src/FOL/IFOL.thy
author wenzelm
Sun Nov 26 23:43:53 2006 +0100 (2006-11-26)
changeset 21539 c5cf9243ad62
parent 21524 7843e2fd14a9
child 22139 539a63b98f76
permissions -rw-r--r--
converted legacy ML scripts;
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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 ("fologic.ML") ("hypsubstdata.ML") ("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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  local
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    val notE = thm "notE"
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    val impE = thm "impE"
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  in
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    fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i
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    fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i
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  end
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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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  assumes "P(a)"
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    and "!!x. P(x) ==> x=a"
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  shows "EX! x. P(x)"
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  apply (unfold ex1_def)
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  apply (assumption | rule assms 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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  assumes ex: "EX x. P(x)"
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    and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
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  shows "EX! x. P(x)"
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  apply (rule ex [THEN exE])
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  apply (assumption | rule ex1I eq)+
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  done
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lemma ex1E:
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  assumes ex1: "EX! x. P(x)"
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    and r: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
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  shows R
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  apply (insert ex1, 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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  local
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    val iffE = thm "iffE"
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    val mp = thm "mp"
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  in
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    fun iff_tac prems i =
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      resolve_tac (prems RL [iffE]) i THEN
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      REPEAT1 (eresolve_tac [asm_rl, mp] i)
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  end
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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')"
wenzelm@21539
   372
  apply (insert assms)
wenzelm@21539
   373
  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   374
  done
wenzelm@21539
   375
wenzelm@21539
   376
lemma imp_cong:
wenzelm@21539
   377
  assumes "P <-> P'"
wenzelm@21539
   378
    and "P' ==> Q <-> Q'"
wenzelm@21539
   379
  shows "(P-->Q) <-> (P'-->Q')"
wenzelm@21539
   380
  apply (insert assms)
wenzelm@21539
   381
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
wenzelm@21539
   382
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   383
  done
wenzelm@21539
   384
wenzelm@21539
   385
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
wenzelm@21539
   386
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   387
  done
wenzelm@21539
   388
wenzelm@21539
   389
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
wenzelm@21539
   390
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
wenzelm@21539
   391
  done
wenzelm@21539
   392
wenzelm@21539
   393
lemma all_cong:
wenzelm@21539
   394
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   395
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   396
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@21539
   397
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   398
  done
wenzelm@21539
   399
wenzelm@21539
   400
lemma ex_cong:
wenzelm@21539
   401
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   402
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   403
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@21539
   404
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   405
  done
wenzelm@21539
   406
wenzelm@21539
   407
lemma ex1_cong:
wenzelm@21539
   408
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   409
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   410
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@21539
   411
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   412
  done
wenzelm@21539
   413
wenzelm@21539
   414
(*** Equality rules ***)
wenzelm@21539
   415
wenzelm@21539
   416
lemma sym: "a=b ==> b=a"
wenzelm@21539
   417
  apply (erule subst)
wenzelm@21539
   418
  apply (rule refl)
wenzelm@21539
   419
  done
wenzelm@21539
   420
wenzelm@21539
   421
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   422
  apply (erule subst, assumption)
wenzelm@21539
   423
  done
wenzelm@21539
   424
wenzelm@21539
   425
(**  **)
wenzelm@21539
   426
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   427
  apply (erule contrapos)
wenzelm@21539
   428
  apply (erule sym)
wenzelm@21539
   429
  done
wenzelm@21539
   430
  
wenzelm@21539
   431
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   432
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   433
wenzelm@21539
   434
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   435
  apply unfold
wenzelm@21539
   436
  apply (rule iff_refl)
wenzelm@21539
   437
  done
wenzelm@21539
   438
wenzelm@21539
   439
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   440
  apply unfold
wenzelm@21539
   441
  apply (rule refl)
wenzelm@21539
   442
  done
wenzelm@21539
   443
wenzelm@21539
   444
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   445
  by unfold (rule iff_refl)
wenzelm@21539
   446
wenzelm@21539
   447
(*substitution*)
wenzelm@21539
   448
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   449
  apply (drule sym)
wenzelm@21539
   450
  apply (erule (1) subst)
wenzelm@21539
   451
  done
wenzelm@21539
   452
wenzelm@21539
   453
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   454
lemma ex1_equalsE:
wenzelm@21539
   455
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   456
  apply (erule ex1E)
wenzelm@21539
   457
  apply (rule trans)
wenzelm@21539
   458
   apply (rule_tac [2] sym)
wenzelm@21539
   459
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   460
  done
wenzelm@21539
   461
wenzelm@21539
   462
(** Polymorphic congruence rules **)
wenzelm@21539
   463
wenzelm@21539
   464
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   465
  apply (erule ssubst)
wenzelm@21539
   466
  apply (rule refl)
wenzelm@21539
   467
  done
wenzelm@21539
   468
wenzelm@21539
   469
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   470
  apply (erule ssubst)+
wenzelm@21539
   471
  apply (rule refl)
wenzelm@21539
   472
  done
wenzelm@21539
   473
wenzelm@21539
   474
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   475
  apply (erule ssubst)+
wenzelm@21539
   476
  apply (rule refl)
wenzelm@21539
   477
  done
wenzelm@21539
   478
wenzelm@21539
   479
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   480
        a = b
wenzelm@21539
   481
        |   |
wenzelm@21539
   482
        c = d   *)
wenzelm@21539
   483
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   484
  apply (rule trans)
wenzelm@21539
   485
   apply (rule trans)
wenzelm@21539
   486
    apply (rule sym)
wenzelm@21539
   487
    apply assumption+
wenzelm@21539
   488
  done
wenzelm@21539
   489
wenzelm@21539
   490
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   491
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   492
  apply (rule trans)
wenzelm@21539
   493
   apply (rule trans)
wenzelm@21539
   494
    apply assumption+
wenzelm@21539
   495
  apply (erule sym)
wenzelm@21539
   496
  done
wenzelm@21539
   497
wenzelm@21539
   498
(** Congruence rules for predicate letters **)
wenzelm@21539
   499
wenzelm@21539
   500
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   501
  apply (rule iffI)
wenzelm@21539
   502
   apply (erule (1) subst)
wenzelm@21539
   503
  apply (erule (1) ssubst)
wenzelm@21539
   504
  done
wenzelm@21539
   505
wenzelm@21539
   506
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   507
  apply (rule iffI)
wenzelm@21539
   508
   apply (erule subst)+
wenzelm@21539
   509
   apply assumption
wenzelm@21539
   510
  apply (erule ssubst)+
wenzelm@21539
   511
  apply assumption
wenzelm@21539
   512
  done
wenzelm@21539
   513
wenzelm@21539
   514
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
wenzelm@21539
   515
  apply (rule iffI)
wenzelm@21539
   516
   apply (erule subst)+
wenzelm@21539
   517
   apply assumption
wenzelm@21539
   518
  apply (erule ssubst)+
wenzelm@21539
   519
  apply assumption
wenzelm@21539
   520
  done
wenzelm@21539
   521
wenzelm@21539
   522
(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
wenzelm@21539
   523
wenzelm@21539
   524
ML {*
wenzelm@21539
   525
bind_thms ("pred_congs",
wenzelm@21539
   526
  List.concat (map (fn c => 
wenzelm@21539
   527
               map (fn th => read_instantiate [("P",c)] th)
wenzelm@21539
   528
                   [thm "pred1_cong", thm "pred2_cong", thm "pred3_cong"])
wenzelm@21539
   529
               (explode"PQRS")))
wenzelm@21539
   530
*}
wenzelm@21539
   531
wenzelm@21539
   532
(*special case for the equality predicate!*)
wenzelm@21539
   533
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   534
  apply (erule (1) pred2_cong)
wenzelm@21539
   535
  done
wenzelm@21539
   536
wenzelm@21539
   537
wenzelm@21539
   538
(*** Simplifications of assumed implications.
wenzelm@21539
   539
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   540
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   541
     intuitionistic propositional logic.  See
wenzelm@21539
   542
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   543
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   544
wenzelm@21539
   545
lemma conj_impE:
wenzelm@21539
   546
  assumes major: "(P&Q)-->S"
wenzelm@21539
   547
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   548
  shows R
wenzelm@21539
   549
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   550
wenzelm@21539
   551
lemma disj_impE:
wenzelm@21539
   552
  assumes major: "(P|Q)-->S"
wenzelm@21539
   553
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   554
  shows R
wenzelm@21539
   555
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   556
wenzelm@21539
   557
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   558
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   559
lemma imp_impE:
wenzelm@21539
   560
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   561
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   562
    and r2: "S ==> R"
wenzelm@21539
   563
  shows R
wenzelm@21539
   564
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   565
wenzelm@21539
   566
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   567
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   568
lemma not_impE:
wenzelm@21539
   569
  assumes major: "~P --> S"
wenzelm@21539
   570
    and r1: "P ==> False"
wenzelm@21539
   571
    and r2: "S ==> R"
wenzelm@21539
   572
  shows R
wenzelm@21539
   573
  apply (assumption | rule notI impI major [THEN mp] r1 r2)+
wenzelm@21539
   574
  done
wenzelm@21539
   575
wenzelm@21539
   576
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   577
lemma iff_impE:
wenzelm@21539
   578
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   579
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   580
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   581
    and r3: "S ==> R"
wenzelm@21539
   582
  shows R
wenzelm@21539
   583
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   584
  done
wenzelm@21539
   585
wenzelm@21539
   586
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   587
lemma all_impE:
wenzelm@21539
   588
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   589
    and r1: "!!x. P(x)"
wenzelm@21539
   590
    and r2: "S ==> R"
wenzelm@21539
   591
  shows R
wenzelm@21539
   592
  apply (assumption | rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   593
  done
wenzelm@21539
   594
wenzelm@21539
   595
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   596
lemma ex_impE:
wenzelm@21539
   597
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   598
    and r: "P(x)-->S ==> R"
wenzelm@21539
   599
  shows R
wenzelm@21539
   600
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   601
  done
wenzelm@21539
   602
wenzelm@21539
   603
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   604
wenzelm@21539
   605
lemma disj_imp_disj:
wenzelm@21539
   606
  assumes major: "P|Q"
wenzelm@21539
   607
    and "P==>R" and "Q==>S"
wenzelm@21539
   608
  shows "R|S"
wenzelm@21539
   609
  apply (rule disjE [OF major])
wenzelm@21539
   610
  apply (rule disjI1) apply assumption
wenzelm@21539
   611
  apply (rule disjI2) apply assumption
wenzelm@21539
   612
  done
wenzelm@11734
   613
wenzelm@18481
   614
ML {*
wenzelm@18481
   615
structure ProjectRule = ProjectRuleFun
wenzelm@18481
   616
(struct
wenzelm@18481
   617
  val conjunct1 = thm "conjunct1";
wenzelm@18481
   618
  val conjunct2 = thm "conjunct2";
wenzelm@18481
   619
  val mp = thm "mp";
wenzelm@18481
   620
end)
wenzelm@18481
   621
*}
wenzelm@18481
   622
wenzelm@7355
   623
use "fologic.ML"
wenzelm@21539
   624
wenzelm@21539
   625
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   626
wenzelm@9886
   627
use "hypsubstdata.ML"
wenzelm@9886
   628
setup hypsubst_setup
wenzelm@7355
   629
use "intprover.ML"
wenzelm@7355
   630
wenzelm@4092
   631
wenzelm@12875
   632
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   633
wenzelm@12349
   634
lemma impE':
wenzelm@12937
   635
  assumes 1: "P --> Q"
wenzelm@12937
   636
    and 2: "Q ==> R"
wenzelm@12937
   637
    and 3: "P --> Q ==> P"
wenzelm@12937
   638
  shows R
wenzelm@12349
   639
proof -
wenzelm@12349
   640
  from 3 and 1 have P .
wenzelm@12368
   641
  with 1 have Q by (rule impE)
wenzelm@12349
   642
  with 2 show R .
wenzelm@12349
   643
qed
wenzelm@12349
   644
wenzelm@12349
   645
lemma allE':
wenzelm@12937
   646
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   647
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   648
  shows Q
wenzelm@12349
   649
proof -
wenzelm@12349
   650
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   651
  from this and 1 show Q by (rule 2)
wenzelm@12349
   652
qed
wenzelm@12349
   653
wenzelm@12937
   654
lemma notE':
wenzelm@12937
   655
  assumes 1: "~ P"
wenzelm@12937
   656
    and 2: "~ P ==> P"
wenzelm@12937
   657
  shows R
wenzelm@12349
   658
proof -
wenzelm@12349
   659
  from 2 and 1 have P .
wenzelm@12349
   660
  with 1 show R by (rule notE)
wenzelm@12349
   661
qed
wenzelm@12349
   662
wenzelm@12349
   663
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   664
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   665
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   666
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   667
wenzelm@18708
   668
setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}
wenzelm@12349
   669
wenzelm@12349
   670
wenzelm@12368
   671
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   672
  by iprover
wenzelm@12368
   673
wenzelm@12368
   674
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   675
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   676
wenzelm@12368
   677
paulson@13435
   678
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   679
apply (rule iffI) 
paulson@13435
   680
apply (erule sym)+
paulson@13435
   681
done
paulson@13435
   682
paulson@13435
   683
wenzelm@11677
   684
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   685
wenzelm@11747
   686
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   687
proof
wenzelm@11677
   688
  assume "!!x. P(x)"
wenzelm@12368
   689
  show "ALL x. P(x)" ..
wenzelm@11677
   690
next
wenzelm@11677
   691
  assume "ALL x. P(x)"
wenzelm@12368
   692
  thus "!!x. P(x)" ..
wenzelm@11677
   693
qed
wenzelm@11677
   694
wenzelm@11747
   695
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   696
proof
wenzelm@12368
   697
  assume "A ==> B"
wenzelm@12368
   698
  thus "A --> B" ..
wenzelm@11677
   699
next
wenzelm@11677
   700
  assume "A --> B" and A
wenzelm@11677
   701
  thus B by (rule mp)
wenzelm@11677
   702
qed
wenzelm@11677
   703
wenzelm@11747
   704
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   705
proof
wenzelm@11677
   706
  assume "x == y"
wenzelm@11677
   707
  show "x = y" by (unfold prems) (rule refl)
wenzelm@11677
   708
next
wenzelm@11677
   709
  assume "x = y"
wenzelm@11677
   710
  thus "x == y" by (rule eq_reflection)
wenzelm@11677
   711
qed
wenzelm@11677
   712
wenzelm@18813
   713
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   714
proof
wenzelm@18813
   715
  assume "A == B"
wenzelm@18813
   716
  show "A <-> B" by (unfold prems) (rule iff_refl)
wenzelm@18813
   717
next
wenzelm@18813
   718
  assume "A <-> B"
wenzelm@18813
   719
  thus "A == B" by (rule iff_reflection)
wenzelm@18813
   720
qed
wenzelm@18813
   721
wenzelm@12875
   722
lemma atomize_conj [atomize]:
wenzelm@19120
   723
  includes meta_conjunction_syntax
wenzelm@19120
   724
  shows "(A && B) == Trueprop (A & B)"
wenzelm@11976
   725
proof
wenzelm@19120
   726
  assume conj: "A && B"
wenzelm@19120
   727
  show "A & B"
wenzelm@19120
   728
  proof (rule conjI)
wenzelm@19120
   729
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   730
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   731
  qed
wenzelm@11953
   732
next
wenzelm@19120
   733
  assume conj: "A & B"
wenzelm@19120
   734
  show "A && B"
wenzelm@19120
   735
  proof -
wenzelm@19120
   736
    from conj show A ..
wenzelm@19120
   737
    from conj show B ..
wenzelm@11953
   738
  qed
wenzelm@11953
   739
qed
wenzelm@11953
   740
wenzelm@12368
   741
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   742
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   743
wenzelm@11848
   744
wenzelm@11848
   745
subsection {* Calculational rules *}
wenzelm@11848
   746
wenzelm@11848
   747
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   748
  by (rule ssubst)
wenzelm@11848
   749
wenzelm@11848
   750
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   751
  by (rule subst)
wenzelm@11848
   752
wenzelm@11848
   753
text {*
wenzelm@11848
   754
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   755
*}
wenzelm@11848
   756
wenzelm@12019
   757
lemmas basic_trans_rules [trans] =
wenzelm@11848
   758
  forw_subst
wenzelm@11848
   759
  back_subst
wenzelm@11848
   760
  rev_mp
wenzelm@11848
   761
  mp
wenzelm@11848
   762
  trans
wenzelm@11848
   763
paulson@13779
   764
subsection {* ``Let'' declarations *}
paulson@13779
   765
paulson@13779
   766
nonterminals letbinds letbind
paulson@13779
   767
paulson@13779
   768
constdefs
wenzelm@14854
   769
  Let :: "['a::{}, 'a => 'b] => ('b::{})"
paulson@13779
   770
    "Let(s, f) == f(s)"
paulson@13779
   771
paulson@13779
   772
syntax
paulson@13779
   773
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   774
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   775
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   776
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   777
paulson@13779
   778
translations
paulson@13779
   779
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
paulson@13779
   780
  "let x = a in e"          == "Let(a, %x. e)"
paulson@13779
   781
paulson@13779
   782
paulson@13779
   783
lemma LetI: 
wenzelm@21539
   784
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   785
  shows "P(let x=t in u(x))"
wenzelm@21539
   786
  apply (unfold Let_def)
wenzelm@21539
   787
  apply (rule refl [THEN assms])
wenzelm@21539
   788
  done
wenzelm@21539
   789
wenzelm@21539
   790
wenzelm@21539
   791
subsection {* ML bindings *}
paulson@13779
   792
wenzelm@21539
   793
ML {*
wenzelm@21539
   794
val refl = thm "refl"
wenzelm@21539
   795
val trans = thm "trans"
wenzelm@21539
   796
val sym = thm "sym"
wenzelm@21539
   797
val subst = thm "subst"
wenzelm@21539
   798
val ssubst = thm "ssubst"
wenzelm@21539
   799
val conjI = thm "conjI"
wenzelm@21539
   800
val conjE = thm "conjE"
wenzelm@21539
   801
val conjunct1 = thm "conjunct1"
wenzelm@21539
   802
val conjunct2 = thm "conjunct2"
wenzelm@21539
   803
val disjI1 = thm "disjI1"
wenzelm@21539
   804
val disjI2 = thm "disjI2"
wenzelm@21539
   805
val disjE = thm "disjE"
wenzelm@21539
   806
val impI = thm "impI"
wenzelm@21539
   807
val impE = thm "impE"
wenzelm@21539
   808
val mp = thm "mp"
wenzelm@21539
   809
val rev_mp = thm "rev_mp"
wenzelm@21539
   810
val TrueI = thm "TrueI"
wenzelm@21539
   811
val FalseE = thm "FalseE"
wenzelm@21539
   812
val iff_refl = thm "iff_refl"
wenzelm@21539
   813
val iff_trans = thm "iff_trans"
wenzelm@21539
   814
val iffI = thm "iffI"
wenzelm@21539
   815
val iffE = thm "iffE"
wenzelm@21539
   816
val iffD1 = thm "iffD1"
wenzelm@21539
   817
val iffD2 = thm "iffD2"
wenzelm@21539
   818
val notI = thm "notI"
wenzelm@21539
   819
val notE = thm "notE"
wenzelm@21539
   820
val allI = thm "allI"
wenzelm@21539
   821
val allE = thm "allE"
wenzelm@21539
   822
val spec = thm "spec"
wenzelm@21539
   823
val exI = thm "exI"
wenzelm@21539
   824
val exE = thm "exE"
wenzelm@21539
   825
val eq_reflection = thm "eq_reflection"
wenzelm@21539
   826
val iff_reflection = thm "iff_reflection"
wenzelm@21539
   827
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"
wenzelm@21539
   828
val meta_eq_to_iff = thm "meta_eq_to_iff"
paulson@13779
   829
*}
paulson@13779
   830
wenzelm@4854
   831
end