author | wenzelm |
Mon, 08 May 2000 20:58:49 +0200 | |
changeset 8839 | 31da5b9790c0 |
parent 8326 | 0e329578b0ef |
child 8913 | 0bc13d5e60b8 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/set |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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1985
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Set theory for higher-order logic. A set is simply a predicate. |
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*) |
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section "Relating predicates and sets"; |
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Addsimps [Collect_mem_eq]; |
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AddIffs [mem_Collect_eq]; |
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|
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Goal "P(a) ==> a : {x. P(x)}"; |
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by (Asm_simp_tac 1); |
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qed "CollectI"; |
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||
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Goal "a : {x. P(x)} ==> P(a)"; |
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by (Asm_full_simp_tac 1); |
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qed "CollectD"; |
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||
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bind_thm ("CollectE", make_elim CollectD); |
23 |
||
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val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B"; |
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by (rtac (prem RS ext RS arg_cong RS box_equals) 1); |
26 |
by (rtac Collect_mem_eq 1); |
|
27 |
by (rtac Collect_mem_eq 1); |
|
28 |
qed "set_ext"; |
|
29 |
||
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val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; |
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by (rtac (prem RS ext RS arg_cong) 1); |
32 |
qed "Collect_cong"; |
|
33 |
||
34 |
val CollectE = make_elim CollectD; |
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AddSIs [CollectI]; |
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AddSEs [CollectE]; |
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|
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section "Bounded quantifiers"; |
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|
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val prems = Goalw [Ball_def] |
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"[| !!x. x:A ==> P(x) |] ==> ! x:A. P(x)"; |
44 |
by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); |
|
45 |
qed "ballI"; |
|
46 |
||
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bind_thms ("strip", [impI, allI, ballI]); |
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||
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Goalw [Ball_def] "[| ! x:A. P(x); x:A |] ==> P(x)"; |
50 |
by (Blast_tac 1); |
|
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qed "bspec"; |
52 |
||
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val major::prems = Goalw [Ball_def] |
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"[| ! x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; |
55 |
by (rtac (major RS spec RS impCE) 1); |
|
56 |
by (REPEAT (eresolve_tac prems 1)); |
|
57 |
qed "ballE"; |
|
58 |
||
59 |
(*Takes assumptions ! x:A.P(x) and a:A; creates assumption P(a)*) |
|
60 |
fun ball_tac i = etac ballE i THEN contr_tac (i+1); |
|
61 |
||
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AddSIs [ballI]; |
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AddEs [ballE]; |
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AddXDs [bspec]; |
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(* gives better instantiation for bound: *) |
66 |
claset_ref() := claset() addWrapper ("bspec", fn tac2 => |
|
67 |
(dtac bspec THEN' atac) APPEND' tac2); |
|
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|
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(*Normally the best argument order: P(x) constrains the choice of x:A*) |
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Goalw [Bex_def] "[| P(x); x:A |] ==> ? x:A. P(x)"; |
71 |
by (Blast_tac 1); |
|
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qed "bexI"; |
73 |
||
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(*The best argument order when there is only one x:A*) |
75 |
Goalw [Bex_def] "[| x:A; P(x) |] ==> ? x:A. P(x)"; |
|
76 |
by (Blast_tac 1); |
|
77 |
qed "rev_bexI"; |
|
78 |
||
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val prems = Goal |
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"[| ! x:A. ~P(x) ==> P(a); a:A |] ==> ? x:A. P(x)"; |
81 |
by (rtac classical 1); |
|
82 |
by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ; |
|
83 |
qed "bexCI"; |
|
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|
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val major::prems = Goalw [Bex_def] |
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"[| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; |
87 |
by (rtac (major RS exE) 1); |
|
88 |
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); |
|
89 |
qed "bexE"; |
|
90 |
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AddIs [bexI]; |
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AddSEs [bexE]; |
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|
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(*Trival rewrite rule*) |
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Goal "(! x:A. P) = ((? x. x:A) --> P)"; |
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by (simp_tac (simpset() addsimps [Ball_def]) 1); |
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qed "ball_triv"; |
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|
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(*Dual form for existentials*) |
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Goal "(? x:A. P) = ((? x. x:A) & P)"; |
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by (simp_tac (simpset() addsimps [Bex_def]) 1); |
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qed "bex_triv"; |
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|
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Addsimps [ball_triv, bex_triv]; |
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|
106 |
(** Congruence rules **) |
|
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||
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val prems = Goalw [Ball_def] |
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
110 |
\ (! x:A. P(x)) = (! x:B. Q(x))"; |
|
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by (asm_simp_tac (simpset() addsimps prems) 1); |
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qed "ball_cong"; |
113 |
||
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val prems = Goalw [Bex_def] |
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"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
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\ (? x:A. P(x)) = (? x:B. Q(x))"; |
|
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by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1); |
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qed "bex_cong"; |
119 |
||
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Addcongs [ball_cong,bex_cong]; |
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||
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section "Subsets"; |
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|
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val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B"; |
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by (REPEAT (ares_tac (prems @ [ballI]) 1)); |
126 |
qed "subsetI"; |
|
127 |
||
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(*Map the type ('a set => anything) to just 'a. |
129 |
For overloading constants whose first argument has type "'a set" *) |
|
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fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type); |
|
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||
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(*While (:) is not, its type must be kept |
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for overloading of = to work.*) |
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Blast.overloaded ("op :", domain_type); |
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|
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overload_1st_set "Ball"; (*need UNION, INTER also?*) |
|
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overload_1st_set "Bex"; |
|
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|
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(*Image: retain the type of the set being expressed*) |
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Blast.overloaded ("image", domain_type); |
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|
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(*Rule in Modus Ponens style*) |
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Goalw [subset_def] "[| A <= B; c:A |] ==> c:B"; |
144 |
by (Blast_tac 1); |
|
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qed "subsetD"; |
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AddXIs [subsetD]; |
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|
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(*The same, with reversed premises for use with etac -- cf rev_mp*) |
|
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Goal "[| c:A; A <= B |] ==> c:B"; |
150 |
by (REPEAT (ares_tac [subsetD] 1)) ; |
|
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qed "rev_subsetD"; |
|
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AddXIs [rev_subsetD]; |
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|
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(*Converts A<=B to x:A ==> x:B*) |
155 |
fun impOfSubs th = th RSN (2, rev_subsetD); |
|
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||
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Goal "[| A <= B; c ~: B |] ==> c ~: A"; |
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by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ; |
|
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qed "contra_subsetD"; |
|
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|
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Goal "[| c ~: B; A <= B |] ==> c ~: A"; |
162 |
by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ; |
|
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qed "rev_contra_subsetD"; |
|
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|
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(*Classical elimination rule*) |
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val major::prems = Goalw [subset_def] |
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"[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; |
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by (rtac (major RS ballE) 1); |
|
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by (REPEAT (eresolve_tac prems 1)); |
|
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qed "subsetCE"; |
|
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(*Takes assumptions A<=B; c:A and creates the assumption c:B *) |
|
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fun set_mp_tac i = etac subsetCE i THEN mp_tac i; |
|
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AddSIs [subsetI]; |
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AddEs [subsetD, subsetCE]; |
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|
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Goal "A <= (A::'a set)"; |
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by (Fast_tac 1); |
|
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qed "subset_refl"; (*Blast_tac would try order_refl and fail*) |
|
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|
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Goal "[| A<=B; B<=C |] ==> A<=(C::'a set)"; |
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by (Blast_tac 1); |
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qed "subset_trans"; |
185 |
||
186 |
||
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section "Equality"; |
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|
189 |
(*Anti-symmetry of the subset relation*) |
|
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Goal "[| A <= B; B <= A |] ==> A = (B::'a set)"; |
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by (rtac set_ext 1); |
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by (blast_tac (claset() addIs [subsetD]) 1); |
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qed "subset_antisym"; |
194 |
val equalityI = subset_antisym; |
|
195 |
||
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AddSIs [equalityI]; |
197 |
||
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(* Equality rules from ZF set theory -- are they appropriate here? *) |
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Goal "A = B ==> A<=(B::'a set)"; |
200 |
by (etac ssubst 1); |
|
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by (rtac subset_refl 1); |
202 |
qed "equalityD1"; |
|
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||
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Goal "A = B ==> B<=(A::'a set)"; |
205 |
by (etac ssubst 1); |
|
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by (rtac subset_refl 1); |
207 |
qed "equalityD2"; |
|
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||
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val prems = Goal |
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"[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; |
211 |
by (resolve_tac prems 1); |
|
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by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); |
|
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qed "equalityE"; |
|
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||
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val major::prems = Goal |
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"[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; |
217 |
by (rtac (major RS equalityE) 1); |
|
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by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); |
|
219 |
qed "equalityCE"; |
|
220 |
||
221 |
(*Lemma for creating induction formulae -- for "pattern matching" on p |
|
222 |
To make the induction hypotheses usable, apply "spec" or "bspec" to |
|
223 |
put universal quantifiers over the free variables in p. *) |
|
5316 | 224 |
val prems = Goal |
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"[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; |
226 |
by (rtac mp 1); |
|
227 |
by (REPEAT (resolve_tac (refl::prems) 1)); |
|
228 |
qed "setup_induction"; |
|
229 |
||
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Goal "A = B ==> (x : A) = (x : B)"; |
231 |
by (Asm_simp_tac 1); |
|
232 |
qed "eqset_imp_iff"; |
|
233 |
||
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|
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235 |
section "The universal set -- UNIV"; |
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236 |
|
7031 | 237 |
Goalw [UNIV_def] "x : UNIV"; |
238 |
by (rtac CollectI 1); |
|
239 |
by (rtac TrueI 1); |
|
240 |
qed "UNIV_I"; |
|
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|
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Addsimps [UNIV_I]; |
243 |
AddIs [UNIV_I]; (*unsafe makes it less likely to cause problems*) |
|
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|
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Goal "A <= UNIV"; |
246 |
by (rtac subsetI 1); |
|
247 |
by (rtac UNIV_I 1); |
|
248 |
qed "subset_UNIV"; |
|
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|
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(** Eta-contracting these two rules (to remove P) causes them to be ignored |
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because of their interaction with congruence rules. **) |
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|
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Goalw [Ball_def] "Ball UNIV P = All P"; |
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by (Simp_tac 1); |
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255 |
qed "ball_UNIV"; |
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|
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Goalw [Bex_def] "Bex UNIV P = Ex P"; |
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by (Simp_tac 1); |
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259 |
qed "bex_UNIV"; |
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260 |
Addsimps [ball_UNIV, bex_UNIV]; |
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|
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262 |
|
2858 | 263 |
section "The empty set -- {}"; |
264 |
||
7007 | 265 |
Goalw [empty_def] "(c : {}) = False"; |
266 |
by (Blast_tac 1) ; |
|
267 |
qed "empty_iff"; |
|
2858 | 268 |
|
269 |
Addsimps [empty_iff]; |
|
270 |
||
7007 | 271 |
Goal "a:{} ==> P"; |
272 |
by (Full_simp_tac 1); |
|
273 |
qed "emptyE"; |
|
2858 | 274 |
|
275 |
AddSEs [emptyE]; |
|
276 |
||
7007 | 277 |
Goal "{} <= A"; |
278 |
by (Blast_tac 1) ; |
|
279 |
qed "empty_subsetI"; |
|
2858 | 280 |
|
5256 | 281 |
(*One effect is to delete the ASSUMPTION {} <= A*) |
282 |
AddIffs [empty_subsetI]; |
|
283 |
||
7031 | 284 |
val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}"; |
7007 | 285 |
by (blast_tac (claset() addIs [prem RS FalseE]) 1) ; |
286 |
qed "equals0I"; |
|
2858 | 287 |
|
5256 | 288 |
(*Use for reasoning about disjointness: A Int B = {} *) |
7007 | 289 |
Goal "A={} ==> a ~: A"; |
290 |
by (Blast_tac 1) ; |
|
291 |
qed "equals0D"; |
|
2858 | 292 |
|
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|
293 |
AddDs [equals0D, sym RS equals0D]; |
5256 | 294 |
|
5069 | 295 |
Goalw [Ball_def] "Ball {} P = True"; |
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by (Simp_tac 1); |
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|
297 |
qed "ball_empty"; |
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|
298 |
|
5069 | 299 |
Goalw [Bex_def] "Bex {} P = False"; |
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by (Simp_tac 1); |
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UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
301 |
qed "bex_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
302 |
Addsimps [ball_empty, bex_empty]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
303 |
|
5069 | 304 |
Goal "UNIV ~= {}"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
305 |
by (blast_tac (claset() addEs [equalityE]) 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
306 |
qed "UNIV_not_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
307 |
AddIffs [UNIV_not_empty]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
308 |
|
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
309 |
|
2858 | 310 |
|
311 |
section "The Powerset operator -- Pow"; |
|
312 |
||
7007 | 313 |
Goalw [Pow_def] "(A : Pow(B)) = (A <= B)"; |
314 |
by (Asm_simp_tac 1); |
|
315 |
qed "Pow_iff"; |
|
2858 | 316 |
|
317 |
AddIffs [Pow_iff]; |
|
318 |
||
7031 | 319 |
Goalw [Pow_def] "A <= B ==> A : Pow(B)"; |
7007 | 320 |
by (etac CollectI 1); |
321 |
qed "PowI"; |
|
2858 | 322 |
|
7031 | 323 |
Goalw [Pow_def] "A : Pow(B) ==> A<=B"; |
7007 | 324 |
by (etac CollectD 1); |
325 |
qed "PowD"; |
|
326 |
||
2858 | 327 |
|
328 |
val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
|
329 |
val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |
|
330 |
||
331 |
||
5931 | 332 |
section "Set complement"; |
923 | 333 |
|
7031 | 334 |
Goalw [Compl_def] "(c : -A) = (c~:A)"; |
335 |
by (Blast_tac 1); |
|
336 |
qed "Compl_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
337 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
338 |
Addsimps [Compl_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
339 |
|
5490 | 340 |
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A"; |
923 | 341 |
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); |
342 |
qed "ComplI"; |
|
343 |
||
344 |
(*This form, with negated conclusion, works well with the Classical prover. |
|
345 |
Negated assumptions behave like formulae on the right side of the notional |
|
346 |
turnstile...*) |
|
5490 | 347 |
Goalw [Compl_def] "c : -A ==> c~:A"; |
5316 | 348 |
by (etac CollectD 1); |
923 | 349 |
qed "ComplD"; |
350 |
||
351 |
val ComplE = make_elim ComplD; |
|
352 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
353 |
AddSIs [ComplI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
354 |
AddSEs [ComplE]; |
1640 | 355 |
|
923 | 356 |
|
1548 | 357 |
section "Binary union -- Un"; |
923 | 358 |
|
7031 | 359 |
Goalw [Un_def] "(c : A Un B) = (c:A | c:B)"; |
360 |
by (Blast_tac 1); |
|
361 |
qed "Un_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
362 |
Addsimps [Un_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
363 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
364 |
Goal "c:A ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
365 |
by (Asm_simp_tac 1); |
923 | 366 |
qed "UnI1"; |
367 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
368 |
Goal "c:B ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
369 |
by (Asm_simp_tac 1); |
923 | 370 |
qed "UnI2"; |
371 |
||
372 |
(*Classical introduction rule: no commitment to A vs B*) |
|
7007 | 373 |
|
7031 | 374 |
val prems = Goal "(c~:B ==> c:A) ==> c : A Un B"; |
7007 | 375 |
by (Simp_tac 1); |
376 |
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ; |
|
377 |
qed "UnCI"; |
|
923 | 378 |
|
5316 | 379 |
val major::prems = Goalw [Un_def] |
923 | 380 |
"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
381 |
by (rtac (major RS CollectD RS disjE) 1); |
|
382 |
by (REPEAT (eresolve_tac prems 1)); |
|
383 |
qed "UnE"; |
|
384 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
385 |
AddSIs [UnCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
386 |
AddSEs [UnE]; |
1640 | 387 |
|
923 | 388 |
|
1548 | 389 |
section "Binary intersection -- Int"; |
923 | 390 |
|
7031 | 391 |
Goalw [Int_def] "(c : A Int B) = (c:A & c:B)"; |
392 |
by (Blast_tac 1); |
|
393 |
qed "Int_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
394 |
Addsimps [Int_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
395 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
396 |
Goal "[| c:A; c:B |] ==> c : A Int B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
397 |
by (Asm_simp_tac 1); |
923 | 398 |
qed "IntI"; |
399 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
400 |
Goal "c : A Int B ==> c:A"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
401 |
by (Asm_full_simp_tac 1); |
923 | 402 |
qed "IntD1"; |
403 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
404 |
Goal "c : A Int B ==> c:B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
405 |
by (Asm_full_simp_tac 1); |
923 | 406 |
qed "IntD2"; |
407 |
||
5316 | 408 |
val [major,minor] = Goal |
923 | 409 |
"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
410 |
by (rtac minor 1); |
|
411 |
by (rtac (major RS IntD1) 1); |
|
412 |
by (rtac (major RS IntD2) 1); |
|
413 |
qed "IntE"; |
|
414 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
415 |
AddSIs [IntI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
416 |
AddSEs [IntE]; |
923 | 417 |
|
1548 | 418 |
section "Set difference"; |
923 | 419 |
|
7031 | 420 |
Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)"; |
421 |
by (Blast_tac 1); |
|
422 |
qed "Diff_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
423 |
Addsimps [Diff_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
424 |
|
7007 | 425 |
Goal "[| c : A; c ~: B |] ==> c : A - B"; |
426 |
by (Asm_simp_tac 1) ; |
|
427 |
qed "DiffI"; |
|
923 | 428 |
|
7007 | 429 |
Goal "c : A - B ==> c : A"; |
430 |
by (Asm_full_simp_tac 1) ; |
|
431 |
qed "DiffD1"; |
|
923 | 432 |
|
7007 | 433 |
Goal "[| c : A - B; c : B |] ==> P"; |
434 |
by (Asm_full_simp_tac 1) ; |
|
435 |
qed "DiffD2"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
436 |
|
7031 | 437 |
val prems = Goal "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P"; |
7007 | 438 |
by (resolve_tac prems 1); |
439 |
by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ; |
|
440 |
qed "DiffE"; |
|
923 | 441 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
442 |
AddSIs [DiffI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
443 |
AddSEs [DiffE]; |
923 | 444 |
|
445 |
||
1548 | 446 |
section "Augmenting a set -- insert"; |
923 | 447 |
|
7031 | 448 |
Goalw [insert_def] "a : insert b A = (a=b | a:A)"; |
449 |
by (Blast_tac 1); |
|
450 |
qed "insert_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
451 |
Addsimps [insert_iff]; |
923 | 452 |
|
7031 | 453 |
Goal "a : insert a B"; |
7007 | 454 |
by (Simp_tac 1); |
455 |
qed "insertI1"; |
|
923 | 456 |
|
7007 | 457 |
Goal "!!a. a : B ==> a : insert b B"; |
458 |
by (Asm_simp_tac 1); |
|
459 |
qed "insertI2"; |
|
460 |
||
461 |
val major::prems = Goalw [insert_def] |
|
462 |
"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P"; |
|
463 |
by (rtac (major RS UnE) 1); |
|
464 |
by (REPEAT (eresolve_tac (prems @ [CollectE]) 1)); |
|
465 |
qed "insertE"; |
|
923 | 466 |
|
467 |
(*Classical introduction rule*) |
|
7031 | 468 |
val prems = Goal "(a~:B ==> a=b) ==> a: insert b B"; |
7007 | 469 |
by (Simp_tac 1); |
470 |
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ; |
|
471 |
qed "insertCI"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
472 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
473 |
AddSIs [insertCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
474 |
AddSEs [insertE]; |
923 | 475 |
|
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
476 |
Goal "A <= insert x B ==> A <= B & x ~: A | (? B'. A = insert x B' & B' <= B)"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
477 |
by (case_tac "x:A" 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
478 |
by (Fast_tac 2); |
7499 | 479 |
by (rtac disjI2 1); |
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
480 |
by (res_inst_tac [("x","A-{x}")] exI 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
481 |
by (Fast_tac 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
482 |
qed "subset_insertD"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
483 |
|
1548 | 484 |
section "Singletons, using insert"; |
923 | 485 |
|
7007 | 486 |
Goal "a : {a}"; |
487 |
by (rtac insertI1 1) ; |
|
488 |
qed "singletonI"; |
|
923 | 489 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
490 |
Goal "b : {a} ==> b=a"; |
2891 | 491 |
by (Blast_tac 1); |
923 | 492 |
qed "singletonD"; |
493 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
494 |
bind_thm ("singletonE", make_elim singletonD); |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
495 |
|
7007 | 496 |
Goal "(b : {a}) = (b=a)"; |
497 |
by (Blast_tac 1); |
|
498 |
qed "singleton_iff"; |
|
923 | 499 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
500 |
Goal "{a}={b} ==> a=b"; |
4089 | 501 |
by (blast_tac (claset() addEs [equalityE]) 1); |
923 | 502 |
qed "singleton_inject"; |
503 |
||
2858 | 504 |
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*) |
505 |
AddSIs [singletonI]; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
506 |
AddSDs [singleton_inject]; |
3718 | 507 |
AddSEs [singletonE]; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
508 |
|
7969 | 509 |
Goal "{b} = insert a A = (a = b & A <= {b})"; |
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
510 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
511 |
qed "singleton_insert_inj_eq"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
512 |
|
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
513 |
Goal "(insert a A = {b}) = (a = b & A <= {b})"; |
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
514 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
7969 | 515 |
qed "singleton_insert_inj_eq'"; |
516 |
||
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
517 |
AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq']; |
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
518 |
|
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
519 |
Goal "A <= {x} ==> A={} | A = {x}"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
520 |
by (Fast_tac 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
521 |
qed "subset_singletonD"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
522 |
|
5069 | 523 |
Goal "{x. x=a} = {a}"; |
4423 | 524 |
by (Blast_tac 1); |
3582 | 525 |
qed "singleton_conv"; |
526 |
Addsimps [singleton_conv]; |
|
1531 | 527 |
|
5600 | 528 |
Goal "{x. a=x} = {a}"; |
6301 | 529 |
by (Blast_tac 1); |
5600 | 530 |
qed "singleton_conv2"; |
531 |
Addsimps [singleton_conv2]; |
|
532 |
||
1531 | 533 |
|
1548 | 534 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; |
923 | 535 |
|
5069 | 536 |
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; |
2891 | 537 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
538 |
qed "UN_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
539 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
540 |
Addsimps [UN_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
541 |
|
923 | 542 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
543 |
Goal "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
544 |
by Auto_tac; |
923 | 545 |
qed "UN_I"; |
546 |
||
5316 | 547 |
val major::prems = Goalw [UNION_def] |
923 | 548 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
549 |
by (rtac (major RS CollectD RS bexE) 1); |
|
550 |
by (REPEAT (ares_tac prems 1)); |
|
551 |
qed "UN_E"; |
|
552 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
553 |
AddIs [UN_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
554 |
AddSEs [UN_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
555 |
|
6291 | 556 |
val prems = Goalw [UNION_def] |
923 | 557 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
558 |
\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
|
6291 | 559 |
by (asm_simp_tac (simpset() addsimps prems) 1); |
923 | 560 |
qed "UN_cong"; |
561 |
||
562 |
||
1548 | 563 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; |
923 | 564 |
|
5069 | 565 |
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
566 |
by Auto_tac; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
567 |
qed "INT_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
568 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
569 |
Addsimps [INT_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
570 |
|
5316 | 571 |
val prems = Goalw [INTER_def] |
923 | 572 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
573 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
|
574 |
qed "INT_I"; |
|
575 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
576 |
Goal "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
577 |
by Auto_tac; |
923 | 578 |
qed "INT_D"; |
579 |
||
580 |
(*"Classical" elimination -- by the Excluded Middle on a:A *) |
|
5316 | 581 |
val major::prems = Goalw [INTER_def] |
923 | 582 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
583 |
by (rtac (major RS CollectD RS ballE) 1); |
|
584 |
by (REPEAT (eresolve_tac prems 1)); |
|
585 |
qed "INT_E"; |
|
586 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
587 |
AddSIs [INT_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
588 |
AddEs [INT_D, INT_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
589 |
|
6291 | 590 |
val prems = Goalw [INTER_def] |
923 | 591 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
592 |
\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
|
6291 | 593 |
by (asm_simp_tac (simpset() addsimps prems) 1); |
923 | 594 |
qed "INT_cong"; |
595 |
||
596 |
||
1548 | 597 |
section "Union"; |
923 | 598 |
|
5069 | 599 |
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; |
2891 | 600 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
601 |
qed "Union_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
602 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
603 |
Addsimps [Union_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
604 |
|
923 | 605 |
(*The order of the premises presupposes that C is rigid; A may be flexible*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
606 |
Goal "[| X:C; A:X |] ==> A : Union(C)"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
607 |
by Auto_tac; |
923 | 608 |
qed "UnionI"; |
609 |
||
5316 | 610 |
val major::prems = Goalw [Union_def] |
923 | 611 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
612 |
by (rtac (major RS UN_E) 1); |
|
613 |
by (REPEAT (ares_tac prems 1)); |
|
614 |
qed "UnionE"; |
|
615 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
616 |
AddIs [UnionI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
617 |
AddSEs [UnionE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
618 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
619 |
|
1548 | 620 |
section "Inter"; |
923 | 621 |
|
5069 | 622 |
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; |
2891 | 623 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
624 |
qed "Inter_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
625 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
626 |
Addsimps [Inter_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
627 |
|
5316 | 628 |
val prems = Goalw [Inter_def] |
923 | 629 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
630 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
|
631 |
qed "InterI"; |
|
632 |
||
633 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
634 |
A:X can hold when X:C does not! This rule is analogous to "spec". *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
635 |
Goal "[| A : Inter(C); X:C |] ==> A:X"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
636 |
by Auto_tac; |
923 | 637 |
qed "InterD"; |
638 |
||
639 |
(*"Classical" elimination rule -- does not require proving X:C *) |
|
5316 | 640 |
val major::prems = Goalw [Inter_def] |
2721 | 641 |
"[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R"; |
923 | 642 |
by (rtac (major RS INT_E) 1); |
643 |
by (REPEAT (eresolve_tac prems 1)); |
|
644 |
qed "InterE"; |
|
645 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
646 |
AddSIs [InterI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
647 |
AddEs [InterD, InterE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
648 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
649 |
|
2912 | 650 |
(*** Image of a set under a function ***) |
651 |
||
652 |
(*Frequently b does not have the syntactic form of f(x).*) |
|
5316 | 653 |
Goalw [image_def] "[| b=f(x); x:A |] ==> b : f``A"; |
654 |
by (Blast_tac 1); |
|
2912 | 655 |
qed "image_eqI"; |
3909 | 656 |
Addsimps [image_eqI]; |
2912 | 657 |
|
658 |
bind_thm ("imageI", refl RS image_eqI); |
|
659 |
||
8025 | 660 |
(*This version's more effective when we already have the required x*) |
661 |
Goalw [image_def] "[| x:A; b=f(x) |] ==> b : f``A"; |
|
662 |
by (Blast_tac 1); |
|
663 |
qed "rev_image_eqI"; |
|
664 |
||
2912 | 665 |
(*The eta-expansion gives variable-name preservation.*) |
5316 | 666 |
val major::prems = Goalw [image_def] |
3842 | 667 |
"[| b : (%x. f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; |
2912 | 668 |
by (rtac (major RS CollectD RS bexE) 1); |
669 |
by (REPEAT (ares_tac prems 1)); |
|
670 |
qed "imageE"; |
|
671 |
||
672 |
AddIs [image_eqI]; |
|
673 |
AddSEs [imageE]; |
|
674 |
||
5069 | 675 |
Goal "f``(A Un B) = f``A Un f``B"; |
2935 | 676 |
by (Blast_tac 1); |
2912 | 677 |
qed "image_Un"; |
678 |
||
5069 | 679 |
Goal "(z : f``A) = (EX x:A. z = f x)"; |
3960 | 680 |
by (Blast_tac 1); |
681 |
qed "image_iff"; |
|
682 |
||
4523 | 683 |
(*This rewrite rule would confuse users if made default.*) |
5069 | 684 |
Goal "(f``A <= B) = (ALL x:A. f(x): B)"; |
4523 | 685 |
by (Blast_tac 1); |
686 |
qed "image_subset_iff"; |
|
687 |
||
688 |
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too |
|
689 |
many existing proofs.*) |
|
5316 | 690 |
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B"; |
4510 | 691 |
by (blast_tac (claset() addIs prems) 1); |
692 |
qed "image_subsetI"; |
|
693 |
||
2912 | 694 |
|
695 |
(*** Range of a function -- just a translation for image! ***) |
|
696 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
697 |
Goal "b=f(x) ==> b : range(f)"; |
2912 | 698 |
by (EVERY1 [etac image_eqI, rtac UNIV_I]); |
699 |
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI)); |
|
700 |
||
701 |
bind_thm ("rangeI", UNIV_I RS imageI); |
|
702 |
||
5316 | 703 |
val [major,minor] = Goal |
3842 | 704 |
"[| b : range(%x. f(x)); !!x. b=f(x) ==> P |] ==> P"; |
2912 | 705 |
by (rtac (major RS imageE) 1); |
706 |
by (etac minor 1); |
|
707 |
qed "rangeE"; |
|
708 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
709 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
710 |
(*** Set reasoning tools ***) |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
711 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
712 |
|
3912 | 713 |
(** Rewrite rules for boolean case-splitting: faster than |
4830 | 714 |
addsplits[split_if] |
3912 | 715 |
**) |
716 |
||
4830 | 717 |
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if); |
718 |
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if); |
|
3912 | 719 |
|
5237 | 720 |
(*Split ifs on either side of the membership relation. |
721 |
Not for Addsimps -- can cause goals to blow up!*) |
|
4830 | 722 |
bind_thm ("split_if_mem1", |
6394 | 723 |
read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. x : ?b")] split_if); |
4830 | 724 |
bind_thm ("split_if_mem2", |
6394 | 725 |
read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. ?a : x")] split_if); |
3912 | 726 |
|
4830 | 727 |
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2, |
728 |
split_if_mem1, split_if_mem2]; |
|
3912 | 729 |
|
730 |
||
4089 | 731 |
(*Each of these has ALREADY been added to simpset() above.*) |
2024
909153d8318f
Rationalized the rewriting of membership for {} and insert
paulson
parents:
1985
diff
changeset
|
732 |
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
733 |
mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]; |
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
734 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
735 |
val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
736 |
|
6291 | 737 |
simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
738 |
|
5256 | 739 |
Addsimps[subset_UNIV, subset_refl]; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
740 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
741 |
|
8001 | 742 |
(*** The 'proper subset' relation (<) ***) |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
743 |
|
5069 | 744 |
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
745 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
746 |
qed "psubsetI"; |
7658 | 747 |
AddXIs [psubsetI]; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
748 |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
749 |
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
750 |
by Auto_tac; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
751 |
qed "psubset_insertD"; |
4059 | 752 |
|
753 |
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq); |
|
6443 | 754 |
|
755 |
bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1); |
|
756 |
||
757 |
Goal"[| (A::'a set) < B; B <= C |] ==> A < C"; |
|
758 |
by (auto_tac (claset(), simpset() addsimps [psubset_eq])); |
|
759 |
qed "psubset_subset_trans"; |
|
760 |
||
761 |
Goal"[| (A::'a set) <= B; B < C|] ==> A < C"; |
|
762 |
by (auto_tac (claset(), simpset() addsimps [psubset_eq])); |
|
763 |
qed "subset_psubset_trans"; |
|
7717
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
764 |
|
8001 | 765 |
Goalw [psubset_def] "A < B ==> EX b. b : (B - A)"; |
766 |
by (Blast_tac 1); |
|
767 |
qed "psubset_imp_ex_mem"; |
|
768 |
||
7717
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
769 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
770 |
(* attributes *) |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
771 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
772 |
local |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
773 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
774 |
fun gen_rulify_prems x = |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
775 |
Attrib.no_args (Drule.rule_attribute (fn _ => (standard o |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
776 |
rule_by_tactic (REPEAT (ALLGOALS (resolve_tac [allI, ballI, impI])))))) x; |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
777 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
778 |
in |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
779 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
780 |
val rulify_prems_attrib_setup = |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
781 |
[Attrib.add_attributes |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
782 |
[("rulify_prems", (gen_rulify_prems, gen_rulify_prems), "put theorem into standard rule form")]]; |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
783 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
784 |
end; |