X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2FQ%2Fq.ma;h=340154979777529f94387cd1c630c6234cf21ab5;hb=97c2d258a5c524eb5c4b85208899d80751a2c82f;hp=0e6678dc935e2910c6abbf2ad3d5e653d62c6c9e;hpb=4d6bf14d95252ecfd1aaca4568184b4d6dee9b24;p=helm.git

diff --git a/helm/matita/library/Q/q.ma b/helm/matita/library/Q/q.ma
index 0e6678dc9..340154979 100644
--- a/helm/matita/library/Q/q.ma
+++ b/helm/matita/library/Q/q.ma
@@ -16,7 +16,6 @@ set "baseuri" "cic:/matita/Q/q".
 
 include "Z/compare.ma".
 include "Z/plus.ma".
-include "higher_order_defs/functions.ma".
 
 (* a fraction is a list of Z-coefficients for primes, in natural
 order. The last coefficient must eventually be different from 0 *)
@@ -45,11 +44,13 @@ theorem fraction_elim2:
 (\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.
-elim g.apply H2.
-apply H4.apply H5.
-apply H3.
-apply H1.
+intros 7.elim f.
+  apply H.
+  apply H1.
+  elim g.
+    apply H2.
+    apply H3.
+    apply H4.apply H5.
 qed. 
 
 (* boolean equality *)
@@ -91,33 +92,33 @@ definition ftl \def
     | (cons x f) \Rightarrow f].
     
 theorem injective_pp : injective nat fraction pp.
-simplify.intros.
-change with (aux (pp x)) = (aux (pp y)).
+unfold injective.intros.
+change with ((aux (pp x)) = (aux (pp y))).
 apply eq_f.assumption.
 qed.
 
 theorem injective_nn : injective nat fraction nn.
-simplify.intros.
-change with (aux (nn x)) = (aux (nn y)).
+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)).
+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)).
+change with ((ftl (cons x f)) = (ftl (cons y g))).
 apply eq_f.assumption.
 qed.
 
-theorem not_eq_pp_nn: \forall n,m:nat. \lnot (pp n) = (nn m).
-intros.simplify. intro.
+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
@@ -128,8 +129,8 @@ qed.
 
 theorem not_eq_pp_cons: 
 \forall n:nat.\forall x:Z. \forall f:fraction. 
-\lnot (pp n) = (cons x f).
-intros.simplify. intro.
+pp n \neq cons x f.
+intros.unfold Not. intro.
 change with match (pp n) with
 [ (pp n) \Rightarrow False
 | (nn n) \Rightarrow True
@@ -140,8 +141,8 @@ qed.
 
 theorem not_eq_nn_cons: 
 \forall n:nat.\forall x:Z. \forall f:fraction. 
-\lnot (nn n) = (cons x f).
-intros.simplify. intro.
+nn n \neq cons x f.
+intros.unfold Not. intro.
 change with match (nn n) with
 [ (pp n) \Rightarrow True
 | (nn n) \Rightarrow False
@@ -152,66 +153,60 @@ qed.
 
 theorem decidable_eq_fraction: \forall f,g:fraction.
 decidable (f = g).
-intros.simplify.
-apply fraction_elim2 (\lambda f,g. Or (f=g) (f=g \to False)).
-intros.elim g1.
-elim ((decidable_eq_nat n n1) : Or (n=n1) (n=n1 \to False)).
-left.apply eq_f. assumption.
-right.intro.apply H.apply injective_pp.assumption.
-right.apply not_eq_pp_cons.
-right.apply not_eq_pp_nn.
-intros. elim g1.
-right.intro.apply not_eq_pp_nn n1 n ?.apply sym_eq. assumption.
-right.apply not_eq_nn_cons.
-elim ((decidable_eq_nat n n1) : Or (n=n1) (n=n1 \to False)).
-left. apply eq_f. assumption.
-right.intro.apply H.apply injective_nn.assumption.
-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) : Or (x=y) (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.
+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 \lnot f=g].
-intros.apply fraction_elim2 
+|false \Rightarrow f \neq g].
+intros.apply (fraction_elim2 
 (\lambda f,g.match (eqfb f g) with
 [true \Rightarrow f=g
-|false \Rightarrow \lnot f=g]).
-intros.elim g1.
-simplify.
-apply eqb_elim.
-intro.simplify.apply eq_f.assumption.
-intro.simplify.intro.apply H.apply injective_pp.assumption.
-simplify.apply not_eq_pp_cons.
-simplify.apply not_eq_pp_nn.
-intros.
-elim g1.simplify.
-intro.apply not_eq_pp_nn n1 n ?.apply sym_eq. assumption.
-simplify.apply not_eq_nn_cons.
-simplify.apply eqb_elim.
-intro.simplify.apply eq_f.assumption.
-intro.simplify.intro.apply H.apply injective_nn.assumption.
-intros.
-simplify.intro.apply not_eq_pp_cons m x f1 ?.apply sym_eq. assumption.
-intros.simplify.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.intro.apply H2.apply eq_cons_to_eq2 x y.assumption.
-intro.simplify.
-intro.apply H1.apply eq_cons_to_eq1 f1 g1.assumption.
+|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
@@ -230,72 +225,66 @@ let rec ftimes f g \def
   match f with
   [ (pp n) \Rightarrow 
     match g with
-    [(pp m) \Rightarrow Z_to_ratio (Zplus (pos n) (pos m))
-    | (nn m) \Rightarrow Z_to_ratio (Zplus (pos n) (neg m))
-    | (cons y g1) \Rightarrow frac (cons (Zplus (pos n) y) g1)]
+    [(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 (Zplus (neg n) (pos m))
-    | (nn m) \Rightarrow Z_to_ratio (Zplus (neg n) (neg m))
-    | (cons y g1) \Rightarrow frac (cons (Zplus (neg n) y) g1)]
+    [(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 (Zplus x (pos m)) f1)
-    | (nn m) \Rightarrow frac (cons (Zplus x (neg m)) f1)
+    [ (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 (Zplus x y)
-        | (frac h) \Rightarrow frac (cons (Zplus x y) h)]]].
+        [ one \Rightarrow Z_to_ratio (x + y)
+        | (frac h) \Rightarrow frac (cons (x + y) h)]]].
         
 theorem symmetric2_ftimes: symmetric2 fraction ratio ftimes.
-simplify. intros.
-apply fraction_elim2 (\lambda f,g.ftimes f g = ftimes g f).
-intros.elim g.
-change with Z_to_ratio (Zplus (pos n) (pos n1)) = Z_to_ratio (Zplus (pos n1) (pos n)).
-apply eq_f.apply sym_Zplus.
-change with frac (cons (Zplus (pos n) z) f) = frac (cons (Zplus z (pos n)) f).
-rewrite < sym_Zplus.reflexivity.
-change with Z_to_ratio (Zplus (pos n) (neg n1)) = Z_to_ratio (Zplus (neg n1) (pos n)).
-apply eq_f.apply sym_Zplus.
-intros.elim g.
-change with Z_to_ratio (Zplus (neg n) (pos n1)) = Z_to_ratio (Zplus (pos n1) (neg n)).
-apply eq_f.apply sym_Zplus.
-change with frac (cons (Zplus (neg n) z) f) = frac (cons (Zplus z (neg n)) f).
-rewrite < sym_Zplus.reflexivity.
-change with Z_to_ratio (Zplus (neg n) (neg n1)) = Z_to_ratio (Zplus (neg n1) (neg n)).
-apply eq_f.apply sym_Zplus.
-intros.
-change with frac (cons (Zplus x1 (pos m)) f) = frac (cons (Zplus (pos m) x1) f).
-rewrite < sym_Zplus.reflexivity.
-intros.
-change with frac (cons (Zplus x1 (neg m)) f) = frac (cons (Zplus (neg m) x1) f).
-rewrite < sym_Zplus.reflexivity.
-intros.
-change with 
-  match ftimes f g with
-  [ one \Rightarrow Z_to_ratio (Zplus x1 y1)
-  | (frac h) \Rightarrow frac (cons (Zplus x1 y1) h)] =
-  match ftimes g f with
-  [ one \Rightarrow Z_to_ratio (Zplus y1 x1)
-  | (frac h) \Rightarrow frac (cons (Zplus y1 x1) h)].
-rewrite < H.rewrite < sym_Zplus.
-reflexivity.
+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 (Zplus (pos n) (Zopp (pos n))) = one.
-rewrite > Zplus_Zopp.reflexivity.
+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 (Zplus z (Zopp z))
- | (frac h) \Rightarrow frac (cons (Zplus z (Zopp z)) h)] = one.
-rewrite > H.
-rewrite > Zplus_Zopp.reflexivity.
-change with Z_to_ratio (Zplus (neg n) (Zopp (neg n))) = one.
-rewrite > Zplus_Zopp.reflexivity.
+  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
@@ -308,7 +297,7 @@ definition rtimes : ratio \to ratio \to ratio \def
       | (frac g) \Rightarrow ftimes f g]].
 
 theorem symmetric_rtimes : symmetric ratio rtimes.
-change with \forall r,s:ratio. rtimes r s = rtimes s r.
+change with (\forall r,s:ratio. rtimes r s = rtimes s r).
 intros.
 elim r. elim s.
 reflexivity.
@@ -328,4 +317,4 @@ theorem rtimes_rinv: \forall r:ratio. rtimes r (rinv r) = one.
 intro.elim r.
 reflexivity.
 simplify.apply ftimes_finv.
-qed.
\ No newline at end of file
+qed.