Theorem sigma_p_knm changed into generic_iter_p_knm.

[helm.git] / helm / software / matita / library / nat / iteration2.ma

diff --git a/helm/software/matita/library/nat/iteration2.ma b/helm/software/matita/library/nat/iteration2.ma
index 3a6bab3969a6dde7c603b1df62fb01773977252a..0230362e72aaa9610ffb25034c21f0028d4e8b69 100644 (file)
--- a/helm/software/matita/library/nat/iteration2.ma
+++ b/helm/software/matita/library/nat/iteration2.ma
@@ -173,7 +173,7 @@ theorem eq_sigma_p_gh:
 (\forall j. j < n1 \to p2 j = true \to p1 (h1 j) = true) \to
 (\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to 
 (\forall j. j < n1 \to p2 j = true \to h1 j < n) \to 
-sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
+sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 p2 g.
 intros.
 unfold sigma_p.
 apply (eq_iter_p_gen_gh nat O plus ? ? ? g h h1 n n1 p1 p2)
@@ -580,3 +580,67 @@ rewrite > (S_pred ((S n)*(S m))) in \vdash (? ? (? % ? ?) ?)
 | apply lt_O_times_S_S
 ]
 qed.
+
+theorem sigma_p_knm: 
+\forall g: nat \to nat.
+\forall h2:nat \to nat \to nat.
+\forall h11,h12:nat \to nat. 
+\forall k,n,m.
+\forall p1,p21:nat \to bool.
+\forall p22:nat \to nat \to bool.
+(\forall x. x < k \to p1 x = true \to 
+p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true
+\land h2 (h11 x) (h12 x) = x 
+\land (h11 x) < n \land (h12 x) < m) \to
+(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to 
+p1 (h2 i j) = true \land 
+h11 (h2 i j) = i \land h12 (h2 i j) = j
+\land h2 i j < k) \to
+sigma_p k p1 g=
+sigma_p n p21 (\lambda x:nat.sigma_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+apply iter_p_gen_knm
+  [apply symmetricIntPlus
+  |apply associative_plus
+  |intro.rewrite < plus_n_O.reflexivity
+  |exact h11
+  |exact h12
+  |assumption
+  |assumption
+  ]
+qed.
+  
+  
+theorem sigma_p2_eq: 
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat. 
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to 
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to 
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) =
+sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?).
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+
+apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intro.
+  rewrite < (plus_n_O).
+  reflexivity
+| assumption
+| assumption
+]
+qed.
\ No newline at end of file