doc-src/Logics/ZF-eg.txt
author clasohm
Thu, 29 Jun 1995 12:28:27 +0200
changeset 1163 c080ff36d24e
parent 115 745affa0262b
child 5151 1e944fe5ce96
permissions -rw-r--r--
changed 'chol' labels to 'hol'; added a few parentheses
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(**** ZF examples ****)
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Pretty.setmargin 72;  (*existing macros just allow this margin*)
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print_depth 0;
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(*** Powerset example ***)
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val [prem] = goal ZF.thy "A<=B  ==>  Pow(A) <= Pow(B)";
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by (resolve_tac [subsetI] 1);
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by (resolve_tac [PowI] 1);
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by (dresolve_tac [PowD] 1);
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by (eresolve_tac [subset_trans] 1);
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by (resolve_tac [prem] 1);
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val Pow_mono = result();
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goal ZF.thy "Pow(A Int B) = Pow(A) Int Pow(B)";
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by (resolve_tac [equalityI] 1);
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by (resolve_tac [Int_greatest] 1);
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by (resolve_tac [Int_lower1 RS Pow_mono] 1);
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by (resolve_tac [Int_lower2 RS Pow_mono] 1);
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by (resolve_tac [subsetI] 1);
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by (eresolve_tac [IntE] 1);
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by (resolve_tac [PowI] 1);
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by (REPEAT (dresolve_tac [PowD] 1));
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by (resolve_tac [Int_greatest] 1);
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by (REPEAT (assume_tac 1));
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choplev 0;
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by (fast_tac (ZF_cs addIs [equalityI]) 1);
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val [prem] = goal ZF.thy "C<=D ==> Union(C) <= Union(D)";
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by (resolve_tac [subsetI] 1);
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by (eresolve_tac [UnionE] 1);
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by (resolve_tac [UnionI] 1);
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by (resolve_tac [prem RS subsetD] 1);
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by (assume_tac 1);
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by (assume_tac 1);
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choplev 0;
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by (resolve_tac [Union_least] 1);
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by (resolve_tac [Union_upper] 1);
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by (eresolve_tac [prem RS subsetD] 1);
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val prems = goal ZF.thy
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    "[| a:A;  f: A->B;  g: C->D;  A Int C = 0 |] ==>  \
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\    (f Un g)`a = f`a";
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by (resolve_tac [apply_equality] 1);
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by (resolve_tac [UnI1] 1);
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by (resolve_tac [apply_Pair] 1);
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by (resolve_tac prems 1);
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by (resolve_tac prems 1);
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by (resolve_tac [fun_disjoint_Un] 1);
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by (resolve_tac prems 1);
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by (resolve_tac prems 1);
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by (resolve_tac prems 1);
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goal ZF.thy "f``(UN x:A. B(x)) = (UN x:A. f``B(x))";
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by (resolve_tac [equalityI] 1);
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by (resolve_tac [subsetI] 1);
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fe imageE;
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goal ZF.thy "(UN x:C. A(x) Int B) = (UN x:C. A(x))  Int  B";
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by (resolve_tac [equalityI] 1);
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by (resolve_tac [Int_greatest] 1);
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fr UN_mono;
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by (resolve_tac [Int_lower1] 1);
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fr UN_least;
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????
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> goal ZF.thy "Pow(A Int B) = Pow(A) Int Pow(B)";
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Level 0
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A Int B) = Pow(A) Int Pow(B)
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> by (resolve_tac [equalityI] 1);
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Level 1
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A Int B) <= Pow(A) Int Pow(B)
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 2. Pow(A) Int Pow(B) <= Pow(A Int B)
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> by (resolve_tac [Int_greatest] 1);
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Level 2
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A Int B) <= Pow(A)
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 2. Pow(A Int B) <= Pow(B)
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 3. Pow(A) Int Pow(B) <= Pow(A Int B)
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> by (resolve_tac [Int_lower1 RS Pow_mono] 1);
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Level 3
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A Int B) <= Pow(B)
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 2. Pow(A) Int Pow(B) <= Pow(A Int B)
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> by (resolve_tac [Int_lower2 RS Pow_mono] 1);
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Level 4
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A) Int Pow(B) <= Pow(A Int B)
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> by (resolve_tac [subsetI] 1);
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Level 5
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)
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> by (eresolve_tac [IntE] 1);
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Level 6
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)
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> by (resolve_tac [PowI] 1);
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Level 7
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B
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> by (REPEAT (dresolve_tac [PowD] 1));
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Level 8
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B
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> by (resolve_tac [Int_greatest] 1);
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Level 9
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. !!x. [| x <= A; x <= B |] ==> x <= A
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 2. !!x. [| x <= A; x <= B |] ==> x <= B
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> by (REPEAT (assume_tac 1));
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Level 10
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Pow(A Int B) = Pow(A) Int Pow(B)
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No subgoals!
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> choplev 0;
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Level 0
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Pow(A Int B) = Pow(A) Int Pow(B)
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 1. Pow(A Int B) = Pow(A) Int Pow(B)
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> by (fast_tac (ZF_cs addIs [equalityI]) 1);
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Level 1
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Pow(A Int B) = Pow(A) Int Pow(B)
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No subgoals!
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> val [prem] = goal ZF.thy "C<=D ==> Union(C) <= Union(D)";
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Level 0
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Union(C) <= Union(D)
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 1. Union(C) <= Union(D)
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> by (resolve_tac [subsetI] 1);
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Level 1
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Union(C) <= Union(D)
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 1. !!x. x : Union(C) ==> x : Union(D)
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> by (eresolve_tac [UnionE] 1);
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Level 2
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Union(C) <= Union(D)
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 1. !!x B. [| x : B; B : C |] ==> x : Union(D)
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> by (resolve_tac [UnionI] 1);
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Level 3
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Union(C) <= Union(D)
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 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : D
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 2. !!x B. [| x : B; B : C |] ==> x : ?B2(x,B)
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> by (resolve_tac [prem RS subsetD] 1);
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Level 4
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Union(C) <= Union(D)
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 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C
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 2. !!x B. [| x : B; B : C |] ==> x : ?B2(x,B)
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> by (assume_tac 1);
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Level 5
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Union(C) <= Union(D)
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 1. !!x B. [| x : B; B : C |] ==> x : B
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> by (assume_tac 1);
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Level 6
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Union(C) <= Union(D)
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No subgoals!
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> val prems = goal ZF.thy
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#     "[| a:A;  f: A->B;  g: C->D;  A Int C = 0 |] ==>  \
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# \    (f Un g)`a = f`a";
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Level 0
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(f Un g) ` a = f ` a
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 1. (f Un g) ` a = f ` a
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> by (resolve_tac [apply_equality] 1);
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Level 1
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(f Un g) ` a = f ` a
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 1. <a,f ` a> : f Un g
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 2. f Un g : (PROD x:?A. ?B(x))
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> by (resolve_tac [UnI1] 1);
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Level 2
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(f Un g) ` a = f ` a
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 1. <a,f ` a> : f
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 2. f Un g : (PROD x:?A. ?B(x))
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> by (resolve_tac [apply_Pair] 1);
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Level 3
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(f Un g) ` a = f ` a
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 1. f : (PROD x:?A2. ?B2(x))
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 2. a : ?A2
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 3. f Un g : (PROD x:?A. ?B(x))
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> by (resolve_tac prems 1);
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Level 4
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(f Un g) ` a = f ` a
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 1. a : A
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 2. f Un g : (PROD x:?A. ?B(x))
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> by (resolve_tac prems 1);
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Level 5
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(f Un g) ` a = f ` a
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 1. f Un g : (PROD x:?A. ?B(x))
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> by (resolve_tac [fun_disjoint_Un] 1);
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Level 6
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(f Un g) ` a = f ` a
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 1. f : ?A3 -> ?B3
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 2. g : ?C3 -> ?D3
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 3. ?A3 Int ?C3 = 0
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> by (resolve_tac prems 1);
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Level 7
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(f Un g) ` a = f ` a
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 1. g : ?C3 -> ?D3
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 2. A Int ?C3 = 0
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> by (resolve_tac prems 1);
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Level 8
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(f Un g) ` a = f ` a
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 1. A Int C = 0
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> by (resolve_tac prems 1);
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Level 9
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(f Un g) ` a = f ` a
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No subgoals!