src/ZF/AC/AC2_AC6.ML

 
 (* ********************************************************************** *)
 (* AC1 ==> AC2                                                            *)
 (* ********************************************************************** *)
 
-goal thy "!!B. [| B:A; f:(PROD X:A. X); 0~:A |]  \
+Goal "!!B. [| B:A; f:(PROD X:A. X); 0~:A |]  \
 \               ==> {f`B} <= B Int {f`C. C:A}";
 by (fast_tac (claset() addSEs [apply_type]) 1);
 val lemma1 = result();
 
-goalw thy [pairwise_disjoint_def]
+Goalw [pairwise_disjoint_def]
         "!!A. [| pairwise_disjoint(A); B:A; C:A; D:B; D:C |] ==> f`B = f`C";
 by (fast_tac (claset() addSEs [equals0D]) 1);
 val lemma2 = result();
 
-goalw thy AC_defs "!!Z. AC1 ==> AC2"; 
+Goalw AC_defs "!!Z. AC1 ==> AC2"; 
 by (rtac allI 1);
 by (rtac impI 1);
 by (REPEAT (eresolve_tac [asm_rl,conjE,allE,exE,impE] 1));
 by (REPEAT (resolve_tac [exI,ballI,equalityI] 1));
 by (rtac lemma1 2 THEN (REPEAT (assume_tac 2)));

 
 (* ********************************************************************** *)
 (* AC2 ==> AC1                                                            *)
 (* ********************************************************************** *)
 
-goal thy "!!A. 0~:A ==> 0 ~: {B*{B}. B:A}";
+Goal "!!A. 0~:A ==> 0 ~: {B*{B}. B:A}";
 by (fast_tac (claset() addSDs [sym RS (Sigma_empty_iff RS iffD1)]
         addSEs [RepFunE, equals0D]) 1);
 val lemma1 = result();
 
-goal thy "!!A. [| X*{X} Int C = {y}; X:A |]  \
+Goal "!!A. [| X*{X} Int C = {y}; X:A |]  \
 \               ==> (THE y. X*{X} Int C = {y}): X*A";
 by (rtac subst_elem 1);
 by (fast_tac (claset() addSIs [the_equality]
                 addSEs [sym RS trans RS (singleton_eq_iff RS iffD1)]) 2);
 by (fast_tac (claset() addSEs [equalityE, make_elim singleton_subsetD]) 1);
 val lemma2 = result();
 
-goal thy "!!A. ALL D:{E*{E}. E:A}. EX y. D Int C = {y}  \
+Goal "!!A. ALL D:{E*{E}. E:A}. EX y. D Int C = {y}  \
 \       ==> (lam x:A. fst(THE z. (x*{x} Int C = {z}))) :  \
 \               (PROD X:A. X) ";
 by (fast_tac (claset() addSEs [lemma2]
                 addSIs [lam_type, RepFunI, fst_type]
                 addSDs [bspec]) 1);
 val lemma3 = result();
 
-goalw thy (AC_defs@AC_aux_defs) "!!Z. AC2 ==> AC1";
+Goalw (AC_defs@AC_aux_defs) "!!Z. AC2 ==> AC1";
 by (REPEAT (resolve_tac [allI, impI] 1));
 by (REPEAT (eresolve_tac [allE, impE] 1));
 by (fast_tac (claset() addSEs [lemma3]) 2);
 by (fast_tac (claset() addSIs [lemma1, equals0I]) 1);
 qed "AC2_AC1";

 
 (* ********************************************************************** *)
 (* AC1 ==> AC4                                                            *)
 (* ********************************************************************** *)
 
-goal thy "!!R. 0 ~: {R``{x}. x:domain(R)}";
+Goal "!!R. 0 ~: {R``{x}. x:domain(R)}";
 by (fast_tac (claset() addEs [sym RS equals0D]) 1);
 val lemma = result();
 
-goalw thy AC_defs "!!Z. AC1 ==> AC4";
+Goalw AC_defs "!!Z. AC1 ==> AC4";
 by (REPEAT (resolve_tac [allI, impI] 1));
 by (REPEAT (eresolve_tac [allE, lemma RSN (2, impE), exE] 1));
 by (best_tac (claset() addSIs [lam_type] addSEs [apply_type]) 1);
 qed "AC1_AC4";
 
 
 (* ********************************************************************** *)
 (* AC4 ==> AC3                                                            *)
 (* ********************************************************************** *)
 
-goal thy "!!f. f:A->B ==> (UN z:A. {z}*f`z) <= A*Union(B)";
+Goal "!!f. f:A->B ==> (UN z:A. {z}*f`z) <= A*Union(B)";
 by (fast_tac (claset() addSDs [apply_type]) 1);
 val lemma1 = result();
 
-goal thy "!!f. domain(UN z:A. {z}*f(z)) = {a:A. f(a)~=0}";
+Goal "!!f. domain(UN z:A. {z}*f(z)) = {a:A. f(a)~=0}";
 by (fast_tac (claset() addSIs [not_emptyI] addDs [range_type]) 1);
 val lemma2 = result();
 
-goal thy "!!f. x:A ==> (UN z:A. {z}*f(z))``{x} = f(x)";
+Goal "!!f. x:A ==> (UN z:A. {z}*f(z))``{x} = f(x)";
 by (Fast_tac 1);
 val lemma3 = result();
 
-goalw thy AC_defs "!!Z. AC4 ==> AC3";
+Goalw AC_defs "!!Z. AC4 ==> AC3";
 by (REPEAT (resolve_tac [allI,ballI] 1));
 by (REPEAT (eresolve_tac [allE,impE] 1));
 by (etac lemma1 1);
 by (asm_full_simp_tac (simpset() addsimps [lemma2, lemma3]
                         addcongs [Pi_cong]) 1);

 
 (* ********************************************************************** *)
 (* AC3 ==> AC1                                                            *)
 (* ********************************************************************** *)
 
-goal thy "!!A. b~:A ==> (PROD x:{a:A. id(A)`a~=b}. id(A)`x) = (PROD x:A. x)";
+Goal "!!A. b~:A ==> (PROD x:{a:A. id(A)`a~=b}. id(A)`x) = (PROD x:A. x)";
 by (asm_full_simp_tac (simpset() addsimps [id_def] addcongs [Pi_cong]) 1);
 by (res_inst_tac [("b","A")] subst_context 1);
 by (Fast_tac 1);
 val lemma = result();
 
-goalw thy AC_defs "!!Z. AC3 ==> AC1";
+Goalw AC_defs "!!Z. AC3 ==> AC1";
 by (REPEAT (resolve_tac [allI, impI] 1));
 by (REPEAT (eresolve_tac [allE, ballE] 1));
 by (fast_tac (claset() addSIs [id_type]) 2);
 by (fast_tac (claset() addEs [lemma RS subst]) 1);
 qed "AC3_AC1";
 
 (* ********************************************************************** *)
 (* AC4 ==> AC5                                                            *)
 (* ********************************************************************** *)
 
-goalw thy (range_def::AC_defs) "!!Z. AC4 ==> AC5";
+Goalw (range_def::AC_defs) "!!Z. AC4 ==> AC5";
 by (REPEAT (resolve_tac [allI,ballI] 1));
 by (REPEAT (eresolve_tac [allE,impE] 1));
 by (eresolve_tac [fun_is_rel RS converse_type] 1);
 by (etac exE 1);
 by (rtac bexI 1);

 
 (* ********************************************************************** *)
 (* AC5 ==> AC4, Rubin & Rubin, p. 11                                      *)
 (* ********************************************************************** *)
 
-goal thy "!!A. R <= A*B ==> (lam x:R. fst(x)) : R -> A";
+Goal "!!A. R <= A*B ==> (lam x:R. fst(x)) : R -> A";
 by (fast_tac (claset() addSIs [lam_type, fst_type]) 1);
 val lemma1 = result();
 
-goalw thy [range_def] "!!A. R <= A*B ==> range(lam x:R. fst(x)) = domain(R)";
+Goalw [range_def] "!!A. R <= A*B ==> range(lam x:R. fst(x)) = domain(R)";
 by (rtac equalityI 1);
 by (fast_tac (claset() addSEs [lamE]
                 addEs [subst_elem]
                 addSDs [Pair_fst_snd_eq]) 1);
 by (rtac subsetI 1);
 by (etac domainE 1);
 by (rtac domainI 1);
 by (fast_tac (claset() addSEs [lamI RS subst_elem] addIs [fst_conv RS ssubst]) 1);
 val lemma2 = result();
 
-goal thy "!!A. [| EX f: A->C. P(f,domain(f)); A=B |] ==>  EX f: B->C. P(f,B)";
+Goal "!!A. [| EX f: A->C. P(f,domain(f)); A=B |] ==>  EX f: B->C. P(f,B)";
 by (etac bexE 1);
 by (forward_tac [domain_of_fun] 1);
 by (Fast_tac 1);
 val lemma3 = result();
 
-goal thy "!!g. [| R <= A*B; g: C->R; ALL x:C. (lam z:R. fst(z))` (g`x) = x |] \
+Goal "!!g. [| R <= A*B; g: C->R; ALL x:C. (lam z:R. fst(z))` (g`x) = x |] \
 \               ==> (lam x:C. snd(g`x)): (PROD x:C. R``{x})";
 by (rtac lam_type 1);
 by (dtac apply_type 1 THEN (assume_tac 1));
 by (dtac bspec 1 THEN (assume_tac 1));
 by (rtac imageI 1);
 by (resolve_tac [subsetD RS Pair_fst_snd_eq RSN (2, subst_elem)] 1
         THEN (REPEAT (assume_tac 1)));
 by (Asm_full_simp_tac 1);
 val lemma4 = result();
 
-goalw thy AC_defs "!!Z. AC5 ==> AC4";
+Goalw AC_defs "!!Z. AC5 ==> AC4";
 by (REPEAT (resolve_tac [allI,impI] 1));
 by (REPEAT (eresolve_tac [allE,ballE] 1));
 by (eresolve_tac [lemma1 RSN (2, notE)] 2 THEN (assume_tac 2));
 by (dresolve_tac [lemma2 RSN (2, lemma3)] 1 THEN (assume_tac 1));
 by (fast_tac (claset() addSEs [lemma4]) 1);

 
 (* ********************************************************************** *)
 (* AC1 <-> AC6                                                            *)
 (* ********************************************************************** *)
 
-goalw thy AC_defs "AC1 <-> AC6";
+Goalw AC_defs "AC1 <-> AC6";
 by (fast_tac (claset() addDs [equals0D] addSEs [not_emptyE]) 1);
 qed "AC1_iff_AC6";