (nn n) \Rightarrow False
-| (cons x f) \Rightarrow True].
-rewrite > H.
-simplify.exact I.
-qed.
-
-theorem decidable_eq_fraction: \forall f,g:fraction.
-decidable (f = g).
-intros.unfold decidable.
-apply (fraction_elim2 (\lambda f,g. f=g \lor (f=g \to False))).
- intros.elim g1.
- elim ((decidable_eq_nat n n1) : n=n1 \lor (n=n1 \to False)).
- left.apply eq_f. assumption.
- right.intro.apply H.apply injective_pp.assumption.
- right.apply not_eq_pp_nn.
- right.apply not_eq_pp_cons.
- intros. elim g1.
- right.intro.apply (not_eq_pp_nn n1 n).apply sym_eq. assumption.
- elim ((decidable_eq_nat n n1) : n=n1 \lor (n=n1 \to False)).
- left. apply eq_f. assumption.
- right.intro.apply H.apply injective_nn.assumption.
- right.apply not_eq_nn_cons.
- intros.right.intro.apply (not_eq_pp_cons m x f1).apply sym_eq.assumption.
- intros.right.intro.apply (not_eq_nn_cons m x f1).apply sym_eq.assumption.
- intros.elim H.
- elim ((decidable_eq_Z x y) : x=y \lor (x=y \to False)).
- left.apply eq_f2.assumption.
- assumption.
- right.intro.apply H2.apply (eq_cons_to_eq1 f1 g1).assumption.
- right.intro.apply H1.apply (eq_cons_to_eq2 x y f1 g1).assumption.
-qed.
-
-theorem eqfb_to_Prop: \forall f,g:fraction.
-match (eqfb f g) with
-[true \Rightarrow f=g
-|false \Rightarrow f \neq g].
-intros.apply (fraction_elim2
-(\lambda f,g.match (eqfb f g) with
-[true \Rightarrow f=g
-|false \Rightarrow f \neq g])).
- intros.elim g1.
- simplify.apply eqb_elim.
- intro.simplify.apply eq_f.assumption.
- intro.simplify.unfold Not.intro.apply H.apply injective_pp.assumption.
- simplify.apply not_eq_pp_nn.
- simplify.apply not_eq_pp_cons.
- intros.elim g1.
- simplify.unfold Not.intro.apply (not_eq_pp_nn n1 n).apply sym_eq. assumption.
- simplify.apply eqb_elim.intro.simplify.apply eq_f.assumption.
- intro.simplify.unfold Not.intro.apply H.apply injective_nn.assumption.
- simplify.apply not_eq_nn_cons.
- intros.simplify.unfold Not.intro.apply (not_eq_pp_cons m x f1).apply sym_eq. assumption.
- intros.simplify.unfold Not.intro.apply (not_eq_nn_cons m x f1).apply sym_eq. assumption.
- intros.
- change in match (eqfb (cons x f1) (cons y g1))
- with (andb (eqZb x y) (eqfb f1 g1)).
- apply eqZb_elim.
- intro.generalize in match H.elim (eqfb f1 g1).
- simplify.apply eq_f2.assumption.
- apply H2.
- simplify.unfold Not.intro.apply H2.apply (eq_cons_to_eq2 x y).assumption.
- intro.simplify.unfold Not.intro.apply H1.apply (eq_cons_to_eq1 f1 g1).assumption.
-qed.
-
-let rec finv f \def
- match f with
- [ (pp n) \Rightarrow (nn n)
- | (nn n) \Rightarrow (pp n)
- | (cons x g) \Rightarrow (cons (Zopp x) (finv g))].
-
-definition Z_to_ratio :Z \to ratio \def
-\lambda x:Z. match x with
-[ OZ \Rightarrow one
-| (pos n) \Rightarrow frac (pp n)
-| (neg n) \Rightarrow frac (nn n)].
-
-let rec ftimes f g \def
- match f with
- [ (pp n) \Rightarrow
- match g with
- [(pp m) \Rightarrow Z_to_ratio (pos n + pos m)
- | (nn m) \Rightarrow Z_to_ratio (pos n + neg m)
- | (cons y g1) \Rightarrow frac (cons (pos n + y) g1)]
- | (nn n) \Rightarrow
- match g with
- [(pp m) \Rightarrow Z_to_ratio (neg n + pos m)
- | (nn m) \Rightarrow Z_to_ratio (neg n + neg m)
- | (cons y g1) \Rightarrow frac (cons (neg n + y) g1)]
- | (cons x f1) \Rightarrow
- match g with
- [ (pp m) \Rightarrow frac (cons (x + pos m) f1)
- | (nn m) \Rightarrow frac (cons (x + neg m) f1)
- | (cons y g1) \Rightarrow
- match ftimes f1 g1 with
- [ one \Rightarrow Z_to_ratio (x + y)
- | (frac h) \Rightarrow frac (cons (x + y) h)]]].
-
-theorem symmetric2_ftimes: symmetric2 fraction ratio ftimes.
-unfold symmetric2. intros.apply (fraction_elim2 (\lambda f,g.ftimes f g = ftimes g f)).
- intros.elim g.
- change with (Z_to_ratio (pos n + pos n1) = Z_to_ratio (pos n1 + pos n)).
- apply eq_f.apply sym_Zplus.
- change with (Z_to_ratio (pos n + neg n1) = Z_to_ratio (neg n1 + pos n)).
- apply eq_f.apply sym_Zplus.
- change with (frac (cons (pos n + z) f) = frac (cons (z + pos n) f)).
- rewrite < sym_Zplus.reflexivity.
- intros.elim g.
- change with (Z_to_ratio (neg n + pos n1) = Z_to_ratio (pos n1 + neg n)).
- apply eq_f.apply sym_Zplus.
- change with (Z_to_ratio (neg n + neg n1) = Z_to_ratio (neg n1 + neg n)).
- apply eq_f.apply sym_Zplus.
- change with (frac (cons (neg n + z) f) = frac (cons (z + neg n) f)).
- rewrite < sym_Zplus.reflexivity.
- intros.change with (frac (cons (x1 + pos m) f) = frac (cons (pos m + x1) f)).
- rewrite < sym_Zplus.reflexivity.
- intros.change with (frac (cons (x1 + neg m) f) = frac (cons (neg m + x1) f)).
- rewrite < sym_Zplus.reflexivity.
- intros.
- change with
- (match ftimes f g with
- [ one \Rightarrow Z_to_ratio (x1 + y1)
- | (frac h) \Rightarrow frac (cons (x1 + y1) h)] =
- match ftimes g f with
- [ one \Rightarrow Z_to_ratio (y1 + x1)
- | (frac h) \Rightarrow frac (cons (y1 + x1) h)]).
- rewrite < H.rewrite < sym_Zplus.reflexivity.
-qed.
-
-theorem ftimes_finv : \forall f:fraction. ftimes f (finv f) = one.
-intro.elim f.
- change with (Z_to_ratio (pos n + - (pos n)) = one).
- rewrite > Zplus_Zopp.reflexivity.
- change with (Z_to_ratio (neg n + - (neg n)) = one).
- rewrite > Zplus_Zopp.reflexivity.
-(* again: we would need something to help finding the right change *)
- change with
- (match ftimes f1 (finv f1) with
- [ one \Rightarrow Z_to_ratio (z + - z)
- | (frac h) \Rightarrow frac (cons (z + - z) h)] = one).
- rewrite > H.rewrite > Zplus_Zopp.reflexivity.
-qed.
-
-definition rtimes : ratio \to ratio \to ratio \def
-\lambda r,s:ratio.
- match r with
- [one \Rightarrow s
- | (frac f) \Rightarrow
- match s with
- [one \Rightarrow frac f
- | (frac g) \Rightarrow ftimes f g]].
-
-theorem symmetric_rtimes : symmetric ratio rtimes.
-change with (\forall r,s:ratio. rtimes r s = rtimes s r).
-intros.
-elim r. elim s.
-reflexivity.
-reflexivity.
-elim s.
-reflexivity.
-simplify.apply symmetric2_ftimes.
-qed.
-
-definition rinv : ratio \to ratio \def
-\lambda r:ratio.
- match r with
- [one \Rightarrow one
- | (frac f) \Rightarrow frac (finv f)].
-
-theorem rtimes_rinv: \forall r:ratio. rtimes r (rinv r) = one.
-intro.elim r.
-reflexivity.
-simplify.apply ftimes_finv.
-qed.
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