src/FOL/IFOL.thy
author krauss
Tue Apr 08 18:30:40 2008 +0200 (2008-04-08)
changeset 26580 c3e597a476fd
parent 26286 3ff5d257f175
child 26956 1309a6a0a29f
permissions -rw-r--r--
Generic conversion and tactic "atomize_elim" to convert elimination rules
to the object logic
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(*  Title:      FOL/IFOL.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson and Markus Wenzel
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*)
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header {* Intuitionistic first-order logic *}
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theory IFOL
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imports Pure
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uses
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  "~~/src/Provers/splitter.ML"
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  "~~/src/Provers/hypsubst.ML"
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  "~~/src/Tools/IsaPlanner/zipper.ML"
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  "~~/src/Tools/IsaPlanner/isand.ML"
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  "~~/src/Tools/IsaPlanner/rw_tools.ML"
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  "~~/src/Tools/IsaPlanner/rw_inst.ML"
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  "~~/src/Provers/eqsubst.ML"
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  "~~/src/Provers/quantifier1.ML"
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  "~~/src/Provers/project_rule.ML"
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  "~~/src/Tools/atomize_elim.ML"
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  ("fologic.ML")
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  ("hypsubstdata.ML")
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  ("intprover.ML")
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begin
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subsection {* Syntax and axiomatic basis *}
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global
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classes "term"
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defaultsort "term"
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typedecl o
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judgment
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  Trueprop      :: "o => prop"                  ("(_)" 5)
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consts
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  True          :: o
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  False         :: o
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  (* Connectives *)
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  "op ="        :: "['a, 'a] => o"              (infixl "=" 50)
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  Not           :: "o => o"                     ("~ _" [40] 40)
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  "op &"        :: "[o, o] => o"                (infixr "&" 35)
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  "op |"        :: "[o, o] => o"                (infixr "|" 30)
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  "op -->"      :: "[o, o] => o"                (infixr "-->" 25)
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  "op <->"      :: "[o, o] => o"                (infixr "<->" 25)
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  (* Quantifiers *)
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  All           :: "('a => o) => o"             (binder "ALL " 10)
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  Ex            :: "('a => o) => o"             (binder "EX " 10)
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  Ex1           :: "('a => o) => o"             (binder "EX! " 10)
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abbreviation
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  not_equal :: "['a, 'a] => o"  (infixl "~=" 50) where
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  "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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  "op &"    (infixr "\<and>" 35) and
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  "op |"    (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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  "op -->"  (infixr "\<longrightarrow>" 25) and
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  "op <->"  (infixr "\<longleftrightarrow>" 25)
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notation (HTML output)
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  Not       ("\<not> _" [40] 40) and
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  "op &"    (infixr "\<and>" 35) and
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  "op |"    (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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local
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finalconsts
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  False All Ex
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  "op ="
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  "op &"
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  "op |"
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  "op -->"
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axioms
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  (* Equality *)
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  refl:         "a=a"
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  (* Propositional logic *)
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  conjI:        "[| P;  Q |] ==> P&Q"
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  conjunct1:    "P&Q ==> P"
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  conjunct2:    "P&Q ==> Q"
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  disjI1:       "P ==> P|Q"
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  disjI2:       "Q ==> P|Q"
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  disjE:        "[| P|Q;  P ==> R;  Q ==> R |] ==> R"
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  impI:         "(P ==> Q) ==> P-->Q"
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  mp:           "[| P-->Q;  P |] ==> Q"
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  FalseE:       "False ==> P"
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  (* Quantifiers *)
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  allI:         "(!!x. P(x)) ==> (ALL x. P(x))"
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  spec:         "(ALL x. P(x)) ==> P(x)"
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  exI:          "P(x) ==> (EX x. P(x))"
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  exE:          "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"
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  (* Reflection *)
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  eq_reflection:  "(x=y)   ==> (x==y)"
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  iff_reflection: "(P<->Q) ==> (P==Q)"
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lemmas strip = impI allI
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text{*Thanks to Stephan Merz*}
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theorem subst:
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  assumes eq: "a = b" and p: "P(a)"
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  shows "P(b)"
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proof -
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  from eq have meta: "a \<equiv> b"
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    by (rule eq_reflection)
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  from p show ?thesis
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    by (unfold meta)
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qed
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defs
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  (* Definitions *)
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  True_def:     "True  == False-->False"
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  not_def:      "~P    == P-->False"
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  iff_def:      "P<->Q == (P-->Q) & (Q-->P)"
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  (* Unique existence *)
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  ex1_def:      "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)"
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subsection {* Lemmas and proof tools *}
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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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   To specify P use something like
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      eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1   *)
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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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(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
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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 *})+
wenzelm@21539
   366
  done
wenzelm@21539
   367
wenzelm@21539
   368
lemma disj_cong:
wenzelm@21539
   369
  assumes "P <-> P'" and "Q <-> Q'"
wenzelm@21539
   370
  shows "(P|Q) <-> (P'|Q')"
wenzelm@21539
   371
  apply (insert assms)
wenzelm@21539
   372
  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   373
  done
wenzelm@21539
   374
wenzelm@21539
   375
lemma imp_cong:
wenzelm@21539
   376
  assumes "P <-> P'"
wenzelm@21539
   377
    and "P' ==> Q <-> Q'"
wenzelm@21539
   378
  shows "(P-->Q) <-> (P'-->Q')"
wenzelm@21539
   379
  apply (insert assms)
wenzelm@21539
   380
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
wenzelm@21539
   381
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   382
  done
wenzelm@21539
   383
wenzelm@21539
   384
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
wenzelm@21539
   385
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   386
  done
wenzelm@21539
   387
wenzelm@21539
   388
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
wenzelm@21539
   389
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
wenzelm@21539
   390
  done
wenzelm@21539
   391
wenzelm@21539
   392
lemma all_cong:
wenzelm@21539
   393
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   394
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   395
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@21539
   396
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   397
  done
wenzelm@21539
   398
wenzelm@21539
   399
lemma ex_cong:
wenzelm@21539
   400
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   401
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   402
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@21539
   403
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   404
  done
wenzelm@21539
   405
wenzelm@21539
   406
lemma ex1_cong:
wenzelm@21539
   407
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   408
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   409
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@21539
   410
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   411
  done
wenzelm@21539
   412
wenzelm@21539
   413
(*** Equality rules ***)
wenzelm@21539
   414
wenzelm@21539
   415
lemma sym: "a=b ==> b=a"
wenzelm@21539
   416
  apply (erule subst)
wenzelm@21539
   417
  apply (rule refl)
wenzelm@21539
   418
  done
wenzelm@21539
   419
wenzelm@21539
   420
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   421
  apply (erule subst, assumption)
wenzelm@21539
   422
  done
wenzelm@21539
   423
wenzelm@21539
   424
(**  **)
wenzelm@21539
   425
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   426
  apply (erule contrapos)
wenzelm@21539
   427
  apply (erule sym)
wenzelm@21539
   428
  done
wenzelm@21539
   429
  
wenzelm@21539
   430
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   431
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   432
wenzelm@21539
   433
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   434
  apply unfold
wenzelm@21539
   435
  apply (rule iff_refl)
wenzelm@21539
   436
  done
wenzelm@21539
   437
wenzelm@21539
   438
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   439
  apply unfold
wenzelm@21539
   440
  apply (rule refl)
wenzelm@21539
   441
  done
wenzelm@21539
   442
wenzelm@21539
   443
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   444
  by unfold (rule iff_refl)
wenzelm@21539
   445
wenzelm@21539
   446
(*substitution*)
wenzelm@21539
   447
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   448
  apply (drule sym)
wenzelm@21539
   449
  apply (erule (1) subst)
wenzelm@21539
   450
  done
wenzelm@21539
   451
wenzelm@21539
   452
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   453
lemma ex1_equalsE:
wenzelm@21539
   454
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   455
  apply (erule ex1E)
wenzelm@21539
   456
  apply (rule trans)
wenzelm@21539
   457
   apply (rule_tac [2] sym)
wenzelm@21539
   458
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   459
  done
wenzelm@21539
   460
wenzelm@21539
   461
(** Polymorphic congruence rules **)
wenzelm@21539
   462
wenzelm@21539
   463
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   464
  apply (erule ssubst)
wenzelm@21539
   465
  apply (rule refl)
wenzelm@21539
   466
  done
wenzelm@21539
   467
wenzelm@21539
   468
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   469
  apply (erule ssubst)+
wenzelm@21539
   470
  apply (rule refl)
wenzelm@21539
   471
  done
wenzelm@21539
   472
wenzelm@21539
   473
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   474
  apply (erule ssubst)+
wenzelm@21539
   475
  apply (rule refl)
wenzelm@21539
   476
  done
wenzelm@21539
   477
wenzelm@21539
   478
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   479
        a = b
wenzelm@21539
   480
        |   |
wenzelm@21539
   481
        c = d   *)
wenzelm@21539
   482
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   483
  apply (rule trans)
wenzelm@21539
   484
   apply (rule trans)
wenzelm@21539
   485
    apply (rule sym)
wenzelm@21539
   486
    apply assumption+
wenzelm@21539
   487
  done
wenzelm@21539
   488
wenzelm@21539
   489
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   490
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   491
  apply (rule trans)
wenzelm@21539
   492
   apply (rule trans)
wenzelm@21539
   493
    apply assumption+
wenzelm@21539
   494
  apply (erule sym)
wenzelm@21539
   495
  done
wenzelm@21539
   496
wenzelm@21539
   497
(** Congruence rules for predicate letters **)
wenzelm@21539
   498
wenzelm@21539
   499
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   500
  apply (rule iffI)
wenzelm@21539
   501
   apply (erule (1) subst)
wenzelm@21539
   502
  apply (erule (1) ssubst)
wenzelm@21539
   503
  done
wenzelm@21539
   504
wenzelm@21539
   505
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   506
  apply (rule iffI)
wenzelm@21539
   507
   apply (erule subst)+
wenzelm@21539
   508
   apply assumption
wenzelm@21539
   509
  apply (erule ssubst)+
wenzelm@21539
   510
  apply assumption
wenzelm@21539
   511
  done
wenzelm@21539
   512
wenzelm@21539
   513
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
wenzelm@21539
   514
  apply (rule iffI)
wenzelm@21539
   515
   apply (erule subst)+
wenzelm@21539
   516
   apply assumption
wenzelm@21539
   517
  apply (erule ssubst)+
wenzelm@21539
   518
  apply assumption
wenzelm@21539
   519
  done
wenzelm@21539
   520
wenzelm@21539
   521
(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
wenzelm@21539
   522
wenzelm@21539
   523
ML {*
wenzelm@21539
   524
bind_thms ("pred_congs",
wenzelm@21539
   525
  List.concat (map (fn c => 
wenzelm@21539
   526
               map (fn th => read_instantiate [("P",c)] th)
wenzelm@22139
   527
                   [@{thm pred1_cong}, @{thm pred2_cong}, @{thm pred3_cong}])
wenzelm@21539
   528
               (explode"PQRS")))
wenzelm@21539
   529
*}
wenzelm@21539
   530
wenzelm@21539
   531
(*special case for the equality predicate!*)
wenzelm@21539
   532
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   533
  apply (erule (1) pred2_cong)
wenzelm@21539
   534
  done
wenzelm@21539
   535
wenzelm@21539
   536
wenzelm@21539
   537
(*** Simplifications of assumed implications.
wenzelm@21539
   538
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   539
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   540
     intuitionistic propositional logic.  See
wenzelm@21539
   541
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   542
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   543
wenzelm@21539
   544
lemma conj_impE:
wenzelm@21539
   545
  assumes major: "(P&Q)-->S"
wenzelm@21539
   546
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   547
  shows R
wenzelm@21539
   548
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   549
wenzelm@21539
   550
lemma disj_impE:
wenzelm@21539
   551
  assumes major: "(P|Q)-->S"
wenzelm@21539
   552
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   553
  shows R
wenzelm@21539
   554
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   555
wenzelm@21539
   556
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   557
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   558
lemma imp_impE:
wenzelm@21539
   559
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   560
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   561
    and r2: "S ==> R"
wenzelm@21539
   562
  shows R
wenzelm@21539
   563
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   564
wenzelm@21539
   565
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   566
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   567
lemma not_impE:
wenzelm@23393
   568
  "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R"
wenzelm@23393
   569
  apply (drule mp)
wenzelm@23393
   570
   apply (rule notI)
wenzelm@23393
   571
   apply assumption
wenzelm@23393
   572
  apply assumption
wenzelm@21539
   573
  done
wenzelm@21539
   574
wenzelm@21539
   575
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   576
lemma iff_impE:
wenzelm@21539
   577
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   578
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   579
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   580
    and r3: "S ==> R"
wenzelm@21539
   581
  shows R
wenzelm@21539
   582
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   583
  done
wenzelm@21539
   584
wenzelm@21539
   585
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   586
lemma all_impE:
wenzelm@21539
   587
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   588
    and r1: "!!x. P(x)"
wenzelm@21539
   589
    and r2: "S ==> R"
wenzelm@21539
   590
  shows R
wenzelm@23393
   591
  apply (rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   592
  done
wenzelm@21539
   593
wenzelm@21539
   594
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   595
lemma ex_impE:
wenzelm@21539
   596
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   597
    and r: "P(x)-->S ==> R"
wenzelm@21539
   598
  shows R
wenzelm@21539
   599
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   600
  done
wenzelm@21539
   601
wenzelm@21539
   602
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   603
wenzelm@21539
   604
lemma disj_imp_disj:
wenzelm@23393
   605
  "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> R|S"
wenzelm@23393
   606
  apply (erule disjE)
wenzelm@21539
   607
  apply (rule disjI1) apply assumption
wenzelm@21539
   608
  apply (rule disjI2) apply assumption
wenzelm@21539
   609
  done
wenzelm@11734
   610
wenzelm@18481
   611
ML {*
wenzelm@18481
   612
structure ProjectRule = ProjectRuleFun
wenzelm@18481
   613
(struct
wenzelm@22139
   614
  val conjunct1 = @{thm conjunct1}
wenzelm@22139
   615
  val conjunct2 = @{thm conjunct2}
wenzelm@22139
   616
  val mp = @{thm mp}
wenzelm@18481
   617
end)
wenzelm@18481
   618
*}
wenzelm@18481
   619
wenzelm@7355
   620
use "fologic.ML"
wenzelm@21539
   621
wenzelm@21539
   622
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   623
wenzelm@9886
   624
use "hypsubstdata.ML"
wenzelm@9886
   625
setup hypsubst_setup
wenzelm@7355
   626
use "intprover.ML"
wenzelm@7355
   627
wenzelm@4092
   628
wenzelm@12875
   629
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   630
wenzelm@12349
   631
lemma impE':
wenzelm@12937
   632
  assumes 1: "P --> Q"
wenzelm@12937
   633
    and 2: "Q ==> R"
wenzelm@12937
   634
    and 3: "P --> Q ==> P"
wenzelm@12937
   635
  shows R
wenzelm@12349
   636
proof -
wenzelm@12349
   637
  from 3 and 1 have P .
wenzelm@12368
   638
  with 1 have Q by (rule impE)
wenzelm@12349
   639
  with 2 show R .
wenzelm@12349
   640
qed
wenzelm@12349
   641
wenzelm@12349
   642
lemma allE':
wenzelm@12937
   643
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   644
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   645
  shows Q
wenzelm@12349
   646
proof -
wenzelm@12349
   647
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   648
  from this and 1 show Q by (rule 2)
wenzelm@12349
   649
qed
wenzelm@12349
   650
wenzelm@12937
   651
lemma notE':
wenzelm@12937
   652
  assumes 1: "~ P"
wenzelm@12937
   653
    and 2: "~ P ==> P"
wenzelm@12937
   654
  shows R
wenzelm@12349
   655
proof -
wenzelm@12349
   656
  from 2 and 1 have P .
wenzelm@12349
   657
  with 1 show R by (rule notE)
wenzelm@12349
   658
qed
wenzelm@12349
   659
wenzelm@12349
   660
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   661
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   662
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   663
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   664
wenzelm@18708
   665
setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}
wenzelm@12349
   666
wenzelm@12349
   667
wenzelm@12368
   668
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   669
  by iprover
wenzelm@12368
   670
wenzelm@12368
   671
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   672
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   673
wenzelm@12368
   674
paulson@13435
   675
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   676
apply (rule iffI) 
paulson@13435
   677
apply (erule sym)+
paulson@13435
   678
done
paulson@13435
   679
paulson@13435
   680
wenzelm@11677
   681
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   682
wenzelm@11747
   683
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   684
proof
wenzelm@11677
   685
  assume "!!x. P(x)"
wenzelm@22931
   686
  then show "ALL x. P(x)" ..
wenzelm@11677
   687
next
wenzelm@11677
   688
  assume "ALL x. P(x)"
wenzelm@22931
   689
  then show "!!x. P(x)" ..
wenzelm@11677
   690
qed
wenzelm@11677
   691
wenzelm@11747
   692
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   693
proof
wenzelm@12368
   694
  assume "A ==> B"
wenzelm@22931
   695
  then show "A --> B" ..
wenzelm@11677
   696
next
wenzelm@11677
   697
  assume "A --> B" and A
wenzelm@22931
   698
  then show B by (rule mp)
wenzelm@11677
   699
qed
wenzelm@11677
   700
wenzelm@11747
   701
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   702
proof
wenzelm@11677
   703
  assume "x == y"
wenzelm@22931
   704
  show "x = y" unfolding `x == y` by (rule refl)
wenzelm@11677
   705
next
wenzelm@11677
   706
  assume "x = y"
wenzelm@22931
   707
  then show "x == y" by (rule eq_reflection)
wenzelm@11677
   708
qed
wenzelm@11677
   709
wenzelm@18813
   710
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   711
proof
wenzelm@18813
   712
  assume "A == B"
wenzelm@22931
   713
  show "A <-> B" unfolding `A == B` by (rule iff_refl)
wenzelm@18813
   714
next
wenzelm@18813
   715
  assume "A <-> B"
wenzelm@22931
   716
  then show "A == B" by (rule iff_reflection)
wenzelm@18813
   717
qed
wenzelm@18813
   718
wenzelm@12875
   719
lemma atomize_conj [atomize]:
wenzelm@19120
   720
  includes meta_conjunction_syntax
wenzelm@19120
   721
  shows "(A && B) == Trueprop (A & B)"
wenzelm@11976
   722
proof
wenzelm@19120
   723
  assume conj: "A && B"
wenzelm@19120
   724
  show "A & B"
wenzelm@19120
   725
  proof (rule conjI)
wenzelm@19120
   726
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   727
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   728
  qed
wenzelm@11953
   729
next
wenzelm@19120
   730
  assume conj: "A & B"
wenzelm@19120
   731
  show "A && B"
wenzelm@19120
   732
  proof -
wenzelm@19120
   733
    from conj show A ..
wenzelm@19120
   734
    from conj show B ..
wenzelm@11953
   735
  qed
wenzelm@11953
   736
qed
wenzelm@11953
   737
wenzelm@12368
   738
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   739
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   740
wenzelm@11848
   741
krauss@26580
   742
subsection {* Atomizing elimination rules *}
krauss@26580
   743
krauss@26580
   744
setup AtomizeElim.setup
krauss@26580
   745
krauss@26580
   746
lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX x. P(x)) ==> Q)"
krauss@26580
   747
by rule iprover+
krauss@26580
   748
krauss@26580
   749
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
krauss@26580
   750
by rule iprover+
krauss@26580
   751
krauss@26580
   752
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
krauss@26580
   753
by rule iprover+
krauss@26580
   754
krauss@26580
   755
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop(A)" ..
krauss@26580
   756
krauss@26580
   757
wenzelm@11848
   758
subsection {* Calculational rules *}
wenzelm@11848
   759
wenzelm@11848
   760
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   761
  by (rule ssubst)
wenzelm@11848
   762
wenzelm@11848
   763
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   764
  by (rule subst)
wenzelm@11848
   765
wenzelm@11848
   766
text {*
wenzelm@11848
   767
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   768
*}
wenzelm@11848
   769
wenzelm@12019
   770
lemmas basic_trans_rules [trans] =
wenzelm@11848
   771
  forw_subst
wenzelm@11848
   772
  back_subst
wenzelm@11848
   773
  rev_mp
wenzelm@11848
   774
  mp
wenzelm@11848
   775
  trans
wenzelm@11848
   776
paulson@13779
   777
subsection {* ``Let'' declarations *}
paulson@13779
   778
paulson@13779
   779
nonterminals letbinds letbind
paulson@13779
   780
paulson@13779
   781
constdefs
wenzelm@14854
   782
  Let :: "['a::{}, 'a => 'b] => ('b::{})"
paulson@13779
   783
    "Let(s, f) == f(s)"
paulson@13779
   784
paulson@13779
   785
syntax
paulson@13779
   786
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   787
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   788
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   789
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   790
paulson@13779
   791
translations
paulson@13779
   792
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
paulson@13779
   793
  "let x = a in e"          == "Let(a, %x. e)"
paulson@13779
   794
paulson@13779
   795
paulson@13779
   796
lemma LetI: 
wenzelm@21539
   797
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   798
  shows "P(let x=t in u(x))"
wenzelm@21539
   799
  apply (unfold Let_def)
wenzelm@21539
   800
  apply (rule refl [THEN assms])
wenzelm@21539
   801
  done
wenzelm@21539
   802
wenzelm@21539
   803
wenzelm@26286
   804
subsection {* Intuitionistic simplification rules *}
wenzelm@26286
   805
wenzelm@26286
   806
lemma conj_simps:
wenzelm@26286
   807
  "P & True <-> P"
wenzelm@26286
   808
  "True & P <-> P"
wenzelm@26286
   809
  "P & False <-> False"
wenzelm@26286
   810
  "False & P <-> False"
wenzelm@26286
   811
  "P & P <-> P"
wenzelm@26286
   812
  "P & P & Q <-> P & Q"
wenzelm@26286
   813
  "P & ~P <-> False"
wenzelm@26286
   814
  "~P & P <-> False"
wenzelm@26286
   815
  "(P & Q) & R <-> P & (Q & R)"
wenzelm@26286
   816
  by iprover+
wenzelm@26286
   817
wenzelm@26286
   818
lemma disj_simps:
wenzelm@26286
   819
  "P | True <-> True"
wenzelm@26286
   820
  "True | P <-> True"
wenzelm@26286
   821
  "P | False <-> P"
wenzelm@26286
   822
  "False | P <-> P"
wenzelm@26286
   823
  "P | P <-> P"
wenzelm@26286
   824
  "P | P | Q <-> P | Q"
wenzelm@26286
   825
  "(P | Q) | R <-> P | (Q | R)"
wenzelm@26286
   826
  by iprover+
wenzelm@26286
   827
wenzelm@26286
   828
lemma not_simps:
wenzelm@26286
   829
  "~(P|Q)  <-> ~P & ~Q"
wenzelm@26286
   830
  "~ False <-> True"
wenzelm@26286
   831
  "~ True <-> False"
wenzelm@26286
   832
  by iprover+
wenzelm@26286
   833
wenzelm@26286
   834
lemma imp_simps:
wenzelm@26286
   835
  "(P --> False) <-> ~P"
wenzelm@26286
   836
  "(P --> True) <-> True"
wenzelm@26286
   837
  "(False --> P) <-> True"
wenzelm@26286
   838
  "(True --> P) <-> P"
wenzelm@26286
   839
  "(P --> P) <-> True"
wenzelm@26286
   840
  "(P --> ~P) <-> ~P"
wenzelm@26286
   841
  by iprover+
wenzelm@26286
   842
wenzelm@26286
   843
lemma iff_simps:
wenzelm@26286
   844
  "(True <-> P) <-> P"
wenzelm@26286
   845
  "(P <-> True) <-> P"
wenzelm@26286
   846
  "(P <-> P) <-> True"
wenzelm@26286
   847
  "(False <-> P) <-> ~P"
wenzelm@26286
   848
  "(P <-> False) <-> ~P"
wenzelm@26286
   849
  by iprover+
wenzelm@26286
   850
wenzelm@26286
   851
(*The x=t versions are needed for the simplification procedures*)
wenzelm@26286
   852
lemma quant_simps:
wenzelm@26286
   853
  "!!P. (ALL x. P) <-> P"
wenzelm@26286
   854
  "(ALL x. x=t --> P(x)) <-> P(t)"
wenzelm@26286
   855
  "(ALL x. t=x --> P(x)) <-> P(t)"
wenzelm@26286
   856
  "!!P. (EX x. P) <-> P"
wenzelm@26286
   857
  "EX x. x=t"
wenzelm@26286
   858
  "EX x. t=x"
wenzelm@26286
   859
  "(EX x. x=t & P(x)) <-> P(t)"
wenzelm@26286
   860
  "(EX x. t=x & P(x)) <-> P(t)"
wenzelm@26286
   861
  by iprover+
wenzelm@26286
   862
wenzelm@26286
   863
(*These are NOT supplied by default!*)
wenzelm@26286
   864
lemma distrib_simps:
wenzelm@26286
   865
  "P & (Q | R) <-> P&Q | P&R"
wenzelm@26286
   866
  "(Q | R) & P <-> Q&P | R&P"
wenzelm@26286
   867
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"
wenzelm@26286
   868
  by iprover+
wenzelm@26286
   869
wenzelm@26286
   870
wenzelm@26286
   871
text {* Conversion into rewrite rules *}
wenzelm@26286
   872
wenzelm@26286
   873
lemma P_iff_F: "~P ==> (P <-> False)" by iprover
wenzelm@26286
   874
lemma iff_reflection_F: "~P ==> (P == False)" by (rule P_iff_F [THEN iff_reflection])
wenzelm@26286
   875
wenzelm@26286
   876
lemma P_iff_T: "P ==> (P <-> True)" by iprover
wenzelm@26286
   877
lemma iff_reflection_T: "P ==> (P == True)" by (rule P_iff_T [THEN iff_reflection])
wenzelm@26286
   878
wenzelm@26286
   879
wenzelm@26286
   880
text {* More rewrite rules *}
wenzelm@26286
   881
wenzelm@26286
   882
lemma conj_commute: "P&Q <-> Q&P" by iprover
wenzelm@26286
   883
lemma conj_left_commute: "P&(Q&R) <-> Q&(P&R)" by iprover
wenzelm@26286
   884
lemmas conj_comms = conj_commute conj_left_commute
wenzelm@26286
   885
wenzelm@26286
   886
lemma disj_commute: "P|Q <-> Q|P" by iprover
wenzelm@26286
   887
lemma disj_left_commute: "P|(Q|R) <-> Q|(P|R)" by iprover
wenzelm@26286
   888
lemmas disj_comms = disj_commute disj_left_commute
wenzelm@26286
   889
wenzelm@26286
   890
lemma conj_disj_distribL: "P&(Q|R) <-> (P&Q | P&R)" by iprover
wenzelm@26286
   891
lemma conj_disj_distribR: "(P|Q)&R <-> (P&R | Q&R)" by iprover
wenzelm@26286
   892
wenzelm@26286
   893
lemma disj_conj_distribL: "P|(Q&R) <-> (P|Q) & (P|R)" by iprover
wenzelm@26286
   894
lemma disj_conj_distribR: "(P&Q)|R <-> (P|R) & (Q|R)" by iprover
wenzelm@26286
   895
wenzelm@26286
   896
lemma imp_conj_distrib: "(P --> (Q&R)) <-> (P-->Q) & (P-->R)" by iprover
wenzelm@26286
   897
lemma imp_conj: "((P&Q)-->R)   <-> (P --> (Q --> R))" by iprover
wenzelm@26286
   898
lemma imp_disj: "(P|Q --> R)   <-> (P-->R) & (Q-->R)" by iprover
wenzelm@26286
   899
wenzelm@26286
   900
lemma de_Morgan_disj: "(~(P | Q)) <-> (~P & ~Q)" by iprover
wenzelm@26286
   901
wenzelm@26286
   902
lemma not_ex: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover
wenzelm@26286
   903
lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover
wenzelm@26286
   904
wenzelm@26286
   905
lemma ex_disj_distrib:
wenzelm@26286
   906
  "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover
wenzelm@26286
   907
wenzelm@26286
   908
lemma all_conj_distrib:
wenzelm@26286
   909
  "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover
wenzelm@26286
   910
wenzelm@26286
   911
wenzelm@26286
   912
subsection {* Legacy ML bindings *}
paulson@13779
   913
wenzelm@21539
   914
ML {*
wenzelm@22139
   915
val refl = @{thm refl}
wenzelm@22139
   916
val trans = @{thm trans}
wenzelm@22139
   917
val sym = @{thm sym}
wenzelm@22139
   918
val subst = @{thm subst}
wenzelm@22139
   919
val ssubst = @{thm ssubst}
wenzelm@22139
   920
val conjI = @{thm conjI}
wenzelm@22139
   921
val conjE = @{thm conjE}
wenzelm@22139
   922
val conjunct1 = @{thm conjunct1}
wenzelm@22139
   923
val conjunct2 = @{thm conjunct2}
wenzelm@22139
   924
val disjI1 = @{thm disjI1}
wenzelm@22139
   925
val disjI2 = @{thm disjI2}
wenzelm@22139
   926
val disjE = @{thm disjE}
wenzelm@22139
   927
val impI = @{thm impI}
wenzelm@22139
   928
val impE = @{thm impE}
wenzelm@22139
   929
val mp = @{thm mp}
wenzelm@22139
   930
val rev_mp = @{thm rev_mp}
wenzelm@22139
   931
val TrueI = @{thm TrueI}
wenzelm@22139
   932
val FalseE = @{thm FalseE}
wenzelm@22139
   933
val iff_refl = @{thm iff_refl}
wenzelm@22139
   934
val iff_trans = @{thm iff_trans}
wenzelm@22139
   935
val iffI = @{thm iffI}
wenzelm@22139
   936
val iffE = @{thm iffE}
wenzelm@22139
   937
val iffD1 = @{thm iffD1}
wenzelm@22139
   938
val iffD2 = @{thm iffD2}
wenzelm@22139
   939
val notI = @{thm notI}
wenzelm@22139
   940
val notE = @{thm notE}
wenzelm@22139
   941
val allI = @{thm allI}
wenzelm@22139
   942
val allE = @{thm allE}
wenzelm@22139
   943
val spec = @{thm spec}
wenzelm@22139
   944
val exI = @{thm exI}
wenzelm@22139
   945
val exE = @{thm exE}
wenzelm@22139
   946
val eq_reflection = @{thm eq_reflection}
wenzelm@22139
   947
val iff_reflection = @{thm iff_reflection}
wenzelm@22139
   948
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22139
   949
val meta_eq_to_iff = @{thm meta_eq_to_iff}
paulson@13779
   950
*}
paulson@13779
   951
wenzelm@4854
   952
end