src/HOL/GroupTheory/PiSets.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/GroupTheory/PiSets.ML	Mon Jul 23 17:37:29 2001 +0200
@@ -0,0 +1,66 @@
+(*  Title:      HOL/ex/PiSets.ML
+    ID:         $Id$
+    Author:     Florian Kammueller, University of Cambridge
+
+Pi sets and their application.
+*)
+
+(*** Bijection between Pi in terms of => and Pi in terms of Sigma ***)
+Goal "f \\<in> Pi A B ==> PiBij A B f <= Sigma A B";
+by (auto_tac (claset(), simpset() addsimps [PiBij_def,Pi_def]));
+qed "PiBij_subset_Sigma";
+
+Goal "f \\<in> Pi A B ==> \\<forall>x \\<in> A. \\<exists>!y. (x, y) \\<in> (PiBij A B f)";
+by (auto_tac (claset(), simpset() addsimps [PiBij_def]));
+qed "PiBij_unique";
+
+Goal "f \\<in> Pi A B ==> PiBij A B f \\<in> Graph A B";
+by (asm_simp_tac (simpset() addsimps [Graph_def,PiBij_unique,
+				      PiBij_subset_Sigma]) 1);
+qed "PiBij_in_Graph";
+
+Goalw [PiBij_def, Graph_def] "PiBij A B \\<in> Pi A B \\<rightarrow> Graph A B";
+by (rtac restrictI 1);
+by (auto_tac (claset(), simpset() addsimps [Pi_def]));
+qed "PiBij_func";
+
+Goal "inj_on (PiBij A B) (Pi A B)";
+by (rtac inj_onI 1);
+by (rtac Pi_extensionality 1);			
+by (assume_tac 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [PiBij_def]) 1);
+by (Blast_tac 1);
+qed "inj_PiBij";
+
+
+Goal "x \\<in> Graph A B \\<Longrightarrow> (lam a:A. SOME y. (a, y) \\<in> x) \\<in> Pi A B";
+by (rtac restrictI 1);
+by (res_inst_tac [("P", "%xa. (a, xa)\\<in>x")] ex1E 1);
+ by (force_tac (claset(), simpset() addsimps [Graph_def]) 1);
+by (full_simp_tac (simpset() addsimps [Graph_def]) 1);
+by (stac some_equality 1);
+  by (assume_tac 1);
+ by (Blast_tac 1);
+by (Blast_tac 1);
+qed "in_Graph_imp_in_Pi";
+
+Goal "PiBij A B ` (Pi A B) = Graph A B";
+by (rtac equalityI 1);
+by (force_tac (claset(), simpset() addsimps [PiBij_in_Graph]) 1);
+by (rtac subsetI 1);
+by (rtac image_eqI 1); 
+by (etac in_Graph_imp_in_Pi 2); 
+(* x = PiBij A B (lam a:A. @ y. (a, y)\\<in>x) *)
+by (asm_simp_tac (simpset() addsimps [in_Graph_imp_in_Pi, PiBij_def]) 1);
+by (auto_tac (claset(), simpset() addsimps [some1_equality, Graph_def])); 
+by (fast_tac (claset() addIs [someI2]) 1);
+qed "surj_PiBij";
+
+Goal "f \\<in> Pi A B ==> Inv (Pi A B) (PiBij A B) (PiBij A B f) = f";
+by (asm_simp_tac (simpset() addsimps [Inv_f_f, inj_PiBij]) 1);
+qed "PiBij_bij1";
+
+Goal "f \\<in> Graph A B ==> PiBij A B (Inv (Pi A B) (PiBij A B) f) = f";
+by (asm_simp_tac (simpset() addsimps [f_Inv_f, surj_PiBij]) 1);
+qed "PiBij_bij2";