src/FOL/IFOL.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 39557 fe5722fce758
child 41310 65631ca437c9
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
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(*  Title:      FOL/IFOL.thy
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    Author:     Lawrence C Paulson and Markus Wenzel
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*)
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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/Tools/eqsubst.ML"
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  "~~/src/Provers/quantifier1.ML"
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  "~~/src/Tools/intuitionistic.ML"
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  "~~/src/Tools/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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setup Pure_Thy.old_appl_syntax_setup
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classes "term"
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default_sort "term"
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typedecl o
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judgment
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  Trueprop      :: "o => prop"                  ("(_)" 5)
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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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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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  subst:        "a=b \<Longrightarrow> P(a) \<Longrightarrow> P(b)"
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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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axioms
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  (* Reflection, admissible *)
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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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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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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 *})+
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  done
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lemma disj_cong:
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  assumes "P <-> P'" and "Q <-> Q'"
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  shows "(P|Q) <-> (P'|Q')"
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  apply (insert assms)
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  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
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  done
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lemma imp_cong:
wenzelm@21539
   364
  assumes "P <-> P'"
wenzelm@21539
   365
    and "P' ==> Q <-> Q'"
wenzelm@21539
   366
  shows "(P-->Q) <-> (P'-->Q')"
wenzelm@21539
   367
  apply (insert assms)
wenzelm@21539
   368
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
wenzelm@39159
   369
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   370
  done
wenzelm@21539
   371
wenzelm@21539
   372
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
wenzelm@21539
   373
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   374
  done
wenzelm@21539
   375
wenzelm@21539
   376
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
wenzelm@21539
   377
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
wenzelm@21539
   378
  done
wenzelm@21539
   379
wenzelm@21539
   380
lemma all_cong:
wenzelm@21539
   381
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   382
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   383
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@39159
   384
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   385
  done
wenzelm@21539
   386
wenzelm@21539
   387
lemma ex_cong:
wenzelm@21539
   388
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   389
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   390
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@39159
   391
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   392
  done
wenzelm@21539
   393
wenzelm@21539
   394
lemma ex1_cong:
wenzelm@21539
   395
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   396
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   397
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@39159
   398
    tactic {* iff_tac @{thms assms} 1 *})+
wenzelm@21539
   399
  done
wenzelm@21539
   400
wenzelm@21539
   401
(*** Equality rules ***)
wenzelm@21539
   402
wenzelm@21539
   403
lemma sym: "a=b ==> b=a"
wenzelm@21539
   404
  apply (erule subst)
wenzelm@21539
   405
  apply (rule refl)
wenzelm@21539
   406
  done
wenzelm@21539
   407
wenzelm@21539
   408
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   409
  apply (erule subst, assumption)
wenzelm@21539
   410
  done
wenzelm@21539
   411
wenzelm@21539
   412
(**  **)
wenzelm@21539
   413
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   414
  apply (erule contrapos)
wenzelm@21539
   415
  apply (erule sym)
wenzelm@21539
   416
  done
wenzelm@21539
   417
  
wenzelm@21539
   418
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   419
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   420
wenzelm@21539
   421
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   422
  apply unfold
wenzelm@21539
   423
  apply (rule iff_refl)
wenzelm@21539
   424
  done
wenzelm@21539
   425
wenzelm@21539
   426
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   427
  apply unfold
wenzelm@21539
   428
  apply (rule refl)
wenzelm@21539
   429
  done
wenzelm@21539
   430
wenzelm@21539
   431
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   432
  by unfold (rule iff_refl)
wenzelm@21539
   433
wenzelm@21539
   434
(*substitution*)
wenzelm@21539
   435
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   436
  apply (drule sym)
wenzelm@21539
   437
  apply (erule (1) subst)
wenzelm@21539
   438
  done
wenzelm@21539
   439
wenzelm@21539
   440
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   441
lemma ex1_equalsE:
wenzelm@21539
   442
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   443
  apply (erule ex1E)
wenzelm@21539
   444
  apply (rule trans)
wenzelm@21539
   445
   apply (rule_tac [2] sym)
wenzelm@21539
   446
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   447
  done
wenzelm@21539
   448
wenzelm@21539
   449
(** Polymorphic congruence rules **)
wenzelm@21539
   450
wenzelm@21539
   451
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   452
  apply (erule ssubst)
wenzelm@21539
   453
  apply (rule refl)
wenzelm@21539
   454
  done
wenzelm@21539
   455
wenzelm@21539
   456
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   457
  apply (erule ssubst)+
wenzelm@21539
   458
  apply (rule refl)
wenzelm@21539
   459
  done
wenzelm@21539
   460
wenzelm@21539
   461
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   462
  apply (erule ssubst)+
wenzelm@21539
   463
  apply (rule refl)
wenzelm@21539
   464
  done
wenzelm@21539
   465
wenzelm@21539
   466
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   467
        a = b
wenzelm@21539
   468
        |   |
wenzelm@21539
   469
        c = d   *)
wenzelm@21539
   470
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   471
  apply (rule trans)
wenzelm@21539
   472
   apply (rule trans)
wenzelm@21539
   473
    apply (rule sym)
wenzelm@21539
   474
    apply assumption+
wenzelm@21539
   475
  done
wenzelm@21539
   476
wenzelm@21539
   477
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   478
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   479
  apply (rule trans)
wenzelm@21539
   480
   apply (rule trans)
wenzelm@21539
   481
    apply assumption+
wenzelm@21539
   482
  apply (erule sym)
wenzelm@21539
   483
  done
wenzelm@21539
   484
wenzelm@21539
   485
(** Congruence rules for predicate letters **)
wenzelm@21539
   486
wenzelm@21539
   487
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   488
  apply (rule iffI)
wenzelm@21539
   489
   apply (erule (1) subst)
wenzelm@21539
   490
  apply (erule (1) ssubst)
wenzelm@21539
   491
  done
wenzelm@21539
   492
wenzelm@21539
   493
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   494
  apply (rule iffI)
wenzelm@21539
   495
   apply (erule subst)+
wenzelm@21539
   496
   apply assumption
wenzelm@21539
   497
  apply (erule ssubst)+
wenzelm@21539
   498
  apply assumption
wenzelm@21539
   499
  done
wenzelm@21539
   500
wenzelm@21539
   501
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
wenzelm@21539
   502
  apply (rule iffI)
wenzelm@21539
   503
   apply (erule subst)+
wenzelm@21539
   504
   apply assumption
wenzelm@21539
   505
  apply (erule ssubst)+
wenzelm@21539
   506
  apply assumption
wenzelm@21539
   507
  done
wenzelm@21539
   508
wenzelm@21539
   509
(*special case for the equality predicate!*)
wenzelm@21539
   510
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   511
  apply (erule (1) pred2_cong)
wenzelm@21539
   512
  done
wenzelm@21539
   513
wenzelm@21539
   514
wenzelm@21539
   515
(*** Simplifications of assumed implications.
wenzelm@21539
   516
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   517
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   518
     intuitionistic propositional logic.  See
wenzelm@21539
   519
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   520
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   521
wenzelm@21539
   522
lemma conj_impE:
wenzelm@21539
   523
  assumes major: "(P&Q)-->S"
wenzelm@21539
   524
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   525
  shows R
wenzelm@21539
   526
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   527
wenzelm@21539
   528
lemma disj_impE:
wenzelm@21539
   529
  assumes major: "(P|Q)-->S"
wenzelm@21539
   530
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   531
  shows R
wenzelm@21539
   532
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   533
wenzelm@21539
   534
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   535
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   536
lemma imp_impE:
wenzelm@21539
   537
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   538
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   539
    and r2: "S ==> R"
wenzelm@21539
   540
  shows R
wenzelm@21539
   541
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   542
wenzelm@21539
   543
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   544
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   545
lemma not_impE:
wenzelm@23393
   546
  "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R"
wenzelm@23393
   547
  apply (drule mp)
wenzelm@23393
   548
   apply (rule notI)
wenzelm@23393
   549
   apply assumption
wenzelm@23393
   550
  apply assumption
wenzelm@21539
   551
  done
wenzelm@21539
   552
wenzelm@21539
   553
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   554
lemma iff_impE:
wenzelm@21539
   555
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   556
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   557
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   558
    and r3: "S ==> R"
wenzelm@21539
   559
  shows R
wenzelm@21539
   560
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   561
  done
wenzelm@21539
   562
wenzelm@21539
   563
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   564
lemma all_impE:
wenzelm@21539
   565
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   566
    and r1: "!!x. P(x)"
wenzelm@21539
   567
    and r2: "S ==> R"
wenzelm@21539
   568
  shows R
wenzelm@23393
   569
  apply (rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   570
  done
wenzelm@21539
   571
wenzelm@21539
   572
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   573
lemma ex_impE:
wenzelm@21539
   574
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   575
    and r: "P(x)-->S ==> R"
wenzelm@21539
   576
  shows R
wenzelm@21539
   577
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   578
  done
wenzelm@21539
   579
wenzelm@21539
   580
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   581
wenzelm@21539
   582
lemma disj_imp_disj:
wenzelm@23393
   583
  "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> R|S"
wenzelm@23393
   584
  apply (erule disjE)
wenzelm@21539
   585
  apply (rule disjI1) apply assumption
wenzelm@21539
   586
  apply (rule disjI2) apply assumption
wenzelm@21539
   587
  done
wenzelm@11734
   588
wenzelm@18481
   589
ML {*
wenzelm@32172
   590
structure Project_Rule = Project_Rule
wenzelm@32172
   591
(
wenzelm@22139
   592
  val conjunct1 = @{thm conjunct1}
wenzelm@22139
   593
  val conjunct2 = @{thm conjunct2}
wenzelm@22139
   594
  val mp = @{thm mp}
wenzelm@32172
   595
)
wenzelm@18481
   596
*}
wenzelm@18481
   597
wenzelm@7355
   598
use "fologic.ML"
wenzelm@21539
   599
wenzelm@21539
   600
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   601
wenzelm@9886
   602
use "hypsubstdata.ML"
wenzelm@9886
   603
setup hypsubst_setup
wenzelm@7355
   604
use "intprover.ML"
wenzelm@7355
   605
wenzelm@4092
   606
wenzelm@12875
   607
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   608
wenzelm@31299
   609
setup {* Intuitionistic.method_setup @{binding iprover} *}
wenzelm@30165
   610
wenzelm@12349
   611
lemma impE':
wenzelm@12937
   612
  assumes 1: "P --> Q"
wenzelm@12937
   613
    and 2: "Q ==> R"
wenzelm@12937
   614
    and 3: "P --> Q ==> P"
wenzelm@12937
   615
  shows R
wenzelm@12349
   616
proof -
wenzelm@12349
   617
  from 3 and 1 have P .
wenzelm@12368
   618
  with 1 have Q by (rule impE)
wenzelm@12349
   619
  with 2 show R .
wenzelm@12349
   620
qed
wenzelm@12349
   621
wenzelm@12349
   622
lemma allE':
wenzelm@12937
   623
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   624
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   625
  shows Q
wenzelm@12349
   626
proof -
wenzelm@12349
   627
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   628
  from this and 1 show Q by (rule 2)
wenzelm@12349
   629
qed
wenzelm@12349
   630
wenzelm@12937
   631
lemma notE':
wenzelm@12937
   632
  assumes 1: "~ P"
wenzelm@12937
   633
    and 2: "~ P ==> P"
wenzelm@12937
   634
  shows R
wenzelm@12349
   635
proof -
wenzelm@12349
   636
  from 2 and 1 have P .
wenzelm@12349
   637
  with 1 show R by (rule notE)
wenzelm@12349
   638
qed
wenzelm@12349
   639
wenzelm@12349
   640
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   641
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   642
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   643
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   644
wenzelm@33369
   645
setup {* Context_Rules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}
wenzelm@12349
   646
wenzelm@12349
   647
wenzelm@12368
   648
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   649
  by iprover
wenzelm@12368
   650
wenzelm@12368
   651
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   652
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   653
wenzelm@12368
   654
paulson@13435
   655
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   656
apply (rule iffI) 
paulson@13435
   657
apply (erule sym)+
paulson@13435
   658
done
paulson@13435
   659
paulson@13435
   660
wenzelm@11677
   661
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   662
wenzelm@11747
   663
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   664
proof
wenzelm@11677
   665
  assume "!!x. P(x)"
wenzelm@22931
   666
  then show "ALL x. P(x)" ..
wenzelm@11677
   667
next
wenzelm@11677
   668
  assume "ALL x. P(x)"
wenzelm@22931
   669
  then show "!!x. P(x)" ..
wenzelm@11677
   670
qed
wenzelm@11677
   671
wenzelm@11747
   672
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   673
proof
wenzelm@12368
   674
  assume "A ==> B"
wenzelm@22931
   675
  then show "A --> B" ..
wenzelm@11677
   676
next
wenzelm@11677
   677
  assume "A --> B" and A
wenzelm@22931
   678
  then show B by (rule mp)
wenzelm@11677
   679
qed
wenzelm@11677
   680
wenzelm@11747
   681
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   682
proof
wenzelm@11677
   683
  assume "x == y"
wenzelm@22931
   684
  show "x = y" unfolding `x == y` by (rule refl)
wenzelm@11677
   685
next
wenzelm@11677
   686
  assume "x = y"
wenzelm@22931
   687
  then show "x == y" by (rule eq_reflection)
wenzelm@11677
   688
qed
wenzelm@11677
   689
wenzelm@18813
   690
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   691
proof
wenzelm@18813
   692
  assume "A == B"
wenzelm@22931
   693
  show "A <-> B" unfolding `A == B` by (rule iff_refl)
wenzelm@18813
   694
next
wenzelm@18813
   695
  assume "A <-> B"
wenzelm@22931
   696
  then show "A == B" by (rule iff_reflection)
wenzelm@18813
   697
qed
wenzelm@18813
   698
wenzelm@28856
   699
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
wenzelm@11976
   700
proof
wenzelm@28856
   701
  assume conj: "A &&& B"
wenzelm@19120
   702
  show "A & B"
wenzelm@19120
   703
  proof (rule conjI)
wenzelm@19120
   704
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   705
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   706
  qed
wenzelm@11953
   707
next
wenzelm@19120
   708
  assume conj: "A & B"
wenzelm@28856
   709
  show "A &&& B"
wenzelm@19120
   710
  proof -
wenzelm@19120
   711
    from conj show A ..
wenzelm@19120
   712
    from conj show B ..
wenzelm@11953
   713
  qed
wenzelm@11953
   714
qed
wenzelm@11953
   715
wenzelm@12368
   716
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   717
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   718
wenzelm@11848
   719
krauss@26580
   720
subsection {* Atomizing elimination rules *}
krauss@26580
   721
krauss@26580
   722
setup AtomizeElim.setup
krauss@26580
   723
krauss@26580
   724
lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX x. P(x)) ==> Q)"
krauss@26580
   725
by rule iprover+
krauss@26580
   726
krauss@26580
   727
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
krauss@26580
   728
by rule iprover+
krauss@26580
   729
krauss@26580
   730
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
krauss@26580
   731
by rule iprover+
krauss@26580
   732
krauss@26580
   733
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop(A)" ..
krauss@26580
   734
krauss@26580
   735
wenzelm@11848
   736
subsection {* Calculational rules *}
wenzelm@11848
   737
wenzelm@11848
   738
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   739
  by (rule ssubst)
wenzelm@11848
   740
wenzelm@11848
   741
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   742
  by (rule subst)
wenzelm@11848
   743
wenzelm@11848
   744
text {*
wenzelm@11848
   745
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   746
*}
wenzelm@11848
   747
wenzelm@12019
   748
lemmas basic_trans_rules [trans] =
wenzelm@11848
   749
  forw_subst
wenzelm@11848
   750
  back_subst
wenzelm@11848
   751
  rev_mp
wenzelm@11848
   752
  mp
wenzelm@11848
   753
  trans
wenzelm@11848
   754
paulson@13779
   755
subsection {* ``Let'' declarations *}
paulson@13779
   756
wenzelm@41229
   757
nonterminal letbinds and letbind
paulson@13779
   758
haftmann@35416
   759
definition Let :: "['a::{}, 'a => 'b] => ('b::{})" where
paulson@13779
   760
    "Let(s, f) == f(s)"
paulson@13779
   761
paulson@13779
   762
syntax
paulson@13779
   763
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   764
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   765
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   766
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   767
paulson@13779
   768
translations
paulson@13779
   769
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
wenzelm@35054
   770
  "let x = a in e"          == "CONST Let(a, %x. e)"
paulson@13779
   771
paulson@13779
   772
paulson@13779
   773
lemma LetI: 
wenzelm@21539
   774
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   775
  shows "P(let x=t in u(x))"
wenzelm@21539
   776
  apply (unfold Let_def)
wenzelm@21539
   777
  apply (rule refl [THEN assms])
wenzelm@21539
   778
  done
wenzelm@21539
   779
wenzelm@21539
   780
wenzelm@26286
   781
subsection {* Intuitionistic simplification rules *}
wenzelm@26286
   782
wenzelm@26286
   783
lemma conj_simps:
wenzelm@26286
   784
  "P & True <-> P"
wenzelm@26286
   785
  "True & P <-> P"
wenzelm@26286
   786
  "P & False <-> False"
wenzelm@26286
   787
  "False & P <-> False"
wenzelm@26286
   788
  "P & P <-> P"
wenzelm@26286
   789
  "P & P & Q <-> P & Q"
wenzelm@26286
   790
  "P & ~P <-> False"
wenzelm@26286
   791
  "~P & P <-> False"
wenzelm@26286
   792
  "(P & Q) & R <-> P & (Q & R)"
wenzelm@26286
   793
  by iprover+
wenzelm@26286
   794
wenzelm@26286
   795
lemma disj_simps:
wenzelm@26286
   796
  "P | True <-> True"
wenzelm@26286
   797
  "True | P <-> True"
wenzelm@26286
   798
  "P | False <-> P"
wenzelm@26286
   799
  "False | P <-> P"
wenzelm@26286
   800
  "P | P <-> P"
wenzelm@26286
   801
  "P | P | Q <-> P | Q"
wenzelm@26286
   802
  "(P | Q) | R <-> P | (Q | R)"
wenzelm@26286
   803
  by iprover+
wenzelm@26286
   804
wenzelm@26286
   805
lemma not_simps:
wenzelm@26286
   806
  "~(P|Q)  <-> ~P & ~Q"
wenzelm@26286
   807
  "~ False <-> True"
wenzelm@26286
   808
  "~ True <-> False"
wenzelm@26286
   809
  by iprover+
wenzelm@26286
   810
wenzelm@26286
   811
lemma imp_simps:
wenzelm@26286
   812
  "(P --> False) <-> ~P"
wenzelm@26286
   813
  "(P --> True) <-> True"
wenzelm@26286
   814
  "(False --> P) <-> True"
wenzelm@26286
   815
  "(True --> P) <-> P"
wenzelm@26286
   816
  "(P --> P) <-> True"
wenzelm@26286
   817
  "(P --> ~P) <-> ~P"
wenzelm@26286
   818
  by iprover+
wenzelm@26286
   819
wenzelm@26286
   820
lemma iff_simps:
wenzelm@26286
   821
  "(True <-> P) <-> P"
wenzelm@26286
   822
  "(P <-> True) <-> P"
wenzelm@26286
   823
  "(P <-> P) <-> True"
wenzelm@26286
   824
  "(False <-> P) <-> ~P"
wenzelm@26286
   825
  "(P <-> False) <-> ~P"
wenzelm@26286
   826
  by iprover+
wenzelm@26286
   827
wenzelm@26286
   828
(*The x=t versions are needed for the simplification procedures*)
wenzelm@26286
   829
lemma quant_simps:
wenzelm@26286
   830
  "!!P. (ALL x. P) <-> P"
wenzelm@26286
   831
  "(ALL x. x=t --> P(x)) <-> P(t)"
wenzelm@26286
   832
  "(ALL x. t=x --> P(x)) <-> P(t)"
wenzelm@26286
   833
  "!!P. (EX x. P) <-> P"
wenzelm@26286
   834
  "EX x. x=t"
wenzelm@26286
   835
  "EX x. t=x"
wenzelm@26286
   836
  "(EX x. x=t & P(x)) <-> P(t)"
wenzelm@26286
   837
  "(EX x. t=x & P(x)) <-> P(t)"
wenzelm@26286
   838
  by iprover+
wenzelm@26286
   839
wenzelm@26286
   840
(*These are NOT supplied by default!*)
wenzelm@26286
   841
lemma distrib_simps:
wenzelm@26286
   842
  "P & (Q | R) <-> P&Q | P&R"
wenzelm@26286
   843
  "(Q | R) & P <-> Q&P | R&P"
wenzelm@26286
   844
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"
wenzelm@26286
   845
  by iprover+
wenzelm@26286
   846
wenzelm@26286
   847
wenzelm@26286
   848
text {* Conversion into rewrite rules *}
wenzelm@26286
   849
wenzelm@26286
   850
lemma P_iff_F: "~P ==> (P <-> False)" by iprover
wenzelm@26286
   851
lemma iff_reflection_F: "~P ==> (P == False)" by (rule P_iff_F [THEN iff_reflection])
wenzelm@26286
   852
wenzelm@26286
   853
lemma P_iff_T: "P ==> (P <-> True)" by iprover
wenzelm@26286
   854
lemma iff_reflection_T: "P ==> (P == True)" by (rule P_iff_T [THEN iff_reflection])
wenzelm@26286
   855
wenzelm@26286
   856
wenzelm@26286
   857
text {* More rewrite rules *}
wenzelm@26286
   858
wenzelm@26286
   859
lemma conj_commute: "P&Q <-> Q&P" by iprover
wenzelm@26286
   860
lemma conj_left_commute: "P&(Q&R) <-> Q&(P&R)" by iprover
wenzelm@26286
   861
lemmas conj_comms = conj_commute conj_left_commute
wenzelm@26286
   862
wenzelm@26286
   863
lemma disj_commute: "P|Q <-> Q|P" by iprover
wenzelm@26286
   864
lemma disj_left_commute: "P|(Q|R) <-> Q|(P|R)" by iprover
wenzelm@26286
   865
lemmas disj_comms = disj_commute disj_left_commute
wenzelm@26286
   866
wenzelm@26286
   867
lemma conj_disj_distribL: "P&(Q|R) <-> (P&Q | P&R)" by iprover
wenzelm@26286
   868
lemma conj_disj_distribR: "(P|Q)&R <-> (P&R | Q&R)" by iprover
wenzelm@26286
   869
wenzelm@26286
   870
lemma disj_conj_distribL: "P|(Q&R) <-> (P|Q) & (P|R)" by iprover
wenzelm@26286
   871
lemma disj_conj_distribR: "(P&Q)|R <-> (P|R) & (Q|R)" by iprover
wenzelm@26286
   872
wenzelm@26286
   873
lemma imp_conj_distrib: "(P --> (Q&R)) <-> (P-->Q) & (P-->R)" by iprover
wenzelm@26286
   874
lemma imp_conj: "((P&Q)-->R)   <-> (P --> (Q --> R))" by iprover
wenzelm@26286
   875
lemma imp_disj: "(P|Q --> R)   <-> (P-->R) & (Q-->R)" by iprover
wenzelm@26286
   876
wenzelm@26286
   877
lemma de_Morgan_disj: "(~(P | Q)) <-> (~P & ~Q)" by iprover
wenzelm@26286
   878
wenzelm@26286
   879
lemma not_ex: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover
wenzelm@26286
   880
lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover
wenzelm@26286
   881
wenzelm@26286
   882
lemma ex_disj_distrib:
wenzelm@26286
   883
  "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover
wenzelm@26286
   884
wenzelm@26286
   885
lemma all_conj_distrib:
wenzelm@26286
   886
  "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover
wenzelm@26286
   887
wenzelm@4854
   888
end