include "Z/compare.ma".
include "Z/plus.ma".
include "nat/factorization.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.
-
-let rec fraction_of_nat_fact n ≝
- match n with
- [ nf_last m ⇒ pp m
- | nf_cons m l ⇒ cons (Z_of_nat (S m)) (fraction_of_nat_fact l)
- ].
-
-(* a fraction is integral if every coefficient is not negative *)
-let rec nat_fact_of_integral_fraction n ≝
- match n with
- [ pp n ⇒ nf_last n
- | nn _ ⇒ nf_last O (* dummy value *)
- | cons z l ⇒
- match z with
- [ OZ ⇒ nf_cons O (nat_fact_of_integral_fraction l)
- | pos n ⇒ nf_cons n (nat_fact_of_integral_fraction l)
- | neg n ⇒ nf_last O (* dummy value *)
- ]
- ].
-
-theorem nat_fact_of_integral_fraction_fraction_of_nat_fact:
- ∀n. nat_fact_of_integral_fraction (fraction_of_nat_fact n) = n.
- intro;
- elim n;
- [ reflexivity;
- | simplify;
- rewrite > H;
- reflexivity
- ]
-qed.
-
-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.
-
-definition Q_of_nat ≝
- λn.
- match factorize n with
- [ nfa_zero ⇒ OQ
- | nfa_one ⇒ Qpos one
- | nfa_proper l ⇒ Qpos (frac (fraction_of_nat_fact l))
- ].
+include "Q/fraction/fraction.ma".
let rec enumerator_integral_fraction l ≝
match l with
]
].
+(*
definition enumerator_of_fraction ≝
λq.
match q with
| Qpos r ⇒ denominator_of_fraction r
| Qneg r ⇒ denominator_of_fraction r
].
-
-(* 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.
- simplify.
- 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.
+*)
\ No newline at end of file
projects / helm.git / blobdiff