src/FOL/IFOL.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58826 2ed2eaabe3df
child 58963 26bf09b95dda
permissions -rw-r--r--
modernized header uniformly as section;
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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 {* Intuitionistic first-order logic *}
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theory IFOL
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imports Pure
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begin
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ML_file "~~/src/Tools/misc_legacy.ML"
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ML_file "~~/src/Provers/splitter.ML"
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ML_file "~~/src/Provers/hypsubst.ML"
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ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
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ML_file "~~/src/Tools/IsaPlanner/isand.ML"
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ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
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ML_file "~~/src/Provers/quantifier1.ML"
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ML_file "~~/src/Tools/intuitionistic.ML"
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ML_file "~~/src/Tools/project_rule.ML"
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ML_file "~~/src/Tools/atomize_elim.ML"
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subsection {* Syntax and axiomatic basis *}
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setup Pure_Thy.old_appl_syntax_setup
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class "term"
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default_sort "term"
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typedecl o
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judgment
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  Trueprop      :: "o => prop"                  ("(_)" 5)
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subsubsection {* Equality *}
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axiomatization
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  eq :: "['a, 'a] => o"  (infixl "=" 50)
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where
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  refl:         "a=a" and
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  subst:        "a=b \<Longrightarrow> P(a) \<Longrightarrow> P(b)"
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subsubsection {* Propositional logic *}
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axiomatization
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  False :: o and
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  conj :: "[o, o] => o"  (infixr "&" 35) and
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  disj :: "[o, o] => o"  (infixr "|" 30) and
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  imp :: "[o, o] => o"  (infixr "-->" 25)
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where
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  conjI: "[| P;  Q |] ==> P&Q" and
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  conjunct1: "P&Q ==> P" and
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  conjunct2: "P&Q ==> Q" and
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  disjI1: "P ==> P|Q" and
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  disjI2: "Q ==> P|Q" and
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  disjE: "[| P|Q;  P ==> R;  Q ==> R |] ==> R" and
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  impI: "(P ==> Q) ==> P-->Q" and
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  mp: "[| P-->Q;  P |] ==> Q" and
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  FalseE: "False ==> P"
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subsubsection {* Quantifiers *}
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axiomatization
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  All :: "('a => o) => o"  (binder "ALL " 10) and
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  Ex :: "('a => o) => o"  (binder "EX " 10)
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where
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  allI: "(!!x. P(x)) ==> (ALL x. P(x))" and
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  spec: "(ALL x. P(x)) ==> P(x)" and
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  exI: "P(x) ==> (EX x. P(x))" and
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  exE: "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"
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subsubsection {* Definitions *}
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definition "True == False-->False"
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definition Not ("~ _" [40] 40) where not_def: "~P == P-->False"
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definition iff  (infixr "<->" 25) where "P<->Q == (P-->Q) & (Q-->P)"
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definition Ex1 :: "('a => o) => o"  (binder "EX! " 10)
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  where ex1_def: "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
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axiomatization where  -- {* Reflection, admissible *}
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  eq_reflection: "(x=y) ==> (x==y)" and
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  iff_reflection: "(P<->Q) ==> (P==Q)"
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subsubsection {* Additional notation *}
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abbreviation not_equal :: "['a, 'a] => o"  (infixl "~=" 50)
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  where "x ~= y == ~ (x = y)"
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notation (xsymbols)
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  not_equal  (infixl "\<noteq>" 50)
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notation (HTML output)
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  not_equal  (infixl "\<noteq>" 50)
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notation (xsymbols)
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  Not  ("\<not> _" [40] 40) and
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  conj  (infixr "\<and>" 35) and
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  disj  (infixr "\<or>" 30) and
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10) and
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  imp  (infixr "\<longrightarrow>" 25) and
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  iff  (infixr "\<longleftrightarrow>" 25)
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notation (HTML output)
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  Not  ("\<not> _" [40] 40) and
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  conj  (infixr "\<and>" 35) and
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  disj  (infixr "\<or>" 30) and
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  All  (binder "\<forall>" 10) and
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  Ex  (binder "\<exists>" 10) and
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  Ex1  (binder "\<exists>!" 10)
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subsection {* Lemmas and proof tools *}
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lemmas strip = impI allI
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lemma TrueI: True
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  unfolding True_def by (rule impI)
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(*** Sequent-style elimination rules for & --> and ALL ***)
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lemma conjE:
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  assumes major: "P & Q"
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    and r: "[| P; Q |] ==> R"
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  shows R
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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: "P --> Q"
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    and P
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  and r: "Q ==> R"
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  shows R
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  apply (rule r)
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  apply (rule major [THEN mp])
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  apply (rule `P`)
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  done
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lemma allE:
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  assumes major: "ALL x. P(x)"
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    and r: "P(x) ==> R"
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  shows R
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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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(*Duplicates the quantifier; for use with eresolve_tac*)
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lemma all_dupE:
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  assumes major: "ALL x. P(x)"
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    and r: "[| P(x); ALL x. P(x) |] ==> R"
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  shows R
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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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(*** Negation rules, which translate between ~P and P-->False ***)
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lemma notI: "(P ==> False) ==> ~P"
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  unfolding not_def by (erule impI)
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lemma notE: "[| ~P;  P |] ==> R"
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  unfolding not_def by (erule mp [THEN FalseE])
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lemma rev_notE: "[| P; ~P |] ==> R"
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  by (erule notE)
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(*This is useful with the special implication rules for each kind of P. *)
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lemma not_to_imp:
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  assumes "~P"
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    and r: "P --> False ==> Q"
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  shows Q
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  apply (rule r)
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  apply (rule impI)
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  apply (erule notE [OF `~P`])
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  done
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(* For substitution into an assumption P, reduce Q to P-->Q, substitute into
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   this implication, then apply impI to move P back into the assumptions.*)
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lemma rev_mp: "[| P;  P --> Q |] ==> Q"
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  by (erule mp)
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(*Contrapositive of an inference rule*)
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lemma contrapos:
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  assumes major: "~Q"
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    and minor: "P ==> Q"
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  shows "~P"
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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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(*** Modus Ponens Tactics ***)
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(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
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ML {*
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  fun mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i  THEN  assume_tac i
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  fun eq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i  THEN  eq_assume_tac i
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*}
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(*** If-and-only-if ***)
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lemma iffI: "[| P ==> Q; Q ==> P |] ==> P<->Q"
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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: "P <-> Q"
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    and r: "P-->Q ==> Q-->P ==> R"
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  shows R
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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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(* Destruct rules for <-> similar to Modus Ponens *)
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lemma iffD1: "[| P <-> Q;  P |] ==> Q"
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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: "[| P <-> Q;  Q |] ==> P"
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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: "[| P; P <-> Q |] ==> Q"
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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: "[| Q; P <-> Q |] ==> P"
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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: "P <-> P"
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  by (rule iffI)
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lemma iff_sym: "Q <-> P ==> P <-> Q"
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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: "[| P <-> Q;  Q<-> R |] ==> P <-> R"
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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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(*** Unique existence.  NOTE THAT the following 2 quantifications
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   EX!x such that [EX!y such that P(x,y)]     (sequential)
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   EX!x,y such that P(x,y)                    (simultaneous)
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 do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
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***)
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lemma ex1I:
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  "P(a) \<Longrightarrow> (!!x. P(x) ==> x=a) \<Longrightarrow> EX! x. P(x)"
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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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(*Sometimes easier to use: the premises have no shared variables.  Safe!*)
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lemma ex_ex1I:
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  "EX x. P(x) \<Longrightarrow> (!!x y. [| P(x); P(y) |] ==> x=y) \<Longrightarrow> EX! x. P(x)"
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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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lemma ex1E:
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  "EX! x. P(x) \<Longrightarrow> (!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R) \<Longrightarrow> R"
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  apply (unfold ex1_def)
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  apply (assumption | erule exE conjE)+
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  done
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(*** <-> congruence rules for simplification ***)
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(*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
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ML {*
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  fun iff_tac prems i =
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    resolve_tac (prems RL @{thms iffE}) i THEN
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    REPEAT1 (eresolve_tac [@{thm asm_rl}, @{thm mp}] i)
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*}
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lemma conj_cong:
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  assumes "P <-> P'"
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    and "P' ==> Q <-> Q'"
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  shows "(P&Q) <-> (P'&Q')"
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  apply (insert assms)
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  apply (assumption | rule iffI conjI | erule iffE conjE mp |
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    tactic {* iff_tac @{thms assms} 1 *})+
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  done
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(*Reversed congruence rule!   Used in ZF/Order*)
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lemma conj_cong2:
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  assumes "P <-> P'"
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    and "P' ==> Q <-> Q'"
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  shows "(Q&P) <-> (Q'&P')"
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  apply (insert assms)
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  apply (assumption | rule iffI conjI | erule iffE conjE mp |
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    tactic {* iff_tac @{thms assms} 1 *})+
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  done
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lemma disj_cong:
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  assumes "P <-> P'" and "Q <-> Q'"
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  shows "(P|Q) <-> (P'|Q')"
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  apply (insert assms)
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  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
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  done
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lemma imp_cong:
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  assumes "P <-> P'"
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    and "P' ==> Q <-> Q'"
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  shows "(P-->Q) <-> (P'-->Q')"
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  apply (insert assms)
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  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
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    tactic {* iff_tac @{thms assms} 1 *})+
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  done
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lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
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  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
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  done
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lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
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  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
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  done
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lemma all_cong:
wenzelm@21539
   356
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   357
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   358
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@39159
   359
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   360
  done
wenzelm@21539
   361
wenzelm@21539
   362
lemma ex_cong:
wenzelm@21539
   363
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   364
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   365
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@39159
   366
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   367
  done
wenzelm@21539
   368
wenzelm@21539
   369
lemma ex1_cong:
wenzelm@21539
   370
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   371
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   372
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@39159
   373
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   374
  done
wenzelm@21539
   375
wenzelm@21539
   376
(*** Equality rules ***)
wenzelm@21539
   377
wenzelm@21539
   378
lemma sym: "a=b ==> b=a"
wenzelm@21539
   379
  apply (erule subst)
wenzelm@21539
   380
  apply (rule refl)
wenzelm@21539
   381
  done
wenzelm@21539
   382
wenzelm@21539
   383
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   384
  apply (erule subst, assumption)
wenzelm@21539
   385
  done
wenzelm@21539
   386
wenzelm@21539
   387
(**  **)
wenzelm@21539
   388
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   389
  apply (erule contrapos)
wenzelm@21539
   390
  apply (erule sym)
wenzelm@21539
   391
  done
wenzelm@21539
   392
  
wenzelm@21539
   393
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   394
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   395
wenzelm@21539
   396
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   397
  apply unfold
wenzelm@21539
   398
  apply (rule iff_refl)
wenzelm@21539
   399
  done
wenzelm@21539
   400
wenzelm@21539
   401
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   402
  apply unfold
wenzelm@21539
   403
  apply (rule refl)
wenzelm@21539
   404
  done
wenzelm@21539
   405
wenzelm@21539
   406
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   407
  by unfold (rule iff_refl)
wenzelm@21539
   408
wenzelm@21539
   409
(*substitution*)
wenzelm@21539
   410
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   411
  apply (drule sym)
wenzelm@21539
   412
  apply (erule (1) subst)
wenzelm@21539
   413
  done
wenzelm@21539
   414
wenzelm@21539
   415
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   416
lemma ex1_equalsE:
wenzelm@21539
   417
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   418
  apply (erule ex1E)
wenzelm@21539
   419
  apply (rule trans)
wenzelm@21539
   420
   apply (rule_tac [2] sym)
wenzelm@21539
   421
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   422
  done
wenzelm@21539
   423
wenzelm@21539
   424
(** Polymorphic congruence rules **)
wenzelm@21539
   425
wenzelm@21539
   426
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   427
  apply (erule ssubst)
wenzelm@21539
   428
  apply (rule refl)
wenzelm@21539
   429
  done
wenzelm@21539
   430
wenzelm@21539
   431
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   432
  apply (erule ssubst)+
wenzelm@21539
   433
  apply (rule refl)
wenzelm@21539
   434
  done
wenzelm@21539
   435
wenzelm@21539
   436
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   437
  apply (erule ssubst)+
wenzelm@21539
   438
  apply (rule refl)
wenzelm@21539
   439
  done
wenzelm@21539
   440
wenzelm@21539
   441
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   442
        a = b
wenzelm@21539
   443
        |   |
wenzelm@21539
   444
        c = d   *)
wenzelm@21539
   445
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   446
  apply (rule trans)
wenzelm@21539
   447
   apply (rule trans)
wenzelm@21539
   448
    apply (rule sym)
wenzelm@21539
   449
    apply assumption+
wenzelm@21539
   450
  done
wenzelm@21539
   451
wenzelm@21539
   452
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   453
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   454
  apply (rule trans)
wenzelm@21539
   455
   apply (rule trans)
wenzelm@21539
   456
    apply assumption+
wenzelm@21539
   457
  apply (erule sym)
wenzelm@21539
   458
  done
wenzelm@21539
   459
wenzelm@21539
   460
(** Congruence rules for predicate letters **)
wenzelm@21539
   461
wenzelm@21539
   462
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   463
  apply (rule iffI)
wenzelm@21539
   464
   apply (erule (1) subst)
wenzelm@21539
   465
  apply (erule (1) ssubst)
wenzelm@21539
   466
  done
wenzelm@21539
   467
wenzelm@21539
   468
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   469
  apply (rule iffI)
wenzelm@21539
   470
   apply (erule subst)+
wenzelm@21539
   471
   apply assumption
wenzelm@21539
   472
  apply (erule ssubst)+
wenzelm@21539
   473
  apply assumption
wenzelm@21539
   474
  done
wenzelm@21539
   475
wenzelm@21539
   476
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
wenzelm@21539
   477
  apply (rule iffI)
wenzelm@21539
   478
   apply (erule subst)+
wenzelm@21539
   479
   apply assumption
wenzelm@21539
   480
  apply (erule ssubst)+
wenzelm@21539
   481
  apply assumption
wenzelm@21539
   482
  done
wenzelm@21539
   483
wenzelm@21539
   484
(*special case for the equality predicate!*)
wenzelm@21539
   485
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   486
  apply (erule (1) pred2_cong)
wenzelm@21539
   487
  done
wenzelm@21539
   488
wenzelm@21539
   489
wenzelm@21539
   490
(*** Simplifications of assumed implications.
wenzelm@21539
   491
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   492
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   493
     intuitionistic propositional logic.  See
wenzelm@21539
   494
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   495
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   496
wenzelm@21539
   497
lemma conj_impE:
wenzelm@21539
   498
  assumes major: "(P&Q)-->S"
wenzelm@21539
   499
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   500
  shows R
wenzelm@21539
   501
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   502
wenzelm@21539
   503
lemma disj_impE:
wenzelm@21539
   504
  assumes major: "(P|Q)-->S"
wenzelm@21539
   505
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   506
  shows R
wenzelm@21539
   507
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   508
wenzelm@21539
   509
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   510
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   511
lemma imp_impE:
wenzelm@21539
   512
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   513
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   514
    and r2: "S ==> R"
wenzelm@21539
   515
  shows R
wenzelm@21539
   516
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   517
wenzelm@21539
   518
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   519
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   520
lemma not_impE:
wenzelm@23393
   521
  "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R"
wenzelm@23393
   522
  apply (drule mp)
wenzelm@23393
   523
   apply (rule notI)
wenzelm@23393
   524
   apply assumption
wenzelm@23393
   525
  apply assumption
wenzelm@21539
   526
  done
wenzelm@21539
   527
wenzelm@21539
   528
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   529
lemma iff_impE:
wenzelm@21539
   530
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   531
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   532
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   533
    and r3: "S ==> R"
wenzelm@21539
   534
  shows R
wenzelm@21539
   535
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   536
  done
wenzelm@21539
   537
wenzelm@21539
   538
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   539
lemma all_impE:
wenzelm@21539
   540
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   541
    and r1: "!!x. P(x)"
wenzelm@21539
   542
    and r2: "S ==> R"
wenzelm@21539
   543
  shows R
wenzelm@23393
   544
  apply (rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   545
  done
wenzelm@21539
   546
wenzelm@21539
   547
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   548
lemma ex_impE:
wenzelm@21539
   549
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   550
    and r: "P(x)-->S ==> R"
wenzelm@21539
   551
  shows R
wenzelm@21539
   552
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   553
  done
wenzelm@21539
   554
wenzelm@21539
   555
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   556
wenzelm@21539
   557
lemma disj_imp_disj:
wenzelm@23393
   558
  "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> R|S"
wenzelm@23393
   559
  apply (erule disjE)
wenzelm@21539
   560
  apply (rule disjI1) apply assumption
wenzelm@21539
   561
  apply (rule disjI2) apply assumption
wenzelm@21539
   562
  done
wenzelm@11734
   563
wenzelm@18481
   564
ML {*
wenzelm@32172
   565
structure Project_Rule = Project_Rule
wenzelm@32172
   566
(
wenzelm@22139
   567
  val conjunct1 = @{thm conjunct1}
wenzelm@22139
   568
  val conjunct2 = @{thm conjunct2}
wenzelm@22139
   569
  val mp = @{thm mp}
wenzelm@32172
   570
)
wenzelm@18481
   571
*}
wenzelm@18481
   572
wenzelm@48891
   573
ML_file "fologic.ML"
wenzelm@21539
   574
wenzelm@42303
   575
lemma thin_refl: "[|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   576
wenzelm@42799
   577
ML {*
wenzelm@42799
   578
structure Hypsubst = Hypsubst
wenzelm@42799
   579
(
wenzelm@42799
   580
  val dest_eq = FOLogic.dest_eq
wenzelm@42799
   581
  val dest_Trueprop = FOLogic.dest_Trueprop
wenzelm@42799
   582
  val dest_imp = FOLogic.dest_imp
wenzelm@42799
   583
  val eq_reflection = @{thm eq_reflection}
wenzelm@42799
   584
  val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
wenzelm@42799
   585
  val imp_intr = @{thm impI}
wenzelm@42799
   586
  val rev_mp = @{thm rev_mp}
wenzelm@42799
   587
  val subst = @{thm subst}
wenzelm@42799
   588
  val sym = @{thm sym}
wenzelm@42799
   589
  val thin_refl = @{thm thin_refl}
wenzelm@42799
   590
);
wenzelm@42799
   591
open Hypsubst;
wenzelm@42799
   592
*}
wenzelm@42799
   593
wenzelm@48891
   594
ML_file "intprover.ML"
wenzelm@7355
   595
wenzelm@4092
   596
wenzelm@12875
   597
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   598
wenzelm@31299
   599
setup {* Intuitionistic.method_setup @{binding iprover} *}
wenzelm@30165
   600
wenzelm@12349
   601
lemma impE':
wenzelm@12937
   602
  assumes 1: "P --> Q"
wenzelm@12937
   603
    and 2: "Q ==> R"
wenzelm@12937
   604
    and 3: "P --> Q ==> P"
wenzelm@12937
   605
  shows R
wenzelm@12349
   606
proof -
wenzelm@12349
   607
  from 3 and 1 have P .
wenzelm@12368
   608
  with 1 have Q by (rule impE)
wenzelm@12349
   609
  with 2 show R .
wenzelm@12349
   610
qed
wenzelm@12349
   611
wenzelm@12349
   612
lemma allE':
wenzelm@12937
   613
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   614
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   615
  shows Q
wenzelm@12349
   616
proof -
wenzelm@12349
   617
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   618
  from this and 1 show Q by (rule 2)
wenzelm@12349
   619
qed
wenzelm@12349
   620
wenzelm@12937
   621
lemma notE':
wenzelm@12937
   622
  assumes 1: "~ P"
wenzelm@12937
   623
    and 2: "~ P ==> P"
wenzelm@12937
   624
  shows R
wenzelm@12349
   625
proof -
wenzelm@12349
   626
  from 2 and 1 have P .
wenzelm@12349
   627
  with 1 show R by (rule notE)
wenzelm@12349
   628
qed
wenzelm@12349
   629
wenzelm@12349
   630
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   631
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   632
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   633
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   634
wenzelm@51798
   635
setup {* Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac ctxt ORELSE' tac) *}
wenzelm@12349
   636
wenzelm@12349
   637
wenzelm@12368
   638
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   639
  by iprover
wenzelm@12368
   640
wenzelm@12368
   641
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   642
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   643
wenzelm@12368
   644
paulson@13435
   645
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   646
apply (rule iffI) 
paulson@13435
   647
apply (erule sym)+
paulson@13435
   648
done
paulson@13435
   649
paulson@13435
   650
wenzelm@11677
   651
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   652
wenzelm@11747
   653
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   654
proof
wenzelm@11677
   655
  assume "!!x. P(x)"
wenzelm@22931
   656
  then show "ALL x. P(x)" ..
wenzelm@11677
   657
next
wenzelm@11677
   658
  assume "ALL x. P(x)"
wenzelm@22931
   659
  then show "!!x. P(x)" ..
wenzelm@11677
   660
qed
wenzelm@11677
   661
wenzelm@11747
   662
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   663
proof
wenzelm@12368
   664
  assume "A ==> B"
wenzelm@22931
   665
  then show "A --> B" ..
wenzelm@11677
   666
next
wenzelm@11677
   667
  assume "A --> B" and A
wenzelm@22931
   668
  then show B by (rule mp)
wenzelm@11677
   669
qed
wenzelm@11677
   670
wenzelm@11747
   671
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   672
proof
wenzelm@11677
   673
  assume "x == y"
wenzelm@22931
   674
  show "x = y" unfolding `x == y` by (rule refl)
wenzelm@11677
   675
next
wenzelm@11677
   676
  assume "x = y"
wenzelm@22931
   677
  then show "x == y" by (rule eq_reflection)
wenzelm@11677
   678
qed
wenzelm@11677
   679
wenzelm@18813
   680
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   681
proof
wenzelm@18813
   682
  assume "A == B"
wenzelm@22931
   683
  show "A <-> B" unfolding `A == B` by (rule iff_refl)
wenzelm@18813
   684
next
wenzelm@18813
   685
  assume "A <-> B"
wenzelm@22931
   686
  then show "A == B" by (rule iff_reflection)
wenzelm@18813
   687
qed
wenzelm@18813
   688
wenzelm@28856
   689
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
wenzelm@11976
   690
proof
wenzelm@28856
   691
  assume conj: "A &&& B"
wenzelm@19120
   692
  show "A & B"
wenzelm@19120
   693
  proof (rule conjI)
wenzelm@19120
   694
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   695
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   696
  qed
wenzelm@11953
   697
next
wenzelm@19120
   698
  assume conj: "A & B"
wenzelm@28856
   699
  show "A &&& B"
wenzelm@19120
   700
  proof -
wenzelm@19120
   701
    from conj show A ..
wenzelm@19120
   702
    from conj show B ..
wenzelm@11953
   703
  qed
wenzelm@11953
   704
qed
wenzelm@11953
   705
wenzelm@12368
   706
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   707
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   708
wenzelm@11848
   709
krauss@26580
   710
subsection {* Atomizing elimination rules *}
krauss@26580
   711
krauss@26580
   712
lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX x. P(x)) ==> Q)"
wenzelm@57948
   713
  by rule iprover+
krauss@26580
   714
krauss@26580
   715
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
wenzelm@57948
   716
  by rule iprover+
krauss@26580
   717
krauss@26580
   718
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
wenzelm@57948
   719
  by rule iprover+
krauss@26580
   720
krauss@26580
   721
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop(A)" ..
krauss@26580
   722
krauss@26580
   723
wenzelm@11848
   724
subsection {* Calculational rules *}
wenzelm@11848
   725
wenzelm@11848
   726
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   727
  by (rule ssubst)
wenzelm@11848
   728
wenzelm@11848
   729
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   730
  by (rule subst)
wenzelm@11848
   731
wenzelm@11848
   732
text {*
wenzelm@11848
   733
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   734
*}
wenzelm@11848
   735
wenzelm@12019
   736
lemmas basic_trans_rules [trans] =
wenzelm@11848
   737
  forw_subst
wenzelm@11848
   738
  back_subst
wenzelm@11848
   739
  rev_mp
wenzelm@11848
   740
  mp
wenzelm@11848
   741
  trans
wenzelm@11848
   742
paulson@13779
   743
subsection {* ``Let'' declarations *}
paulson@13779
   744
wenzelm@41229
   745
nonterminal letbinds and letbind
paulson@13779
   746
haftmann@35416
   747
definition Let :: "['a::{}, 'a => 'b] => ('b::{})" where
paulson@13779
   748
    "Let(s, f) == f(s)"
paulson@13779
   749
paulson@13779
   750
syntax
paulson@13779
   751
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   752
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   753
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   754
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   755
paulson@13779
   756
translations
paulson@13779
   757
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
wenzelm@35054
   758
  "let x = a in e"          == "CONST Let(a, %x. e)"
paulson@13779
   759
paulson@13779
   760
paulson@13779
   761
lemma LetI: 
wenzelm@21539
   762
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   763
  shows "P(let x=t in u(x))"
wenzelm@21539
   764
  apply (unfold Let_def)
wenzelm@21539
   765
  apply (rule refl [THEN assms])
wenzelm@21539
   766
  done
wenzelm@21539
   767
wenzelm@21539
   768
wenzelm@26286
   769
subsection {* Intuitionistic simplification rules *}
wenzelm@26286
   770
wenzelm@26286
   771
lemma conj_simps:
wenzelm@26286
   772
  "P & True <-> P"
wenzelm@26286
   773
  "True & P <-> P"
wenzelm@26286
   774
  "P & False <-> False"
wenzelm@26286
   775
  "False & P <-> False"
wenzelm@26286
   776
  "P & P <-> P"
wenzelm@26286
   777
  "P & P & Q <-> P & Q"
wenzelm@26286
   778
  "P & ~P <-> False"
wenzelm@26286
   779
  "~P & P <-> False"
wenzelm@26286
   780
  "(P & Q) & R <-> P & (Q & R)"
wenzelm@26286
   781
  by iprover+
wenzelm@26286
   782
wenzelm@26286
   783
lemma disj_simps:
wenzelm@26286
   784
  "P | True <-> True"
wenzelm@26286
   785
  "True | P <-> True"
wenzelm@26286
   786
  "P | False <-> P"
wenzelm@26286
   787
  "False | P <-> P"
wenzelm@26286
   788
  "P | P <-> P"
wenzelm@26286
   789
  "P | P | Q <-> P | Q"
wenzelm@26286
   790
  "(P | Q) | R <-> P | (Q | R)"
wenzelm@26286
   791
  by iprover+
wenzelm@26286
   792
wenzelm@26286
   793
lemma not_simps:
wenzelm@26286
   794
  "~(P|Q)  <-> ~P & ~Q"
wenzelm@26286
   795
  "~ False <-> True"
wenzelm@26286
   796
  "~ True <-> False"
wenzelm@26286
   797
  by iprover+
wenzelm@26286
   798
wenzelm@26286
   799
lemma imp_simps:
wenzelm@26286
   800
  "(P --> False) <-> ~P"
wenzelm@26286
   801
  "(P --> True) <-> True"
wenzelm@26286
   802
  "(False --> P) <-> True"
wenzelm@26286
   803
  "(True --> P) <-> P"
wenzelm@26286
   804
  "(P --> P) <-> True"
wenzelm@26286
   805
  "(P --> ~P) <-> ~P"
wenzelm@26286
   806
  by iprover+
wenzelm@26286
   807
wenzelm@26286
   808
lemma iff_simps:
wenzelm@26286
   809
  "(True <-> P) <-> P"
wenzelm@26286
   810
  "(P <-> True) <-> P"
wenzelm@26286
   811
  "(P <-> P) <-> True"
wenzelm@26286
   812
  "(False <-> P) <-> ~P"
wenzelm@26286
   813
  "(P <-> False) <-> ~P"
wenzelm@26286
   814
  by iprover+
wenzelm@26286
   815
wenzelm@26286
   816
(*The x=t versions are needed for the simplification procedures*)
wenzelm@26286
   817
lemma quant_simps:
wenzelm@26286
   818
  "!!P. (ALL x. P) <-> P"
wenzelm@26286
   819
  "(ALL x. x=t --> P(x)) <-> P(t)"
wenzelm@26286
   820
  "(ALL x. t=x --> P(x)) <-> P(t)"
wenzelm@26286
   821
  "!!P. (EX x. P) <-> P"
wenzelm@26286
   822
  "EX x. x=t"
wenzelm@26286
   823
  "EX x. t=x"
wenzelm@26286
   824
  "(EX x. x=t & P(x)) <-> P(t)"
wenzelm@26286
   825
  "(EX x. t=x & P(x)) <-> P(t)"
wenzelm@26286
   826
  by iprover+
wenzelm@26286
   827
wenzelm@26286
   828
(*These are NOT supplied by default!*)
wenzelm@26286
   829
lemma distrib_simps:
wenzelm@26286
   830
  "P & (Q | R) <-> P&Q | P&R"
wenzelm@26286
   831
  "(Q | R) & P <-> Q&P | R&P"
wenzelm@26286
   832
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"
wenzelm@26286
   833
  by iprover+
wenzelm@26286
   834
wenzelm@26286
   835
wenzelm@26286
   836
text {* Conversion into rewrite rules *}
wenzelm@26286
   837
wenzelm@26286
   838
lemma P_iff_F: "~P ==> (P <-> False)" by iprover
wenzelm@26286
   839
lemma iff_reflection_F: "~P ==> (P == False)" by (rule P_iff_F [THEN iff_reflection])
wenzelm@26286
   840
wenzelm@26286
   841
lemma P_iff_T: "P ==> (P <-> True)" by iprover
wenzelm@26286
   842
lemma iff_reflection_T: "P ==> (P == True)" by (rule P_iff_T [THEN iff_reflection])
wenzelm@26286
   843
wenzelm@26286
   844
wenzelm@26286
   845
text {* More rewrite rules *}
wenzelm@26286
   846
wenzelm@26286
   847
lemma conj_commute: "P&Q <-> Q&P" by iprover
wenzelm@26286
   848
lemma conj_left_commute: "P&(Q&R) <-> Q&(P&R)" by iprover
wenzelm@26286
   849
lemmas conj_comms = conj_commute conj_left_commute
wenzelm@26286
   850
wenzelm@26286
   851
lemma disj_commute: "P|Q <-> Q|P" by iprover
wenzelm@26286
   852
lemma disj_left_commute: "P|(Q|R) <-> Q|(P|R)" by iprover
wenzelm@26286
   853
lemmas disj_comms = disj_commute disj_left_commute
wenzelm@26286
   854
wenzelm@26286
   855
lemma conj_disj_distribL: "P&(Q|R) <-> (P&Q | P&R)" by iprover
wenzelm@26286
   856
lemma conj_disj_distribR: "(P|Q)&R <-> (P&R | Q&R)" by iprover
wenzelm@26286
   857
wenzelm@26286
   858
lemma disj_conj_distribL: "P|(Q&R) <-> (P|Q) & (P|R)" by iprover
wenzelm@26286
   859
lemma disj_conj_distribR: "(P&Q)|R <-> (P|R) & (Q|R)" by iprover
wenzelm@26286
   860
wenzelm@26286
   861
lemma imp_conj_distrib: "(P --> (Q&R)) <-> (P-->Q) & (P-->R)" by iprover
wenzelm@26286
   862
lemma imp_conj: "((P&Q)-->R)   <-> (P --> (Q --> R))" by iprover
wenzelm@26286
   863
lemma imp_disj: "(P|Q --> R)   <-> (P-->R) & (Q-->R)" by iprover
wenzelm@26286
   864
wenzelm@26286
   865
lemma de_Morgan_disj: "(~(P | Q)) <-> (~P & ~Q)" by iprover
wenzelm@26286
   866
wenzelm@26286
   867
lemma not_ex: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover
wenzelm@26286
   868
lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover
wenzelm@26286
   869
wenzelm@26286
   870
lemma ex_disj_distrib:
wenzelm@26286
   871
  "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover
wenzelm@26286
   872
wenzelm@26286
   873
lemma all_conj_distrib:
wenzelm@26286
   874
  "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover
wenzelm@26286
   875
wenzelm@4854
   876
end