--- a/src/ZF/AC/HH.ML Mon Jun 22 17:12:27 1998 +0200
+++ b/src/ZF/AC/HH.ML Mon Jun 22 17:13:09 1998 +0200
@@ -14,25 +14,25 @@
(* Lemmas useful in each of the three proofs *)
(* ********************************************************************** *)
-goal thy "HH(f,x,a) = \
+Goal "HH(f,x,a) = \
\ (let z = x - (UN b:a. HH(f,x,b)) \
\ in if(f`z:Pow(z)-{0}, f`z, {x}))";
by (resolve_tac [HH_def RS def_transrec RS trans] 1);
by (Simp_tac 1);
qed "HH_def_satisfies_eq";
-goal thy "HH(f,x,a) : Pow(x)-{0} | HH(f,x,a)={x}";
+Goal "HH(f,x,a) : Pow(x)-{0} | HH(f,x,a)={x}";
by (resolve_tac [HH_def_satisfies_eq RS ssubst] 1);
by (simp_tac (simpset() addsimps [Let_def, Diff_subset RS PowI]
setloop split_tac [expand_if]) 1);
by (Fast_tac 1);
qed "HH_values";
-goal thy "!!A. B<=A ==> X-(UN a:A. P(a)) = X-(UN a:A-B. P(a))-(UN b:B. P(b))";
+Goal "!!A. B<=A ==> X-(UN a:A. P(a)) = X-(UN a:A-B. P(a))-(UN b:B. P(b))";
by (Fast_tac 1);
qed "subset_imp_Diff_eq";
-goal thy "!!c. [| c:a-b; b<a |] ==> c=b | b<c & c<a";
+Goal "!!c. [| c:a-b; b<a |] ==> c=b | b<c & c<a";
by (etac ltE 1);
by (dtac Ord_linear 1);
by (fast_tac (claset() addSIs [ltI] addIs [Ord_in_Ord]) 2);
@@ -44,14 +44,14 @@
by (fast_tac (claset() addSDs [prem] addSEs [mem_irrefl]) 1);
qed "Diff_UN_eq_self";
-goal thy "!!a. x - (UN b:a. HH(f,x,b)) = x - (UN b:a1. HH(f,x,b)) \
+Goal "!!a. x - (UN b:a. HH(f,x,b)) = x - (UN b:a1. HH(f,x,b)) \
\ ==> HH(f,x,a) = HH(f,x,a1)";
by (resolve_tac [HH_def_satisfies_eq RS
(HH_def_satisfies_eq RS sym RSN (3, trans RS trans))] 1);
by (etac subst_context 1);
qed "HH_eq";
-goal thy "!!a. [| HH(f,x,b)={x}; b<a |] ==> HH(f,x,a)={x}";
+Goal "!!a. [| HH(f,x,b)={x}; b<a |] ==> HH(f,x,a)={x}";
by (res_inst_tac [("P","b<a")] impE 1 THEN REPEAT (assume_tac 2));
by (eresolve_tac [lt_Ord2 RS trans_induct] 1);
by (rtac impI 1);
@@ -64,14 +64,14 @@
by (fast_tac (claset() addEs [ltE]) 1);
qed "HH_is_x_gt_too";
-goal thy "!!a. [| HH(f,x,a) : Pow(x)-{0}; b<a |] ==> HH(f,x,b) : Pow(x)-{0}";
+Goal "!!a. [| HH(f,x,a) : Pow(x)-{0}; b<a |] ==> HH(f,x,b) : Pow(x)-{0}";
by (resolve_tac [HH_values RS disjE] 1 THEN (assume_tac 1));
by (dtac HH_is_x_gt_too 1 THEN (assume_tac 1));
by (dtac subst 1 THEN (assume_tac 1));
by (fast_tac (claset() addSEs [mem_irrefl]) 1);
qed "HH_subset_x_lt_too";
-goal thy "!!a. HH(f,x,a) : Pow(x)-{0} \
+Goal "!!a. HH(f,x,a) : Pow(x)-{0} \
\ ==> HH(f,x,a) : Pow(x - (UN b:a. HH(f,x,b)))-{0}";
by (dresolve_tac [HH_def_satisfies_eq RS subst] 1);
by (resolve_tac [HH_def_satisfies_eq RS ssubst] 1);
@@ -81,14 +81,14 @@
by (fast_tac (subset_cs addSEs [mem_irrefl]) 1);
qed "HH_subset_x_imp_subset_Diff_UN";
-goal thy "!!x. [| HH(f,x,v)=HH(f,x,w); HH(f,x,v): Pow(x)-{0}; v:w |] ==> P";
+Goal "!!x. [| HH(f,x,v)=HH(f,x,w); HH(f,x,v): Pow(x)-{0}; v:w |] ==> P";
by (forw_inst_tac [("P","%y. y: Pow(x)-{0}")] subst 1 THEN (assume_tac 1));
by (dres_inst_tac [("a","w")] HH_subset_x_imp_subset_Diff_UN 1);
by (dtac subst_elem 1 THEN (assume_tac 1));
by (fast_tac (claset() addSIs [singleton_iff RS iffD2, equals0I]) 1);
qed "HH_eq_arg_lt";
-goal thy "!!x. [| HH(f,x,v)=HH(f,x,w); HH(f,x,w): Pow(x)-{0}; \
+Goal "!!x. [| HH(f,x,v)=HH(f,x,w); HH(f,x,w): Pow(x)-{0}; \
\ Ord(v); Ord(w) |] ==> v=w";
by (res_inst_tac [("j","w")] Ord_linear_lt 1 THEN TRYALL assume_tac);
by (resolve_tac [sym RS (ltD RSN (3, HH_eq_arg_lt))] 2
@@ -97,7 +97,7 @@
by (fast_tac (FOL_cs addDs [ltD] addSEs [HH_eq_arg_lt]) 1);
qed "HH_eq_imp_arg_eq";
-goalw thy [lepoll_def, inj_def]
+Goalw [lepoll_def, inj_def]
"!!i. [| HH(f, x, i) : Pow(x)-{0}; Ord(i) |] ==> i lepoll Pow(x)-{0}";
by (res_inst_tac [("x","lam j:i. HH(f, x, j)")] exI 1);
by (Asm_simp_tac 1);
@@ -105,17 +105,17 @@
addSIs [lam_type, ballI, ltI] addIs [bexI]) 1);
qed "HH_subset_x_imp_lepoll";
-goal thy "HH(f, x, Hartog(Pow(x)-{0})) = {x}";
+Goal "HH(f, x, Hartog(Pow(x)-{0})) = {x}";
by (resolve_tac [HH_values RS disjE] 1 THEN (assume_tac 2));
by (fast_tac (FOL_cs addSDs [HH_subset_x_imp_lepoll]
addSIs [Ord_Hartog] addSEs [HartogE]) 1);
qed "HH_Hartog_is_x";
-goal thy "HH(f, x, LEAST i. HH(f, x, i) = {x}) = {x}";
+Goal "HH(f, x, LEAST i. HH(f, x, i) = {x}) = {x}";
by (fast_tac (claset() addSIs [Ord_Hartog, HH_Hartog_is_x, LeastI]) 1);
qed "HH_Least_eq_x";
-goal thy "!!a. a:(LEAST i. HH(f,x,i)={x}) ==> HH(f,x,a) : Pow(x)-{0}";
+Goal "!!a. a:(LEAST i. HH(f,x,i)={x}) ==> HH(f,x,a) : Pow(x)-{0}";
by (resolve_tac [HH_values RS disjE] 1 THEN (assume_tac 1));
by (rtac less_LeastE 1);
by (eresolve_tac [Ord_Least RSN (2, ltI)] 2);
@@ -126,7 +126,7 @@
(* Lemmas used in the proofs of AC1 ==> WO2 and AC17 ==> AC1 *)
(* ********************************************************************** *)
-goalw thy [inj_def]
+Goalw [inj_def]
"(lam a:(LEAST i. HH(f,x,i)={x}). HH(f,x,a)) : \
\ inj(LEAST i. HH(f,x,i)={x}, Pow(x)-{0})";
by (Asm_full_simp_tac 1);
@@ -134,14 +134,14 @@
addSEs [HH_eq_imp_arg_eq, Ord_Least RS Ord_in_Ord]) 1);
qed "lam_Least_HH_inj_Pow";
-goal thy "!!x. ALL a:(LEAST i. HH(f,x,i)={x}). EX z:x. HH(f,x,a) = {z} \
+Goal "!!x. ALL a:(LEAST i. HH(f,x,i)={x}). EX z:x. HH(f,x,a) = {z} \
\ ==> (lam a:(LEAST i. HH(f,x,i)={x}). HH(f,x,a)) \
\ : inj(LEAST i. HH(f,x,i)={x}, {{y}. y:x})";
by (resolve_tac [lam_Least_HH_inj_Pow RS inj_strengthen_type] 1);
by (Asm_full_simp_tac 1);
qed "lam_Least_HH_inj";
-goalw thy [surj_def]
+Goalw [surj_def]
"!!x. [| x - (UN a:A. F(a)) = 0; \
\ ALL a:A. EX z:x. F(a) = {z} |] \
\ ==> (lam a:A. F(a)) : surj(A, {{y}. y:x})";
@@ -151,12 +151,12 @@
by (deepen_tac (claset() addSIs [bexI] addSEs [equalityE]) 4 1);
qed "lam_surj_sing";
-goal thy "!!x. y:Pow(x)-{0} ==> x ~= 0";
+Goal "!!x. y:Pow(x)-{0} ==> x ~= 0";
by (fast_tac (claset() addSIs [equals0I, singletonI RS subst_elem]
addSDs [equals0D]) 1);
qed "not_emptyI2";
-goal thy "!!f. f`(x - (UN j:i. HH(f,x,j))): Pow(x - (UN j:i. HH(f,x,j)))-{0} \
+Goal "!!f. f`(x - (UN j:i. HH(f,x,j))): Pow(x - (UN j:i. HH(f,x,j)))-{0} \
\ ==> HH(f, x, i) : Pow(x) - {0}";
by (resolve_tac [HH_def_satisfies_eq RS ssubst] 1);
by (asm_full_simp_tac (simpset() addsimps [Let_def, Diff_subset RS PowI,
@@ -174,20 +174,20 @@
addSEs [mem_irrefl]) 1);
qed "f_subsets_imp_UN_HH_eq_x";
-goal thy "HH(f,x,i)=f`(x - (UN j:i. HH(f,x,j))) | HH(f,x,i)={x}";
+Goal "HH(f,x,i)=f`(x - (UN j:i. HH(f,x,j))) | HH(f,x,i)={x}";
by (resolve_tac [HH_def_satisfies_eq RS ssubst] 1);
by (simp_tac (simpset() addsimps [Let_def, Diff_subset RS PowI]
setloop split_tac [expand_if]) 1);
qed "HH_values2";
-goal thy
+Goal
"!!f. HH(f,x,i): Pow(x)-{0} ==> HH(f,x,i)=f`(x - (UN j:i. HH(f,x,j)))";
by (resolve_tac [HH_values2 RS disjE] 1 THEN (assume_tac 1));
by (fast_tac (claset() addSEs [equalityE, mem_irrefl]
addSDs [singleton_subsetD]) 1);
qed "HH_subset_imp_eq";
-goal thy "!!f. [| f : (Pow(x)-{0}) -> {{z}. z:x}; \
+Goal "!!f. [| f : (Pow(x)-{0}) -> {{z}. z:x}; \
\ a:(LEAST i. HH(f,x,i)={x}) |] ==> EX z:x. HH(f,x,a) = {z}";
by (dtac less_Least_subset_x 1);
by (forward_tac [HH_subset_imp_eq] 1);
@@ -198,7 +198,7 @@
by (fast_tac (claset() addss (simpset())) 1);
qed "f_sing_imp_HH_sing";
-goalw thy [bij_def]
+Goalw [bij_def]
"!!f. [| x - (UN j: (LEAST i. HH(f,x,i)={x}). HH(f,x,j)) = 0; \
\ f : (Pow(x)-{0}) -> {{z}. z:x} |] \
\ ==> (lam a:(LEAST i. HH(f,x,i)={x}). HH(f,x,a)) \
@@ -207,7 +207,7 @@
f_sing_imp_HH_sing]) 1);
qed "f_sing_lam_bij";
-goal thy "!!f. f: (PROD X: Pow(x)-{0}. F(X)) \
+Goal "!!f. f: (PROD X: Pow(x)-{0}. F(X)) \
\ ==> (lam X:Pow(x)-{0}. {f`X}) : (PROD X: Pow(x)-{0}. {{z}. z:F(X)})";
by (fast_tac (FOL_cs addSIs [lam_type, RepFun_eqI, singleton_eq_iff RS iffD2]
addDs [apply_type]) 1);