X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Ftests%2Fdecl.ma;h=569a2f8ffa12cf96d24d095c7062c294b26f7ec0;hb=5649890273cf8e660bba744e84ce5fee1e5efe69;hp=e2035d182a11bc87a7bc308e5b4ab18a52ff9cd7;hpb=bd277786d78d49594b3ada1c3c9c28cba5dc03b9;p=helm.git

diff --git a/helm/software/matita/tests/decl.ma b/helm/software/matita/tests/decl.ma
index e2035d182..569a2f8ff 100644
--- a/helm/software/matita/tests/decl.ma
+++ b/helm/software/matita/tests/decl.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/test/decl".
+
 
 include "nat/times.ma".
 include "nat/orders.ma".
@@ -72,16 +72,49 @@ theorem easy2: ∀n,m. n * m = O → n = O ∨ m = O.
   ]
 qed.
 
+theorem easy15: ∀n,m. n * m = O → n = O ∨ m = O.
+ assume n: nat.
+ assume m: nat.
+ (* base case *)
+ by _ we proved (O = O) (trivial).
+ by _ we proved (O = O ∨ m = O) (trivial2).
+ by _ we proved (O*m=O → O=O ∨ m=O) (base_case).
+ (* inductive case *)
+ we need to prove
+  (∀n1. (n1 * m = O → n1 = O ∨ m = O) → (S n1) * m = O → (S n1) = O ∨ m = O)
+  (inductive_case).
+   assume n1: nat.
+   suppose (n1 * m = O → n1 = O ∨ m = O) (inductive_hyp).
+   (* base case *)
+   by _ we proved (S n1 = O ∨ O = O) (pre_base_case2).
+   by _ we proved (S n1*O = O → S n1 = O ∨ O = O) (base_case2).
+   (* inductive case *)
+   we need to prove
+    (∀m1. (S n1 * m1 = O → S n1 = O ∨ m1 = O) →
+      (S n1 * S m1 = O → S n1 = O ∨ S m1 = O)) (inductive_hyp2).
+     assume m1: nat.
+     suppose (S n1 * m1 = O → S n1 = O ∨ m1 = O) (useless).
+     suppose (S n1 * S m1 = O) (absurd_hyp).
+     simplify in absurd_hyp.
+     by _ we proved (O = S (m1+n1*S m1)) (absurd_hyp').
+     (* BUG: automation failure *)
+     by (not_eq_O_S ? absurd_hyp') we proved False (the_absurd).
+     (* BUG: automation failure *)
+     by (False_ind ? the_absurd)
+   done.
+   (* the induction *)
+   by (nat_ind (λm.S n1 * m = O → S n1 = O ∨ m = O) base_case2 inductive_hyp2 m)
+ done.
+ (* the induction *)
+ by (nat_ind (λn.n*m=O → n=O ∨ m=O) base_case inductive_case n)
+done.
+qed.
 
 theorem easy3: ∀A:Prop. (A ∧ ∃n:nat.n ≠ n) → True.
  assume P: Prop.
  suppose (P ∧ ∃m:nat.m ≠ m) (H).
  by H we have P (H1) and (∃x:nat.x≠x) (H2).
- (*BUG:
  by H2 let q:nat such that (q ≠ q) (Ineq).
- *)
- (* the next line is wrong, but for the moment it does the job *)
- by H2 let q:nat such that False (Ineq).
  by I done.
 qed.
 
@@ -92,8 +125,78 @@ assume p:nat.
 suppose (n=m) (H).
 suppose (S m = S p) (K).
 suppose (n = S p) (L).
-obtain (S n) = (S m) by (eq_f ? ? ? ? ? H).
+conclude (S n) = (S m) by H.
              = (S p) by K.
-             = n by (sym_eq ? ? ? L)
+             = n by L
+done.
+qed.
+
+theorem easy45: ∀n,m,p. n = m → S m = S p → n = S p → S n = n.
+assume n: nat.
+assume m:nat.
+assume p:nat.
+suppose (n=m) (H).
+suppose (S m = S p) (K).
+suppose (n = S p) (L).
+conclude (S n) = (S m).
+             = (S p).
+             = n
 done.
 qed.
+
+theorem easy5: ∀n:nat. n*O=O.
+assume n: nat.
+(* Bug here: False should be n*0=0 *)
+we proceed by induction on n to prove False. 
+ case O.
+   the thesis becomes (O*O=O).
+   by (refl_eq ? O) done.
+ case S (m:nat).
+  by induction hypothesis we know (m*O=O) (I).
+  the thesis becomes (S m * O = O).
+  (* Bug here: missing that is equivalent to *)
+  simplify.
+  by I done.
+qed.
+
+inductive tree : Type ≝
+   Empty: tree
+ | Node: tree → tree → tree.
+ 
+let rec size t ≝
+ match t with
+  [ Empty ⇒ O
+  | (Node t1 t2) ⇒ S ((size t1) + (size t2))
+  ].
+  
+theorem easy6: ∀t. O ≮ O → O < size t → t ≠ Empty. 
+ assume t: tree.
+ suppose (O ≮ O) (trivial).
+ (*Bug here: False should be something else *)
+ we proceed by induction on t to prove False.
+  case Empty.
+    the thesis becomes (O < size Empty → Empty ≠ Empty).
+     suppose (O < size Empty) (absurd)
+     that is equivalent to (O < O).
+     (* Here the "natural" language is not natural at all *)
+     we proceed by induction on (trivial absurd) to prove False.
+  (*Bug here: this is what we want
+  case Node (t1:tree) (t2:tree).
+     by induction hypothesis we know (O < size t1 → t1 ≠ Empty) (Ht1).
+     by induction hypothesis we know (O < size t2 → t2 ≠ Empty) (Ht2). *)
+  (*This is the best we can do right now*)
+  case Node.
+   assume t1: tree.
+   by induction hypothesis we know (O < size t1 → t1 ≠ Empty) (Ht1).
+   assume t2: tree.
+   by induction hypothesis we know (O < size t2 → t2 ≠ Empty) (Ht2).
+   suppose (O < size (Node t1 t2)) (useless).
+   we need to prove (Node t1 t2 ≠ Empty) (final)
+   or equivalently (Node t1 t2 = Empty → False).
+     suppose (Node t1 t2 = Empty) (absurd).
+     (* Discriminate should really generate a theorem to be useful with
+        declarative tactics *)
+     destruct absurd.
+     by final
+   done.
+qed.