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(* Title: HOL/ex/PiSets.thy


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ID: $Id$


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Author: Florian Kammueller, University of Cambridge


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Pi sets and their application.


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*)


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(*** Bijection between Pi in terms of => and Pi in terms of Sigma ***)


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Goal "f: Pi A B ==> PiBij A B f <= Sigma A B";


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by (auto_tac (claset(),


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simpset() addsimps [PiBij_def,Pi_def,restrict_apply1]));


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qed "PiBij_subset_Sigma";

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Goal "f: Pi A B ==> (! x: A. (?! y. (x, y): (PiBij A B f)))";


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by (auto_tac (claset(),


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simpset() addsimps [PiBij_def,restrict_apply1]));


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qed "PiBij_unique";

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Goal "f: Pi A B ==> PiBij A B f : Graph A B";


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by (asm_simp_tac (simpset() addsimps [Graph_def,PiBij_unique,


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PiBij_subset_Sigma]) 1);


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qed "PiBij_in_Graph";

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Goalw [PiBij_def, Graph_def] "PiBij A B: Pi A B > Graph A B";

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by (rtac restrictI 1);

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by (auto_tac (claset(), simpset() addsimps [Pi_def]));


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qed "PiBij_func";

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Goal "inj_on (PiBij A B) (Pi A B)";

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by (rtac inj_onI 1);


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by (rtac Pi_extensionality 1);


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by (assume_tac 1);


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by (assume_tac 1);

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by (rotate_tac 1 1);


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by (asm_full_simp_tac (simpset() addsimps [PiBij_def,restrict_apply1]) 1);


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by (blast_tac (claset() addEs [equalityE]) 1);


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qed "inj_PiBij";

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Goal "PiBij A B `` (Pi A B) = Graph A B";

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by (rtac equalityI 1);

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by (force_tac (claset(), simpset() addsimps [image_def,PiBij_in_Graph]) 1);

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by (rtac subsetI 1);

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by (asm_full_simp_tac (simpset() addsimps [image_def]) 1);

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by (res_inst_tac [("x","lam a: A. @ y. (a, y): x")] bexI 1);

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by (rtac restrictI 2);


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by (res_inst_tac [("P", "%xa. (a, xa) : x")] ex1E 2);


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by (force_tac (claset(), simpset() addsimps [Graph_def]) 2);


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by (full_simp_tac (simpset() addsimps [Graph_def]) 2);


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by (stac select_equality 2);

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by (assume_tac 2);


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by (Blast_tac 2);


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by (Blast_tac 2);

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(* x = PiBij A B (lam a:A. @ y. (a, y) : x) *)

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by (full_simp_tac (simpset() addsimps [PiBij_def,Graph_def]) 1);


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by (stac restrict_apply1 1);


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by (rtac restrictI 1);


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by (blast_tac (claset() addSDs [[select_eq_Ex, ex1_implies_ex] MRS iffD2]) 1);


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(** LEVEL 17 **)

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by (rtac equalityI 1);


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by (rtac subsetI 1);

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by (split_all_tac 1);


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by (subgoal_tac "a: A" 1);


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by (Blast_tac 2);


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by (asm_full_simp_tac (simpset() addsimps [restrict_apply1]) 1);


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(*Blast_tac: PROOF FAILED for depth 5*)


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by (fast_tac (claset() addSIs [select1_equality RS sym]) 1);

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(* {(xa,y). xa : A & y = (lam a:A. @ y. (a, y) : x) xa} <= x *)

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by (Clarify_tac 1);


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by (asm_full_simp_tac (simpset() addsimps [restrict_apply1]) 1);


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by (fast_tac (claset() addIs [selectI2]) 1);


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qed "surj_PiBij";

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Goal "f: Pi A B ==> \


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\ (lam y: Graph A B. (Inv (Pi A B)(PiBij A B)) y)(PiBij A B f) = f";


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by (asm_simp_tac


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(simpset() addsimps [Inv_f_f, PiBij_func, inj_PiBij, surj_PiBij]) 1);


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qed "PiBij_bij1";

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Goal "[ f: Graph A B ] ==> \

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\ (PiBij A B) ((lam y: (Graph A B). (Inv (Pi A B)(PiBij A B)) y) f) = f";

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by (rtac (PiBij_func RS f_Inv_f) 1);


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by (asm_full_simp_tac (simpset() addsimps [surj_PiBij]) 1);

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by (assume_tac 1);

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qed "PiBij_bij2";

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