src/FOL/IFOL.thy
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(*  Title:      FOL/IFOL.thy
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    Author:     Lawrence C Paulson and Markus Wenzel
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*)
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section \<open>Intuitionistic first-order logic\<close>
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theory IFOL
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imports Pure
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begin
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ML_file \<open>~~/src/Tools/misc_legacy.ML\<close>
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ML_file \<open>~~/src/Provers/splitter.ML\<close>
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ML_file \<open>~~/src/Provers/hypsubst.ML\<close>
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ML_file \<open>~~/src/Tools/IsaPlanner/zipper.ML\<close>
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ML_file \<open>~~/src/Tools/IsaPlanner/isand.ML\<close>
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ML_file \<open>~~/src/Tools/IsaPlanner/rw_inst.ML\<close>
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ML_file \<open>~~/src/Provers/quantifier1.ML\<close>
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ML_file \<open>~~/src/Tools/intuitionistic.ML\<close>
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ML_file \<open>~~/src/Tools/project_rule.ML\<close>
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ML_file \<open>~~/src/Tools/atomize_elim.ML\<close>
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subsection \<open>Syntax and axiomatic basis\<close>
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setup Pure_Thy.old_appl_syntax_setup
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setup \<open>Proofterm.set_preproc (Proof_Rewrite_Rules.standard_preproc [])\<close>
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class "term"
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default_sort \<open>term\<close>
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typedecl o
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judgment
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  Trueprop :: \<open>o \<Rightarrow> prop\<close>  (\<open>(_)\<close> 5)
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subsubsection \<open>Equality\<close>
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axiomatization
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  eq :: \<open>['a, 'a] \<Rightarrow> o\<close>  (infixl \<open>=\<close> 50)
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where
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  refl: \<open>a = a\<close> and
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  subst: \<open>a = b \<Longrightarrow> P(a) \<Longrightarrow> P(b)\<close>
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subsubsection \<open>Propositional logic\<close>
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axiomatization
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  False :: \<open>o\<close> and
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  conj :: \<open>[o, o] => o\<close>  (infixr \<open>\<and>\<close> 35) and
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  disj :: \<open>[o, o] => o\<close>  (infixr \<open>\<or>\<close> 30) and
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  imp :: \<open>[o, o] => o\<close>  (infixr \<open>\<longrightarrow>\<close> 25)
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where
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  conjI: \<open>\<lbrakk>P;  Q\<rbrakk> \<Longrightarrow> P \<and> Q\<close> and
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  conjunct1: \<open>P \<and> Q \<Longrightarrow> P\<close> and
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  conjunct2: \<open>P \<and> Q \<Longrightarrow> Q\<close> and
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  disjI1: \<open>P \<Longrightarrow> P \<or> Q\<close> and
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  disjI2: \<open>Q \<Longrightarrow> P \<or> Q\<close> and
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  disjE: \<open>\<lbrakk>P \<or> Q; P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R\<close> and
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  impI: \<open>(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q\<close> and
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  mp: \<open>\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q\<close> and
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  FalseE: \<open>False \<Longrightarrow> P\<close>
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subsubsection \<open>Quantifiers\<close>
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axiomatization
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  All :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close>  (binder \<open>\<forall>\<close> 10) and
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  Ex :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close>  (binder \<open>\<exists>\<close> 10)
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where
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  allI: \<open>(\<And>x. P(x)) \<Longrightarrow> (\<forall>x. P(x))\<close> and
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  spec: \<open>(\<forall>x. P(x)) \<Longrightarrow> P(x)\<close> and
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  exI: \<open>P(x) \<Longrightarrow> (\<exists>x. P(x))\<close> and
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  exE: \<open>\<lbrakk>\<exists>x. P(x); \<And>x. P(x) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R\<close>
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subsubsection \<open>Definitions\<close>
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definition \<open>True \<equiv> False \<longrightarrow> False\<close>
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definition Not (\<open>\<not> _\<close> [40] 40)
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  where not_def: \<open>\<not> P \<equiv> P \<longrightarrow> False\<close>
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definition iff  (infixr \<open>\<longleftrightarrow>\<close> 25)
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  where \<open>P \<longleftrightarrow> Q \<equiv> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)\<close>
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definition Ex1 :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close>  (binder \<open>\<exists>!\<close> 10)
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  where ex1_def: \<open>\<exists>!x. P(x) \<equiv> \<exists>x. P(x) \<and> (\<forall>y. P(y) \<longrightarrow> y = x)\<close>
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axiomatization where  \<comment> \<open>Reflection, admissible\<close>
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  eq_reflection: \<open>(x = y) \<Longrightarrow> (x \<equiv> y)\<close> and
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  iff_reflection: \<open>(P \<longleftrightarrow> Q) \<Longrightarrow> (P \<equiv> Q)\<close>
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abbreviation not_equal :: \<open>['a, 'a] \<Rightarrow> o\<close>  (infixl \<open>\<noteq>\<close> 50)
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  where \<open>x \<noteq> y \<equiv> \<not> (x = y)\<close>
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subsubsection \<open>Old-style ASCII syntax\<close>
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notation (ASCII)
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  not_equal  (infixl \<open>~=\<close> 50) and
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  Not  (\<open>~ _\<close> [40] 40) and
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  conj  (infixr \<open>&\<close> 35) and
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  disj  (infixr \<open>|\<close> 30) and
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  All  (binder \<open>ALL \<close> 10) and
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  Ex  (binder \<open>EX \<close> 10) and
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  Ex1  (binder \<open>EX! \<close> 10) and
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  imp  (infixr \<open>-->\<close> 25) and
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  iff  (infixr \<open><->\<close> 25)
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subsection \<open>Lemmas and proof tools\<close>
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lemmas strip = impI allI
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lemma TrueI: \<open>True\<close>
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  unfolding True_def by (rule impI)
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subsubsection \<open>Sequent-style elimination rules for \<open>\<and>\<close> \<open>\<longrightarrow>\<close> and \<open>\<forall>\<close>\<close>
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lemma conjE:
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  assumes major: \<open>P \<and> Q\<close>
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    and r: \<open>\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R\<close>
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  shows \<open>R\<close>
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  apply (rule r)
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   apply (rule major [THEN conjunct1])
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  apply (rule major [THEN conjunct2])
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  done
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lemma impE:
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  assumes major: \<open>P \<longrightarrow> Q\<close>
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    and \<open>P\<close>
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  and r: \<open>Q \<Longrightarrow> R\<close>
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  shows \<open>R\<close>
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  apply (rule r)
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  apply (rule major [THEN mp])
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  apply (rule \<open>P\<close>)
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  done
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lemma allE:
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  assumes major: \<open>\<forall>x. P(x)\<close>
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    and r: \<open>P(x) \<Longrightarrow> R\<close>
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  shows \<open>R\<close>
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  apply (rule r)
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  apply (rule major [THEN spec])
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  done
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text \<open>Duplicates the quantifier; for use with \<^ML>\<open>eresolve_tac\<close>.\<close>
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lemma all_dupE:
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  assumes major: \<open>\<forall>x. P(x)\<close>
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    and r: \<open>\<lbrakk>P(x); \<forall>x. P(x)\<rbrakk> \<Longrightarrow> R\<close>
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  shows \<open>R\<close>
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  apply (rule r)
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   apply (rule major [THEN spec])
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  apply (rule major)
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  done
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subsubsection \<open>Negation rules, which translate between \<open>\<not> P\<close> and \<open>P \<longrightarrow> False\<close>\<close>
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lemma notI: \<open>(P \<Longrightarrow> False) \<Longrightarrow> \<not> P\<close>
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  unfolding not_def by (erule impI)
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lemma notE: \<open>\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R\<close>
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  unfolding not_def by (erule mp [THEN FalseE])
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lemma rev_notE: \<open>\<lbrakk>P; \<not> P\<rbrakk> \<Longrightarrow> R\<close>
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  by (erule notE)
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text \<open>This is useful with the special implication rules for each kind of \<open>P\<close>.\<close>
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lemma not_to_imp:
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  assumes \<open>\<not> P\<close>
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    and r: \<open>P \<longrightarrow> False \<Longrightarrow> Q\<close>
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  shows \<open>Q\<close>
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  apply (rule r)
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  apply (rule impI)
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  apply (erule notE [OF \<open>\<not> P\<close>])
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  done
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text \<open>
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  For substitution into an assumption \<open>P\<close>, reduce \<open>Q\<close> to \<open>P \<longrightarrow> Q\<close>, substitute into this implication, then apply \<open>impI\<close> to
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  move \<open>P\<close> back into the assumptions.
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\<close>
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lemma rev_mp: \<open>\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q\<close>
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  by (erule mp)
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text \<open>Contrapositive of an inference rule.\<close>
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lemma contrapos:
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  assumes major: \<open>\<not> Q\<close>
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    and minor: \<open>P \<Longrightarrow> Q\<close>
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  shows \<open>\<not> P\<close>
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  apply (rule major [THEN notE, THEN notI])
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  apply (erule minor)
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  done
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subsubsection \<open>Modus Ponens Tactics\<close>
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text \<open>
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  Finds \<open>P \<longrightarrow> Q\<close> and P in the assumptions, replaces implication by
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  \<open>Q\<close>.
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\<close>
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ML \<open>
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  fun mp_tac ctxt i =
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    eresolve_tac ctxt @{thms notE impE} i THEN assume_tac ctxt i;
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  fun eq_mp_tac ctxt i =
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    eresolve_tac ctxt @{thms notE impE} i THEN eq_assume_tac i;
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\<close>
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subsection \<open>If-and-only-if\<close>
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lemma iffI: \<open>\<lbrakk>P \<Longrightarrow> Q; Q \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P \<longleftrightarrow> Q\<close>
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  apply (unfold iff_def)
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  apply (rule conjI)
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   apply (erule impI)
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  apply (erule impI)
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  done
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lemma iffE:
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  assumes major: \<open>P \<longleftrightarrow> Q\<close>
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    and r: \<open>P \<longrightarrow> Q \<Longrightarrow> Q \<longrightarrow> P \<Longrightarrow> R\<close>
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  shows \<open>R\<close>
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  apply (insert major, unfold iff_def)
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  apply (erule conjE)
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  apply (erule r)
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  apply assumption
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  done
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subsubsection \<open>Destruct rules for \<open>\<longleftrightarrow>\<close> similar to Modus Ponens\<close>
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lemma iffD1: \<open>\<lbrakk>P \<longleftrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q\<close>
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  apply (unfold iff_def)
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  apply (erule conjunct1 [THEN mp])
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  apply assumption
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  done
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lemma iffD2: \<open>\<lbrakk>P \<longleftrightarrow> Q; Q\<rbrakk> \<Longrightarrow> P\<close>
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  apply (unfold iff_def)
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  apply (erule conjunct2 [THEN mp])
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  apply assumption
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  done
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lemma rev_iffD1: \<open>\<lbrakk>P; P \<longleftrightarrow> Q\<rbrakk> \<Longrightarrow> Q\<close>
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  apply (erule iffD1)
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  apply assumption
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  done
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lemma rev_iffD2: \<open>\<lbrakk>Q; P \<longleftrightarrow> Q\<rbrakk> \<Longrightarrow> P\<close>
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  apply (erule iffD2)
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  apply assumption
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  done
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lemma iff_refl: \<open>P \<longleftrightarrow> P\<close>
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  by (rule iffI)
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lemma iff_sym: \<open>Q \<longleftrightarrow> P \<Longrightarrow> P \<longleftrightarrow> Q\<close>
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  apply (erule iffE)
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  apply (rule iffI)
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  apply (assumption | erule mp)+
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  done
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lemma iff_trans: \<open>\<lbrakk>P \<longleftrightarrow> Q; Q \<longleftrightarrow> R\<rbrakk> \<Longrightarrow> P \<longleftrightarrow> R\<close>
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  apply (rule iffI)
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  apply (assumption | erule iffE | erule (1) notE impE)+
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  done
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subsection \<open>Unique existence\<close>
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text \<open>
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  NOTE THAT the following 2 quantifications:
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    \<^item> \<open>\<exists>!x\<close> such that [\<open>\<exists>!y\<close> such that P(x,y)]   (sequential)
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    \<^item> \<open>\<exists>!x,y\<close> such that P(x,y)                   (simultaneous)
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  do NOT mean the same thing. The parser treats \<open>\<exists>!x y.P(x,y)\<close> as sequential.
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\<close>
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lemma ex1I: \<open>P(a) \<Longrightarrow> (\<And>x. P(x) \<Longrightarrow> x = a) \<Longrightarrow> \<exists>!x. P(x)\<close>
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  apply (unfold ex1_def)
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  apply (assumption | rule exI conjI allI impI)+
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  done
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text \<open>Sometimes easier to use: the premises have no shared variables. Safe!\<close>
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lemma ex_ex1I: \<open>\<exists>x. P(x) \<Longrightarrow> (\<And>x y. \<lbrakk>P(x); P(y)\<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> \<exists>!x. P(x)\<close>
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  apply (erule exE)
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  apply (rule ex1I)
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   apply assumption
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  apply assumption
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  done
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   297
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lemma ex1E: \<open>\<exists>! x. P(x) \<Longrightarrow> (\<And>x. \<lbrakk>P(x); \<forall>y. P(y) \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R\<close>
23393
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  apply (unfold ex1_def)
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  apply (assumption | erule exE conjE)+
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   301
  done
c5cf9243ad62 converted legacy ML scripts;
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   302
c5cf9243ad62 converted legacy ML scripts;
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   303
62020
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subsubsection \<open>\<open>\<longleftrightarrow>\<close> congruence rules for simplification\<close>
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62020
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   306
text \<open>Use \<open>iffE\<close> on a premise. For \<open>conj_cong\<close>, \<open>imp_cong\<close>, \<open>all_cong\<close>, \<open>ex_cong\<close>.\<close>
60770
240563fbf41d isabelle update_cartouches;
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ML \<open>
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  fun iff_tac ctxt prems i =
d881f5288d5a proper context and method setup;
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    resolve_tac ctxt (prems RL @{thms iffE}) i THEN
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    REPEAT1 (eresolve_tac ctxt @{thms asm_rl mp} i);
60770
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\<close>
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59529
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method_setup iff =
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  \<open>Attrib.thms >>
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    (fn prems => fn ctxt => SIMPLE_METHOD' (iff_tac ctxt prems))\<close>
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lemma conj_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
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   318
  assumes \<open>P \<longleftrightarrow> P'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
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   319
    and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
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  shows \<open>(P \<and> Q) \<longleftrightarrow> (P' \<and> Q')\<close>
21539
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diff changeset
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  apply (insert assms)
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   322
  apply (assumption | rule iffI conjI | erule iffE conjE mp | iff assms)+
21539
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diff changeset
   323
  done
c5cf9243ad62 converted legacy ML scripts;
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diff changeset
   324
61487
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   325
text \<open>Reversed congruence rule!  Used in ZF/Order.\<close>
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   326
lemma conj_cong2:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
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   327
  assumes \<open>P \<longleftrightarrow> P'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   328
    and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   329
  shows \<open>(Q \<and> P) \<longleftrightarrow> (Q' \<and> P')\<close>
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diff changeset
   330
  apply (insert assms)
59529
d881f5288d5a proper context and method setup;
wenzelm
parents: 59498
diff changeset
   331
  apply (assumption | rule iffI conjI | erule iffE conjE mp | iff assms)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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diff changeset
   332
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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diff changeset
   333
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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   334
lemma disj_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   335
  assumes \<open>P \<longleftrightarrow> P'\<close> and \<open>Q \<longleftrightarrow> Q'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   336
  shows \<open>(P \<or> Q) \<longleftrightarrow> (P' \<or> Q')\<close>
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diff changeset
   337
  apply (insert assms)
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   338
  apply (erule iffE disjE disjI1 disjI2 |
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
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diff changeset
   339
    assumption | rule iffI | erule (1) notE impE)+
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diff changeset
   340
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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diff changeset
   341
c5cf9243ad62 converted legacy ML scripts;
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diff changeset
   342
lemma imp_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
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diff changeset
   343
  assumes \<open>P \<longleftrightarrow> P'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   344
    and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   345
  shows \<open>(P \<longrightarrow> Q) \<longleftrightarrow> (P' \<longrightarrow> Q')\<close>
21539
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diff changeset
   346
  apply (insert assms)
59529
d881f5288d5a proper context and method setup;
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parents: 59498
diff changeset
   347
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE | iff assms)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   348
  done
c5cf9243ad62 converted legacy ML scripts;
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parents: 21524
diff changeset
   349
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
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   350
lemma iff_cong: \<open>\<lbrakk>P \<longleftrightarrow> P'; Q \<longleftrightarrow> Q'\<rbrakk> \<Longrightarrow> (P \<longleftrightarrow> Q) \<longleftrightarrow> (P' \<longleftrightarrow> Q')\<close>
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diff changeset
   351
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   352
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   353
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
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diff changeset
   354
lemma not_cong: \<open>P \<longleftrightarrow> P' \<Longrightarrow> \<not> P \<longleftrightarrow> \<not> P'\<close>
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wenzelm
parents: 21524
diff changeset
   355
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   356
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   357
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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diff changeset
   358
lemma all_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   359
  assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   360
  shows \<open>(\<forall>x. P(x)) \<longleftrightarrow> (\<forall>x. Q(x))\<close>
59529
d881f5288d5a proper context and method setup;
wenzelm
parents: 59498
diff changeset
   361
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE | iff assms)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   362
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   363
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   364
lemma ex_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   365
  assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   366
  shows \<open>(\<exists>x. P(x)) \<longleftrightarrow> (\<exists>x. Q(x))\<close>
59529
d881f5288d5a proper context and method setup;
wenzelm
parents: 59498
diff changeset
   367
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE | iff assms)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   368
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   369
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   370
lemma ex1_cong:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   371
  assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   372
  shows \<open>(\<exists>!x. P(x)) \<longleftrightarrow> (\<exists>!x. Q(x))\<close>
59529
d881f5288d5a proper context and method setup;
wenzelm
parents: 59498
diff changeset
   373
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE | iff assms)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   374
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
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diff changeset
   375
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   376
61487
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   377
subsection \<open>Equality rules\<close>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
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parents: 61378
diff changeset
   378
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   379
lemma sym: \<open>a = b \<Longrightarrow> b = a\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
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parents: 21524
diff changeset
   380
  apply (erule subst)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   381
  apply (rule refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   382
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   383
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   384
lemma trans: \<open>\<lbrakk>a = b; b = c\<rbrakk> \<Longrightarrow> a = c\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   385
  apply (erule subst, assumption)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   386
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   387
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   388
lemma not_sym: \<open>b \<noteq> a \<Longrightarrow> a \<noteq> b\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   389
  apply (erule contrapos)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   390
  apply (erule sym)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   391
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   392
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   393
text \<open>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   394
  Two theorems for rewriting only one instance of a definition:
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
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parents: 61378
diff changeset
   395
  the first for definitions of formulae and the second for terms.
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   396
\<close>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   397
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   398
lemma def_imp_iff: \<open>(A \<equiv> B) \<Longrightarrow> A \<longleftrightarrow> B\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   399
  apply unfold
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   400
  apply (rule iff_refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   401
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   402
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   403
lemma meta_eq_to_obj_eq: \<open>(A \<equiv> B) \<Longrightarrow> A = B\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   404
  apply unfold
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   405
  apply (rule refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   406
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   407
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   408
lemma meta_eq_to_iff: \<open>x \<equiv> y \<Longrightarrow> x \<longleftrightarrow> y\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   409
  by unfold (rule iff_refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   410
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   411
text \<open>Substitution.\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   412
lemma ssubst: \<open>\<lbrakk>b = a; P(a)\<rbrakk> \<Longrightarrow> P(b)\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   413
  apply (drule sym)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   414
  apply (erule (1) subst)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   415
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   416
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   417
text \<open>A special case of \<open>ex1E\<close> that would otherwise need quantifier
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   418
  expansion.\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   419
lemma ex1_equalsE: \<open>\<lbrakk>\<exists>!x. P(x); P(a); P(b)\<rbrakk> \<Longrightarrow> a = b\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   420
  apply (erule ex1E)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   421
  apply (rule trans)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   422
   apply (rule_tac [2] sym)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   423
   apply (assumption | erule spec [THEN mp])+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   424
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   425
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   426
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   427
subsubsection \<open>Polymorphic congruence rules\<close>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   428
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   429
lemma subst_context: \<open>a = b \<Longrightarrow> t(a) = t(b)\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   430
  apply (erule ssubst)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   431
  apply (rule refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   432
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   433
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   434
lemma subst_context2: \<open>\<lbrakk>a = b; c = d\<rbrakk> \<Longrightarrow> t(a,c) = t(b,d)\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   435
  apply (erule ssubst)+
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   436
  apply (rule refl)
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   437
  done
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   438
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   439
lemma subst_context3: \<open>\<lbrakk>a = b; c = d; e = f\<rbrakk> \<Longrightarrow> t(a,c,e) = t(b,d,f)\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   440
  apply (erule ssubst)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   441
  apply (rule refl)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   442
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   443
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   444
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69590
diff changeset
   445
  Useful with \<^ML>\<open>eresolve_tac\<close> for proving equalities from known
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   446
  equalities.
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   447
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   448
        a = b
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   449
        |   |
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   450
        c = d
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   451
\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   452
lemma box_equals: \<open>\<lbrakk>a = b; a = c; b = d\<rbrakk> \<Longrightarrow> c = d\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   453
  apply (rule trans)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   454
   apply (rule trans)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   455
    apply (rule sym)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   456
    apply assumption+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   457
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   458
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   459
text \<open>Dual of \<open>box_equals\<close>: for proving equalities backwards.\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   460
lemma simp_equals: \<open>\<lbrakk>a = c; b = d; c = d\<rbrakk> \<Longrightarrow> a = b\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   461
  apply (rule trans)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   462
   apply (rule trans)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   463
    apply assumption+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   464
  apply (erule sym)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   465
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   466
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   467
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   468
subsubsection \<open>Congruence rules for predicate letters\<close>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   469
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   470
lemma pred1_cong: \<open>a = a' \<Longrightarrow> P(a) \<longleftrightarrow> P(a')\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   471
  apply (rule iffI)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   472
   apply (erule (1) subst)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   473
  apply (erule (1) ssubst)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   474
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   475
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   476
lemma pred2_cong: \<open>\<lbrakk>a = a'; b = b'\<rbrakk> \<Longrightarrow> P(a,b) \<longleftrightarrow> P(a',b')\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   477
  apply (rule iffI)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   478
   apply (erule subst)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   479
   apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   480
  apply (erule ssubst)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   481
  apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   482
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   483
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   484
lemma pred3_cong: \<open>\<lbrakk>a = a'; b = b'; c = c'\<rbrakk> \<Longrightarrow> P(a,b,c) \<longleftrightarrow> P(a',b',c')\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   485
  apply (rule iffI)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   486
   apply (erule subst)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   487
   apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   488
  apply (erule ssubst)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   489
  apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   490
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   491
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   492
text \<open>Special case for the equality predicate!\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   493
lemma eq_cong: \<open>\<lbrakk>a = a'; b = b'\<rbrakk> \<Longrightarrow> a = b \<longleftrightarrow> a' = b'\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   494
  apply (erule (1) pred2_cong)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   495
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   496
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   497
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   498
subsection \<open>Simplifications of assumed implications\<close>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   499
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   500
text \<open>
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   501
  Roy Dyckhoff has proved that \<open>conj_impE\<close>, \<open>disj_impE\<close>, and
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69590
diff changeset
   502
  \<open>imp_impE\<close> used with \<^ML>\<open>mp_tac\<close> (restricted to atomic formulae) is
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   503
  COMPLETE for intuitionistic propositional logic.
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   504
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   505
  See R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   506
  (preprint, University of St Andrews, 1991).
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   507
\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   508
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   509
lemma conj_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   510
  assumes major: \<open>(P \<and> Q) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   511
    and r: \<open>P \<longrightarrow> (Q \<longrightarrow> S) \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   512
  shows \<open>R\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   513
  by (assumption | rule conjI impI major [THEN mp] r)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   514
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   515
lemma disj_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   516
  assumes major: \<open>(P \<or> Q) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   517
    and r: \<open>\<lbrakk>P \<longrightarrow> S; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   518
  shows \<open>R\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   519
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   520
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   521
text \<open>Simplifies the implication.  Classical version is stronger.
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   522
  Still UNSAFE since Q must be provable -- backtracking needed.\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   523
lemma imp_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   524
  assumes major: \<open>(P \<longrightarrow> Q) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   525
    and r1: \<open>\<lbrakk>P; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   526
    and r2: \<open>S \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   527
  shows \<open>R\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   528
  by (assumption | rule impI major [THEN mp] r1 r2)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   529
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   530
text \<open>Simplifies the implication.  Classical version is stronger.
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   531
  Still UNSAFE since ~P must be provable -- backtracking needed.\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   532
lemma not_impE: \<open>\<not> P \<longrightarrow> S \<Longrightarrow> (P \<Longrightarrow> False) \<Longrightarrow> (S \<Longrightarrow> R) \<Longrightarrow> R\<close>
23393
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   533
  apply (drule mp)
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   534
   apply (rule notI)
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   535
   apply assumption
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   536
  apply assumption
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   537
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   538
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   539
text \<open>Simplifies the implication. UNSAFE.\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   540
lemma iff_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   541
  assumes major: \<open>(P \<longleftrightarrow> Q) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   542
    and r1: \<open>\<lbrakk>P; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   543
    and r2: \<open>\<lbrakk>Q; P \<longrightarrow> S\<rbrakk> \<Longrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   544
    and r3: \<open>S \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   545
  shows \<open>R\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   546
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   547
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   548
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   549
text \<open>What if \<open>(\<forall>x. \<not> \<not> P(x)) \<longrightarrow> \<not> \<not> (\<forall>x. P(x))\<close> is an assumption?
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   550
  UNSAFE.\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   551
lemma all_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   552
  assumes major: \<open>(\<forall>x. P(x)) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   553
    and r1: \<open>\<And>x. P(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   554
    and r2: \<open>S \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   555
  shows \<open>R\<close>
23393
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   556
  apply (rule allI impI major [THEN mp] r1 r2)+
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   557
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   558
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   559
text \<open>
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   560
  Unsafe: \<open>\<exists>x. P(x)) \<longrightarrow> S\<close> is equivalent
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   561
  to \<open>\<forall>x. P(x) \<longrightarrow> S\<close>.\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   562
lemma ex_impE:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   563
  assumes major: \<open>(\<exists>x. P(x)) \<longrightarrow> S\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   564
    and r: \<open>P(x) \<longrightarrow> S \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   565
  shows \<open>R\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   566
  apply (assumption | rule exI impI major [THEN mp] r)+
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   567
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   568
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   569
text \<open>Courtesy of Krzysztof Grabczewski.\<close>
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   570
lemma disj_imp_disj: \<open>P \<or> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> S) \<Longrightarrow> R \<or> S\<close>
23393
31781b2de73d tuned proofs: avoid implicit prems;
wenzelm
parents: 23171
diff changeset
   571
  apply (erule disjE)
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   572
  apply (rule disjI1) apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   573
  apply (rule disjI2) apply assumption
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   574
  done
11734
7a21bf539412 declare impE iffD1 iffD2 ad elim of Pure;
wenzelm
parents: 11677
diff changeset
   575
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   576
ML \<open>
32172
c4e55f30d527 renamed functor ProjectRuleFun to Project_Rule;
wenzelm
parents: 31299
diff changeset
   577
structure Project_Rule = Project_Rule
c4e55f30d527 renamed functor ProjectRuleFun to Project_Rule;
wenzelm
parents: 31299
diff changeset
   578
(
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21539
diff changeset
   579
  val conjunct1 = @{thm conjunct1}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21539
diff changeset
   580
  val conjunct2 = @{thm conjunct2}
539a63b98f76 tuned ML setup;
wenzelm
parents: 21539
diff changeset
   581
  val mp = @{thm mp}
32172
c4e55f30d527 renamed functor ProjectRuleFun to Project_Rule;
wenzelm
parents: 31299
diff changeset
   582
)
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   583
\<close>
18481
b75ce99617c7 structure ProjectRule;
wenzelm
parents: 17702
diff changeset
   584
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 69593
diff changeset
   585
ML_file \<open>fologic.ML\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   586
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   587
lemma thin_refl: \<open>\<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W\<close> .
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   588
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   589
ML \<open>
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   590
structure Hypsubst = Hypsubst
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   591
(
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   592
  val dest_eq = FOLogic.dest_eq
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   593
  val dest_Trueprop = FOLogic.dest_Trueprop
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   594
  val dest_imp = FOLogic.dest_imp
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   595
  val eq_reflection = @{thm eq_reflection}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   596
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   597
  val imp_intr = @{thm impI}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   598
  val rev_mp = @{thm rev_mp}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   599
  val subst = @{thm subst}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   600
  val sym = @{thm sym}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   601
  val thin_refl = @{thm thin_refl}
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   602
);
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   603
open Hypsubst;
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   604
\<close>
42799
4e33894aec6d modernized functor names;
wenzelm
parents: 42303
diff changeset
   605
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 69593
diff changeset
   606
ML_file \<open>intprover.ML\<close>
7355
4c43090659ca proper bootstrap of IFOL/FOL theories and packages;
wenzelm
parents: 6340
diff changeset
   607
4092
9faf228771dc added simpset thy_data;
wenzelm
parents: 3932
diff changeset
   608
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   609
subsection \<open>Intuitionistic Reasoning\<close>
12368
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   610
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69590
diff changeset
   611
setup \<open>Intuitionistic.method_setup \<^binding>\<open>iprover\<close>\<close>
30165
6ee87f67d9cd moved generic intuitionistic prover to src/Tools/intuitionistic.ML;
wenzelm
parents: 30160
diff changeset
   612
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   613
lemma impE':
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   614
  assumes 1: \<open>P \<longrightarrow> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   615
    and 2: \<open>Q \<Longrightarrow> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   616
    and 3: \<open>P \<longrightarrow> Q \<Longrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   617
  shows \<open>R\<close>
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   618
proof -
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   619
  from 3 and 1 have \<open>P\<close> .
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   620
  with 1 have \<open>Q\<close> by (rule impE)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   621
  with 2 show \<open>R\<close> .
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   622
qed
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   623
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   624
lemma allE':
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   625
  assumes 1: \<open>\<forall>x. P(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   626
    and 2: \<open>P(x) \<Longrightarrow> \<forall>x. P(x) \<Longrightarrow> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   627
  shows \<open>Q\<close>
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   628
proof -
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   629
  from 1 have \<open>P(x)\<close> by (rule spec)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   630
  from this and 1 show \<open>Q\<close> by (rule 2)
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   631
qed
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   632
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12875
diff changeset
   633
lemma notE':
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   634
  assumes 1: \<open>\<not> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   635
    and 2: \<open>\<not> P \<Longrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   636
  shows \<open>R\<close>
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   637
proof -
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   638
  from 2 and 1 have \<open>P\<close> .
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   639
  with 1 show \<open>R\<close> by (rule notE)
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   640
qed
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   641
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   642
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   643
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   644
  and [Pure.elim 2] = allE notE' impE'
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   645
  and [Pure.intro] = exI disjI2 disjI1
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   646
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   647
setup \<open>
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   648
  Context_Rules.addSWrapper
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   649
    (fn ctxt => fn tac => hyp_subst_tac ctxt ORELSE' tac)
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   650
\<close>
12349
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   651
94e812f9683e setup "rules" method;
wenzelm
parents: 12114
diff changeset
   652
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   653
lemma iff_not_sym: \<open>\<not> (Q \<longleftrightarrow> P) \<Longrightarrow> \<not> (P \<longleftrightarrow> Q)\<close>
17591
33d409318266 renamed rules to iprover
nipkow
parents: 17276
diff changeset
   654
  by iprover
12368
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   655
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   656
lemmas [sym] = sym iff_sym not_sym iff_not_sym
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   657
  and [Pure.elim?] = iffD1 iffD2 impE
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   658
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   659
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   660
lemma eq_commute: \<open>a = b \<longleftrightarrow> b = a\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   661
  apply (rule iffI)
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   662
  apply (erule sym)+
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   663
  done
13435
05631e8f0258 new theorem eq_commute
paulson
parents: 12937
diff changeset
   664
05631e8f0258 new theorem eq_commute
paulson
parents: 12937
diff changeset
   665
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   666
subsection \<open>Atomizing meta-level rules\<close>
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   667
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   668
lemma atomize_all [atomize]: \<open>(\<And>x. P(x)) \<equiv> Trueprop (\<forall>x. P(x))\<close>
11976
075df6e46cef equal_intr_rule already declared in Pure;
wenzelm
parents: 11953
diff changeset
   669
proof
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   670
  assume \<open>\<And>x. P(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   671
  then show \<open>\<forall>x. P(x)\<close> ..
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   672
next
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   673
  assume \<open>\<forall>x. P(x)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   674
  then show \<open>\<And>x. P(x)\<close> ..
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   675
qed
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   676
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   677
lemma atomize_imp [atomize]: \<open>(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)\<close>
11976
075df6e46cef equal_intr_rule already declared in Pure;
wenzelm
parents: 11953
diff changeset
   678
proof
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   679
  assume \<open>A \<Longrightarrow> B\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   680
  then show \<open>A \<longrightarrow> B\<close> ..
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   681
next
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   682
  assume \<open>A \<longrightarrow> B\<close> and \<open>A\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   683
  then show \<open>B\<close> by (rule mp)
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   684
qed
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   685
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   686
lemma atomize_eq [atomize]: \<open>(x \<equiv> y) \<equiv> Trueprop (x = y)\<close>
11976
075df6e46cef equal_intr_rule already declared in Pure;
wenzelm
parents: 11953
diff changeset
   687
proof
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   688
  assume \<open>x \<equiv> y\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   689
  show \<open>x = y\<close> unfolding \<open>x \<equiv> y\<close> by (rule refl)
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   690
next
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   691
  assume \<open>x = y\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   692
  then show \<open>x \<equiv> y\<close> by (rule eq_reflection)
11677
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   693
qed
ee12f18599e5 atomize stuff from theory FOL;
wenzelm
parents: 9886
diff changeset
   694
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   695
lemma atomize_iff [atomize]: \<open>(A \<equiv> B) \<equiv> Trueprop (A \<longleftrightarrow> B)\<close>
18813
fc35dcc2558b added atomize_iff;
wenzelm
parents: 18708
diff changeset
   696
proof
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   697
  assume \<open>A \<equiv> B\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   698
  show \<open>A \<longleftrightarrow> B\<close> unfolding \<open>A \<equiv> B\<close> by (rule iff_refl)
18813
fc35dcc2558b added atomize_iff;
wenzelm
parents: 18708
diff changeset
   699
next
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   700
  assume \<open>A \<longleftrightarrow> B\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   701
  then show \<open>A \<equiv> B\<close> by (rule iff_reflection)
18813
fc35dcc2558b added atomize_iff;
wenzelm
parents: 18708
diff changeset
   702
qed
fc35dcc2558b added atomize_iff;
wenzelm
parents: 18708
diff changeset
   703
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   704
lemma atomize_conj [atomize]: \<open>(A &&& B) \<equiv> Trueprop (A \<and> B)\<close>
11976
075df6e46cef equal_intr_rule already declared in Pure;
wenzelm
parents: 11953
diff changeset
   705
proof
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   706
  assume conj: \<open>A &&& B\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   707
  show \<open>A \<and> B\<close>
19120
353d349740de not_equal: replaced syntax translation by abbreviation;
wenzelm
parents: 18861
diff changeset
   708
  proof (rule conjI)
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   709
    from conj show \<open>A\<close> by (rule conjunctionD1)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   710
    from conj show \<open>B\<close> by (rule conjunctionD2)
19120
353d349740de not_equal: replaced syntax translation by abbreviation;
wenzelm
parents: 18861
diff changeset
   711
  qed
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11848
diff changeset
   712
next
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   713
  assume conj: \<open>A \<and> B\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   714
  show \<open>A &&& B\<close>
19120
353d349740de not_equal: replaced syntax translation by abbreviation;
wenzelm
parents: 18861
diff changeset
   715
  proof -
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   716
    from conj show \<open>A\<close> ..
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   717
    from conj show \<open>B\<close> ..
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11848
diff changeset
   718
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11848
diff changeset
   719
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11848
diff changeset
   720
12368
2af9ad81ea56 sym declarations;
wenzelm
parents: 12352
diff changeset
   721
lemmas [symmetric, rulify] = atomize_all atomize_imp
18861
38269ef5175a declare defn rules;
wenzelm
parents: 18813
diff changeset
   722
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
11771
b7b100a2de1d moved rulify to ObjectLogic;
wenzelm
parents: 11747
diff changeset
   723
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   724
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   725
subsection \<open>Atomizing elimination rules\<close>
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   726
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   727
lemma atomize_exL[atomize_elim]: \<open>(\<And>x. P(x) \<Longrightarrow> Q) \<equiv> ((\<exists>x. P(x)) \<Longrightarrow> Q)\<close>
57948
75724d71013c modernized module name and setup;
wenzelm
parents: 55380
diff changeset
   728
  by rule iprover+
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   729
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   730
lemma atomize_conjL[atomize_elim]: \<open>(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)\<close>
57948
75724d71013c modernized module name and setup;
wenzelm
parents: 55380
diff changeset
   731
  by rule iprover+
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   732
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   733
lemma atomize_disjL[atomize_elim]: \<open>((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)\<close>
57948
75724d71013c modernized module name and setup;
wenzelm
parents: 55380
diff changeset
   734
  by rule iprover+
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   735
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   736
lemma atomize_elimL[atomize_elim]: \<open>(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop(A)\<close> ..
26580
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   737
c3e597a476fd Generic conversion and tactic "atomize_elim" to convert elimination rules
krauss
parents: 26286
diff changeset
   738
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   739
subsection \<open>Calculational rules\<close>
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   740
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   741
lemma forw_subst: \<open>a = b \<Longrightarrow> P(b) \<Longrightarrow> P(a)\<close>
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   742
  by (rule ssubst)
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   743
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   744
lemma back_subst: \<open>P(a) \<Longrightarrow> a = b \<Longrightarrow> P(b)\<close>
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   745
  by (rule subst)
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   746
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   747
text \<open>
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   748
  Note that this list of rules is in reverse order of priorities.
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   749
\<close>
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   750
12019
abe9b7c6016e transitive declared in Pure;
wenzelm
parents: 11976
diff changeset
   751
lemmas basic_trans_rules [trans] =
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   752
  forw_subst
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   753
  back_subst
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   754
  rev_mp
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   755
  mp
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   756
  trans
6e3017adb8c0 calculational rules moved from FOL to IFOL;
wenzelm
parents: 11771
diff changeset
   757
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   758
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   759
subsection \<open>``Let'' declarations\<close>
13779
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   760
41229
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 39557
diff changeset
   761
nonterminal letbinds and letbind
13779
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   762
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   763
definition Let :: \<open>['a::{}, 'a => 'b] \<Rightarrow> ('b::{})\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   764
  where \<open>Let(s, f) \<equiv> f(s)\<close>
13779
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   765
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   766
syntax
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   767
  "_bind"       :: \<open>[pttrn, 'a] => letbind\<close>           (\<open>(2_ =/ _)\<close> 10)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   768
  ""            :: \<open>letbind => letbinds\<close>              (\<open>_\<close>)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   769
  "_binds"      :: \<open>[letbind, letbinds] => letbinds\<close>  (\<open>_;/ _\<close>)
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   770
  "_Let"        :: \<open>[letbinds, 'a] => 'a\<close>             (\<open>(let (_)/ in (_))\<close> 10)
13779
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   771
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   772
translations
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   773
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   774
  "let x = a in e"          == "CONST Let(a, \<lambda>x. e)"
13779
2a34dc5cf79e moving "let" from ZF to FOL
paulson
parents: 13435
diff changeset
   775
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   776
lemma LetI:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   777
  assumes \<open>\<And>x. x = t \<Longrightarrow> P(u(x))\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   778
  shows \<open>P(let x = t in u(x))\<close>
21539
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   779
  apply (unfold Let_def)
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   780
  apply (rule refl [THEN assms])
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   781
  done
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   782
c5cf9243ad62 converted legacy ML scripts;
wenzelm
parents: 21524
diff changeset
   783
60770
240563fbf41d isabelle update_cartouches;
wenzelm
parents: 59529
diff changeset
   784
subsection \<open>Intuitionistic simplification rules\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   785
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   786
lemma conj_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   787
  \<open>P \<and> True \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   788
  \<open>True \<and> P \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   789
  \<open>P \<and> False \<longleftrightarrow> False\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   790
  \<open>False \<and> P \<longleftrightarrow> False\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   791
  \<open>P \<and> P \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   792
  \<open>P \<and> P \<and> Q \<longleftrightarrow> P \<and> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   793
  \<open>P \<and> \<not> P \<longleftrightarrow> False\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   794
  \<open>\<not> P \<and> P \<longleftrightarrow> False\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   795
  \<open>(P \<and> Q) \<and> R \<longleftrightarrow> P \<and> (Q \<and> R)\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   796
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   797
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   798
lemma disj_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   799
  \<open>P \<or> True \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   800
  \<open>True \<or> P \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   801
  \<open>P \<or> False \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   802
  \<open>False \<or> P \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   803
  \<open>P \<or> P \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   804
  \<open>P \<or> P \<or> Q \<longleftrightarrow> P \<or> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   805
  \<open>(P \<or> Q) \<or> R \<longleftrightarrow> P \<or> (Q \<or> R)\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   806
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   807
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   808
lemma not_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   809
  \<open>\<not> (P \<or> Q) \<longleftrightarrow> \<not> P \<and> \<not> Q\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   810
  \<open>\<not> False \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   811
  \<open>\<not> True \<longleftrightarrow> False\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   812
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   813
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   814
lemma imp_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   815
  \<open>(P \<longrightarrow> False) \<longleftrightarrow> \<not> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   816
  \<open>(P \<longrightarrow> True) \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   817
  \<open>(False \<longrightarrow> P) \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   818
  \<open>(True \<longrightarrow> P) \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   819
  \<open>(P \<longrightarrow> P) \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   820
  \<open>(P \<longrightarrow> \<not> P) \<longleftrightarrow> \<not> P\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   821
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   822
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   823
lemma iff_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   824
  \<open>(True \<longleftrightarrow> P) \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   825
  \<open>(P \<longleftrightarrow> True) \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   826
  \<open>(P \<longleftrightarrow> P) \<longleftrightarrow> True\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   827
  \<open>(False \<longleftrightarrow> P) \<longleftrightarrow> \<not> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   828
  \<open>(P \<longleftrightarrow> False) \<longleftrightarrow> \<not> P\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   829
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   830
62020
5d208fd2507d isabelle update_cartouches -c -t;
wenzelm
parents: 61490
diff changeset
   831
text \<open>The \<open>x = t\<close> versions are needed for the simplification
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   832
  procedures.\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   833
lemma quant_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   834
  \<open>\<And>P. (\<forall>x. P) \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   835
  \<open>(\<forall>x. x = t \<longrightarrow> P(x)) \<longleftrightarrow> P(t)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   836
  \<open>(\<forall>x. t = x \<longrightarrow> P(x)) \<longleftrightarrow> P(t)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   837
  \<open>\<And>P. (\<exists>x. P) \<longleftrightarrow> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   838
  \<open>\<exists>x. x = t\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   839
  \<open>\<exists>x. t = x\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   840
  \<open>(\<exists>x. x = t \<and> P(x)) \<longleftrightarrow> P(t)\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   841
  \<open>(\<exists>x. t = x \<and> P(x)) \<longleftrightarrow> P(t)\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   842
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   843
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   844
text \<open>These are NOT supplied by default!\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   845
lemma distrib_simps:
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   846
  \<open>P \<and> (Q \<or> R) \<longleftrightarrow> P \<and> Q \<or> P \<and> R\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   847
  \<open>(Q \<or> R) \<and> P \<longleftrightarrow> Q \<and> P \<or> R \<and> P\<close>
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   848
  \<open>(P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   849
  by iprover+
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   850
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   851
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   852
subsubsection \<open>Conversion into rewrite rules\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   853
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   854
lemma P_iff_F: \<open>\<not> P \<Longrightarrow> (P \<longleftrightarrow> False)\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   855
  by iprover
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   856
lemma iff_reflection_F: \<open>\<not> P \<Longrightarrow> (P \<equiv> False)\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   857
  by (rule P_iff_F [THEN iff_reflection])
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   858
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   859
lemma P_iff_T: \<open>P \<Longrightarrow> (P \<longleftrightarrow> True)\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   860
  by iprover
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   861
lemma iff_reflection_T: \<open>P \<Longrightarrow> (P \<equiv> True)\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   862
  by (rule P_iff_T [THEN iff_reflection])
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   863
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   864
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   865
subsubsection \<open>More rewrite rules\<close>
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   866
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   867
lemma conj_commute: \<open>P \<and> Q \<longleftrightarrow> Q \<and> P\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   868
lemma conj_left_commute: \<open>P \<and> (Q \<and> R) \<longleftrightarrow> Q \<and> (P \<and> R)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   869
lemmas conj_comms = conj_commute conj_left_commute
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   870
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   871
lemma disj_commute: \<open>P \<or> Q \<longleftrightarrow> Q \<or> P\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   872
lemma disj_left_commute: \<open>P \<or> (Q \<or> R) \<longleftrightarrow> Q \<or> (P \<or> R)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   873
lemmas disj_comms = disj_commute disj_left_commute
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   874
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   875
lemma conj_disj_distribL: \<open>P \<and> (Q \<or> R) \<longleftrightarrow> (P \<and> Q \<or> P \<and> R)\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   876
lemma conj_disj_distribR: \<open>(P \<or> Q) \<and> R \<longleftrightarrow> (P \<and> R \<or> Q \<and> R)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   877
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   878
lemma disj_conj_distribL: \<open>P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   879
lemma disj_conj_distribR: \<open>(P \<and> Q) \<or> R \<longleftrightarrow> (P \<or> R) \<and> (Q \<or> R)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   880
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   881
lemma imp_conj_distrib: \<open>(P \<longrightarrow> (Q \<and> R)) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (P \<longrightarrow> R)\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   882
lemma imp_conj: \<open>((P \<and> Q) \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> (Q \<longrightarrow> R))\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   883
lemma imp_disj: \<open>(P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   884
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   885
lemma de_Morgan_disj: \<open>(\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   886
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   887
lemma not_ex: \<open>(\<not> (\<exists>x. P(x))) \<longleftrightarrow> (\<forall>x. \<not> P(x))\<close> by iprover
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   888
lemma imp_ex: \<open>((\<exists>x. P(x)) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P(x) \<longrightarrow> Q)\<close> by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   889
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   890
lemma ex_disj_distrib: \<open>(\<exists>x. P(x) \<or> Q(x)) \<longleftrightarrow> ((\<exists>x. P(x)) \<or> (\<exists>x. Q(x)))\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   891
  by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   892
69590
e65314985426 isabelle update_inner_syntax_cartouches;
wenzelm
parents: 69587
diff changeset
   893
lemma all_conj_distrib: \<open>(\<forall>x. P(x) \<and> Q(x)) \<longleftrightarrow> ((\<forall>x. P(x)) \<and> (\<forall>x. Q(x)))\<close>
61487
f8cb97e0fd0b more symbols, with swapped defaults: old-style ASCII syntax uses "ASCII" print mode;
wenzelm
parents: 61378
diff changeset
   894
  by iprover
26286
3ff5d257f175 added lemmas from simpdata.ML;
wenzelm
parents: 24830
diff changeset
   895
4854
d1850e0964f2 tuned setup;
wenzelm
parents: 4793
diff changeset
   896
end