src/FOL/FOL.thy
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(*  Title:      FOL/FOL.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson and Markus Wenzel
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*)
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header {* Classical first-order logic *}
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theory FOL
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imports IFOL
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uses ("FOL_lemmas1.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML")
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begin
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subsection {* The classical axiom *}
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axioms
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  classical: "(~P ==> P) ==> P"
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subsection {* Lemmas and proof tools *}
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use "FOL_lemmas1.ML"
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theorems case_split = case_split_thm [case_names True False, cases type: o]
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use "cladata.ML"
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setup Cla.setup
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setup cla_setup
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setup case_setup
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use "blastdata.ML"
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setup Blast.setup
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lemma ex1_functional: "[| EX! z. P(a,z);  P(a,b);  P(a,c) |] ==> b = c"
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by blast
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ML {*
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val ex1_functional = thm "ex1_functional";
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*}
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use "simpdata.ML"
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setup simpsetup
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setup "Simplifier.method_setup Splitter.split_modifiers"
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setup Splitter.setup
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setup Clasimp.setup
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setup EqSubst.setup
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subsection {* Other simple lemmas *}
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lemma [simp]: "((P-->R) <-> (Q-->R)) <-> ((P<->Q) | R)"
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by blast
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lemma [simp]: "((P-->Q) <-> (P-->R)) <-> (P --> (Q<->R))"
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by blast
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lemma not_disj_iff_imp: "~P | Q <-> (P-->Q)"
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by blast
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(** Monotonicity of implications **)
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lemma conj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1&P2) --> (Q1&Q2)"
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by fast (*or (IntPr.fast_tac 1)*)
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lemma disj_mono: "[| P1-->Q1; P2-->Q2 |] ==> (P1|P2) --> (Q1|Q2)"
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by fast (*or (IntPr.fast_tac 1)*)
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lemma imp_mono: "[| Q1-->P1; P2-->Q2 |] ==> (P1-->P2)-->(Q1-->Q2)"
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by fast (*or (IntPr.fast_tac 1)*)
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lemma imp_refl: "P-->P"
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by (rule impI, assumption)
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(*The quantifier monotonicity rules are also intuitionistically valid*)
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lemma ex_mono: "(!!x. P(x) --> Q(x)) ==> (EX x. P(x)) --> (EX x. Q(x))"
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by blast
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lemma all_mono: "(!!x. P(x) --> Q(x)) ==> (ALL x. P(x)) --> (ALL x. Q(x))"
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by blast
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subsection {* Proof by cases and induction *}
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text {* Proper handling of non-atomic rule statements. *}
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constdefs
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  induct_forall where "induct_forall(P) == \<forall>x. P(x)"
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  induct_implies where "induct_implies(A, B) == A \<longrightarrow> B"
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  induct_equal where "induct_equal(x, y) == x = y"
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  induct_conj where "induct_conj(A, B) == A \<and> B"
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lemma induct_forall_eq: "(!!x. P(x)) == Trueprop(induct_forall(\<lambda>x. P(x)))"
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  by (unfold atomize_all induct_forall_def)
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lemma induct_implies_eq: "(A ==> B) == Trueprop(induct_implies(A, B))"
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  by (unfold atomize_imp induct_implies_def)
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lemma induct_equal_eq: "(x == y) == Trueprop(induct_equal(x, y))"
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  by (unfold atomize_eq induct_equal_def)
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lemma induct_conj_eq:
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  includes meta_conjunction_syntax
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  shows "(A && B) == Trueprop(induct_conj(A, B))"
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  by (unfold atomize_conj induct_conj_def)
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lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
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lemmas induct_rulify [symmetric, standard] = induct_atomize
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lemmas induct_rulify_fallback =
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  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
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hide const induct_forall induct_implies induct_equal induct_conj
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text {* Method setup. *}
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ML {*
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  structure InductMethod = InductMethodFun
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  (struct
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    val cases_default = thm "case_split";
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    val atomize = thms "induct_atomize";
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    val rulify = thms "induct_rulify";
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    val rulify_fallback = thms "induct_rulify_fallback";
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  end);
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*}
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setup InductMethod.setup
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end