src/FOL/IFOL.thy
author haftmann
Fri Oct 24 17:48:33 2008 +0200 (2008-10-24)
changeset 28681 e8664643f543
parent 27151 bd387c1a0536
child 28699 32b6a8f12c1c
permissions -rw-r--r--
subst is a proper axiom again
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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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setup PureThy.old_appl_syntax_setup
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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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  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)+
wenzelm@21539
   365
  done
wenzelm@21539
   366
wenzelm@21539
   367
lemma imp_cong:
wenzelm@21539
   368
  assumes "P <-> P'"
wenzelm@21539
   369
    and "P' ==> Q <-> Q'"
wenzelm@21539
   370
  shows "(P-->Q) <-> (P'-->Q')"
wenzelm@21539
   371
  apply (insert assms)
wenzelm@21539
   372
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
wenzelm@21539
   373
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   374
  done
wenzelm@21539
   375
wenzelm@21539
   376
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
wenzelm@21539
   377
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   378
  done
wenzelm@21539
   379
wenzelm@21539
   380
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
wenzelm@21539
   381
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
wenzelm@21539
   382
  done
wenzelm@21539
   383
wenzelm@21539
   384
lemma all_cong:
wenzelm@21539
   385
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   386
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   387
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@21539
   388
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   389
  done
wenzelm@21539
   390
wenzelm@21539
   391
lemma ex_cong:
wenzelm@21539
   392
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   393
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   394
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@21539
   395
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   396
  done
wenzelm@21539
   397
wenzelm@21539
   398
lemma ex1_cong:
wenzelm@21539
   399
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   400
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   401
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@21539
   402
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   403
  done
wenzelm@21539
   404
wenzelm@21539
   405
(*** Equality rules ***)
wenzelm@21539
   406
wenzelm@21539
   407
lemma sym: "a=b ==> b=a"
wenzelm@21539
   408
  apply (erule subst)
wenzelm@21539
   409
  apply (rule refl)
wenzelm@21539
   410
  done
wenzelm@21539
   411
wenzelm@21539
   412
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   413
  apply (erule subst, assumption)
wenzelm@21539
   414
  done
wenzelm@21539
   415
wenzelm@21539
   416
(**  **)
wenzelm@21539
   417
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   418
  apply (erule contrapos)
wenzelm@21539
   419
  apply (erule sym)
wenzelm@21539
   420
  done
wenzelm@21539
   421
  
wenzelm@21539
   422
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   423
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   424
wenzelm@21539
   425
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   426
  apply unfold
wenzelm@21539
   427
  apply (rule iff_refl)
wenzelm@21539
   428
  done
wenzelm@21539
   429
wenzelm@21539
   430
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   431
  apply unfold
wenzelm@21539
   432
  apply (rule refl)
wenzelm@21539
   433
  done
wenzelm@21539
   434
wenzelm@21539
   435
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   436
  by unfold (rule iff_refl)
wenzelm@21539
   437
wenzelm@21539
   438
(*substitution*)
wenzelm@21539
   439
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   440
  apply (drule sym)
wenzelm@21539
   441
  apply (erule (1) subst)
wenzelm@21539
   442
  done
wenzelm@21539
   443
wenzelm@21539
   444
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   445
lemma ex1_equalsE:
wenzelm@21539
   446
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   447
  apply (erule ex1E)
wenzelm@21539
   448
  apply (rule trans)
wenzelm@21539
   449
   apply (rule_tac [2] sym)
wenzelm@21539
   450
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   451
  done
wenzelm@21539
   452
wenzelm@21539
   453
(** Polymorphic congruence rules **)
wenzelm@21539
   454
wenzelm@21539
   455
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   456
  apply (erule ssubst)
wenzelm@21539
   457
  apply (rule refl)
wenzelm@21539
   458
  done
wenzelm@21539
   459
wenzelm@21539
   460
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   461
  apply (erule ssubst)+
wenzelm@21539
   462
  apply (rule refl)
wenzelm@21539
   463
  done
wenzelm@21539
   464
wenzelm@21539
   465
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   466
  apply (erule ssubst)+
wenzelm@21539
   467
  apply (rule refl)
wenzelm@21539
   468
  done
wenzelm@21539
   469
wenzelm@21539
   470
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   471
        a = b
wenzelm@21539
   472
        |   |
wenzelm@21539
   473
        c = d   *)
wenzelm@21539
   474
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   475
  apply (rule trans)
wenzelm@21539
   476
   apply (rule trans)
wenzelm@21539
   477
    apply (rule sym)
wenzelm@21539
   478
    apply assumption+
wenzelm@21539
   479
  done
wenzelm@21539
   480
wenzelm@21539
   481
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   482
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   483
  apply (rule trans)
wenzelm@21539
   484
   apply (rule trans)
wenzelm@21539
   485
    apply assumption+
wenzelm@21539
   486
  apply (erule sym)
wenzelm@21539
   487
  done
wenzelm@21539
   488
wenzelm@21539
   489
(** Congruence rules for predicate letters **)
wenzelm@21539
   490
wenzelm@21539
   491
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   492
  apply (rule iffI)
wenzelm@21539
   493
   apply (erule (1) subst)
wenzelm@21539
   494
  apply (erule (1) ssubst)
wenzelm@21539
   495
  done
wenzelm@21539
   496
wenzelm@21539
   497
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   498
  apply (rule iffI)
wenzelm@21539
   499
   apply (erule subst)+
wenzelm@21539
   500
   apply assumption
wenzelm@21539
   501
  apply (erule ssubst)+
wenzelm@21539
   502
  apply assumption
wenzelm@21539
   503
  done
wenzelm@21539
   504
wenzelm@21539
   505
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
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
(*special case for the equality predicate!*)
wenzelm@21539
   514
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   515
  apply (erule (1) pred2_cong)
wenzelm@21539
   516
  done
wenzelm@21539
   517
wenzelm@21539
   518
wenzelm@21539
   519
(*** Simplifications of assumed implications.
wenzelm@21539
   520
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   521
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   522
     intuitionistic propositional logic.  See
wenzelm@21539
   523
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   524
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   525
wenzelm@21539
   526
lemma conj_impE:
wenzelm@21539
   527
  assumes major: "(P&Q)-->S"
wenzelm@21539
   528
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   529
  shows R
wenzelm@21539
   530
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   531
wenzelm@21539
   532
lemma disj_impE:
wenzelm@21539
   533
  assumes major: "(P|Q)-->S"
wenzelm@21539
   534
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   535
  shows R
wenzelm@21539
   536
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   537
wenzelm@21539
   538
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   539
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   540
lemma imp_impE:
wenzelm@21539
   541
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   542
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   543
    and r2: "S ==> R"
wenzelm@21539
   544
  shows R
wenzelm@21539
   545
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   546
wenzelm@21539
   547
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   548
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   549
lemma not_impE:
wenzelm@23393
   550
  "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R"
wenzelm@23393
   551
  apply (drule mp)
wenzelm@23393
   552
   apply (rule notI)
wenzelm@23393
   553
   apply assumption
wenzelm@23393
   554
  apply assumption
wenzelm@21539
   555
  done
wenzelm@21539
   556
wenzelm@21539
   557
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   558
lemma iff_impE:
wenzelm@21539
   559
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   560
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   561
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   562
    and r3: "S ==> R"
wenzelm@21539
   563
  shows R
wenzelm@21539
   564
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   565
  done
wenzelm@21539
   566
wenzelm@21539
   567
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   568
lemma all_impE:
wenzelm@21539
   569
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   570
    and r1: "!!x. P(x)"
wenzelm@21539
   571
    and r2: "S ==> R"
wenzelm@21539
   572
  shows R
wenzelm@23393
   573
  apply (rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   574
  done
wenzelm@21539
   575
wenzelm@21539
   576
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   577
lemma ex_impE:
wenzelm@21539
   578
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   579
    and r: "P(x)-->S ==> R"
wenzelm@21539
   580
  shows R
wenzelm@21539
   581
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   582
  done
wenzelm@21539
   583
wenzelm@21539
   584
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   585
wenzelm@21539
   586
lemma disj_imp_disj:
wenzelm@23393
   587
  "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> R|S"
wenzelm@23393
   588
  apply (erule disjE)
wenzelm@21539
   589
  apply (rule disjI1) apply assumption
wenzelm@21539
   590
  apply (rule disjI2) apply assumption
wenzelm@21539
   591
  done
wenzelm@11734
   592
wenzelm@18481
   593
ML {*
wenzelm@18481
   594
structure ProjectRule = ProjectRuleFun
wenzelm@18481
   595
(struct
wenzelm@22139
   596
  val conjunct1 = @{thm conjunct1}
wenzelm@22139
   597
  val conjunct2 = @{thm conjunct2}
wenzelm@22139
   598
  val mp = @{thm mp}
wenzelm@18481
   599
end)
wenzelm@18481
   600
*}
wenzelm@18481
   601
wenzelm@7355
   602
use "fologic.ML"
wenzelm@21539
   603
wenzelm@21539
   604
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   605
wenzelm@9886
   606
use "hypsubstdata.ML"
wenzelm@9886
   607
setup hypsubst_setup
wenzelm@7355
   608
use "intprover.ML"
wenzelm@7355
   609
wenzelm@4092
   610
wenzelm@12875
   611
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   612
wenzelm@12349
   613
lemma impE':
wenzelm@12937
   614
  assumes 1: "P --> Q"
wenzelm@12937
   615
    and 2: "Q ==> R"
wenzelm@12937
   616
    and 3: "P --> Q ==> P"
wenzelm@12937
   617
  shows R
wenzelm@12349
   618
proof -
wenzelm@12349
   619
  from 3 and 1 have P .
wenzelm@12368
   620
  with 1 have Q by (rule impE)
wenzelm@12349
   621
  with 2 show R .
wenzelm@12349
   622
qed
wenzelm@12349
   623
wenzelm@12349
   624
lemma allE':
wenzelm@12937
   625
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   626
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   627
  shows Q
wenzelm@12349
   628
proof -
wenzelm@12349
   629
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   630
  from this and 1 show Q by (rule 2)
wenzelm@12349
   631
qed
wenzelm@12349
   632
wenzelm@12937
   633
lemma notE':
wenzelm@12937
   634
  assumes 1: "~ P"
wenzelm@12937
   635
    and 2: "~ P ==> P"
wenzelm@12937
   636
  shows R
wenzelm@12349
   637
proof -
wenzelm@12349
   638
  from 2 and 1 have P .
wenzelm@12349
   639
  with 1 show R by (rule notE)
wenzelm@12349
   640
qed
wenzelm@12349
   641
wenzelm@12349
   642
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   643
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   644
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   645
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   646
wenzelm@18708
   647
setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}
wenzelm@12349
   648
wenzelm@12349
   649
wenzelm@12368
   650
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   651
  by iprover
wenzelm@12368
   652
wenzelm@12368
   653
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   654
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   655
wenzelm@12368
   656
paulson@13435
   657
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   658
apply (rule iffI) 
paulson@13435
   659
apply (erule sym)+
paulson@13435
   660
done
paulson@13435
   661
paulson@13435
   662
wenzelm@11677
   663
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   664
wenzelm@11747
   665
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   666
proof
wenzelm@11677
   667
  assume "!!x. P(x)"
wenzelm@22931
   668
  then show "ALL x. P(x)" ..
wenzelm@11677
   669
next
wenzelm@11677
   670
  assume "ALL x. P(x)"
wenzelm@22931
   671
  then show "!!x. P(x)" ..
wenzelm@11677
   672
qed
wenzelm@11677
   673
wenzelm@11747
   674
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   675
proof
wenzelm@12368
   676
  assume "A ==> B"
wenzelm@22931
   677
  then show "A --> B" ..
wenzelm@11677
   678
next
wenzelm@11677
   679
  assume "A --> B" and A
wenzelm@22931
   680
  then show B by (rule mp)
wenzelm@11677
   681
qed
wenzelm@11677
   682
wenzelm@11747
   683
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   684
proof
wenzelm@11677
   685
  assume "x == y"
wenzelm@22931
   686
  show "x = y" unfolding `x == y` by (rule refl)
wenzelm@11677
   687
next
wenzelm@11677
   688
  assume "x = y"
wenzelm@22931
   689
  then show "x == y" by (rule eq_reflection)
wenzelm@11677
   690
qed
wenzelm@11677
   691
wenzelm@18813
   692
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   693
proof
wenzelm@18813
   694
  assume "A == B"
wenzelm@22931
   695
  show "A <-> B" unfolding `A == B` by (rule iff_refl)
wenzelm@18813
   696
next
wenzelm@18813
   697
  assume "A <-> B"
wenzelm@22931
   698
  then show "A == B" by (rule iff_reflection)
wenzelm@18813
   699
qed
wenzelm@18813
   700
wenzelm@12875
   701
lemma atomize_conj [atomize]:
wenzelm@19120
   702
  includes meta_conjunction_syntax
wenzelm@19120
   703
  shows "(A && B) == Trueprop (A & B)"
wenzelm@11976
   704
proof
wenzelm@19120
   705
  assume conj: "A && B"
wenzelm@19120
   706
  show "A & B"
wenzelm@19120
   707
  proof (rule conjI)
wenzelm@19120
   708
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   709
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   710
  qed
wenzelm@11953
   711
next
wenzelm@19120
   712
  assume conj: "A & B"
wenzelm@19120
   713
  show "A && B"
wenzelm@19120
   714
  proof -
wenzelm@19120
   715
    from conj show A ..
wenzelm@19120
   716
    from conj show B ..
wenzelm@11953
   717
  qed
wenzelm@11953
   718
qed
wenzelm@11953
   719
wenzelm@12368
   720
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   721
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   722
wenzelm@11848
   723
krauss@26580
   724
subsection {* Atomizing elimination rules *}
krauss@26580
   725
krauss@26580
   726
setup AtomizeElim.setup
krauss@26580
   727
krauss@26580
   728
lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX x. P(x)) ==> Q)"
krauss@26580
   729
by rule iprover+
krauss@26580
   730
krauss@26580
   731
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
krauss@26580
   732
by rule iprover+
krauss@26580
   733
krauss@26580
   734
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
krauss@26580
   735
by rule iprover+
krauss@26580
   736
krauss@26580
   737
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop(A)" ..
krauss@26580
   738
krauss@26580
   739
wenzelm@11848
   740
subsection {* Calculational rules *}
wenzelm@11848
   741
wenzelm@11848
   742
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   743
  by (rule ssubst)
wenzelm@11848
   744
wenzelm@11848
   745
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   746
  by (rule subst)
wenzelm@11848
   747
wenzelm@11848
   748
text {*
wenzelm@11848
   749
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   750
*}
wenzelm@11848
   751
wenzelm@12019
   752
lemmas basic_trans_rules [trans] =
wenzelm@11848
   753
  forw_subst
wenzelm@11848
   754
  back_subst
wenzelm@11848
   755
  rev_mp
wenzelm@11848
   756
  mp
wenzelm@11848
   757
  trans
wenzelm@11848
   758
paulson@13779
   759
subsection {* ``Let'' declarations *}
paulson@13779
   760
paulson@13779
   761
nonterminals letbinds letbind
paulson@13779
   762
paulson@13779
   763
constdefs
wenzelm@14854
   764
  Let :: "['a::{}, 'a => 'b] => ('b::{})"
paulson@13779
   765
    "Let(s, f) == f(s)"
paulson@13779
   766
paulson@13779
   767
syntax
paulson@13779
   768
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   769
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   770
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   771
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   772
paulson@13779
   773
translations
paulson@13779
   774
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
paulson@13779
   775
  "let x = a in e"          == "Let(a, %x. e)"
paulson@13779
   776
paulson@13779
   777
paulson@13779
   778
lemma LetI: 
wenzelm@21539
   779
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   780
  shows "P(let x=t in u(x))"
wenzelm@21539
   781
  apply (unfold Let_def)
wenzelm@21539
   782
  apply (rule refl [THEN assms])
wenzelm@21539
   783
  done
wenzelm@21539
   784
wenzelm@21539
   785
wenzelm@26286
   786
subsection {* Intuitionistic simplification rules *}
wenzelm@26286
   787
wenzelm@26286
   788
lemma conj_simps:
wenzelm@26286
   789
  "P & True <-> P"
wenzelm@26286
   790
  "True & P <-> P"
wenzelm@26286
   791
  "P & False <-> False"
wenzelm@26286
   792
  "False & P <-> False"
wenzelm@26286
   793
  "P & P <-> P"
wenzelm@26286
   794
  "P & P & Q <-> P & Q"
wenzelm@26286
   795
  "P & ~P <-> False"
wenzelm@26286
   796
  "~P & P <-> False"
wenzelm@26286
   797
  "(P & Q) & R <-> P & (Q & R)"
wenzelm@26286
   798
  by iprover+
wenzelm@26286
   799
wenzelm@26286
   800
lemma disj_simps:
wenzelm@26286
   801
  "P | True <-> True"
wenzelm@26286
   802
  "True | P <-> True"
wenzelm@26286
   803
  "P | False <-> P"
wenzelm@26286
   804
  "False | P <-> P"
wenzelm@26286
   805
  "P | P <-> P"
wenzelm@26286
   806
  "P | P | Q <-> P | Q"
wenzelm@26286
   807
  "(P | Q) | R <-> P | (Q | R)"
wenzelm@26286
   808
  by iprover+
wenzelm@26286
   809
wenzelm@26286
   810
lemma not_simps:
wenzelm@26286
   811
  "~(P|Q)  <-> ~P & ~Q"
wenzelm@26286
   812
  "~ False <-> True"
wenzelm@26286
   813
  "~ True <-> False"
wenzelm@26286
   814
  by iprover+
wenzelm@26286
   815
wenzelm@26286
   816
lemma imp_simps:
wenzelm@26286
   817
  "(P --> False) <-> ~P"
wenzelm@26286
   818
  "(P --> True) <-> True"
wenzelm@26286
   819
  "(False --> P) <-> True"
wenzelm@26286
   820
  "(True --> P) <-> P"
wenzelm@26286
   821
  "(P --> P) <-> True"
wenzelm@26286
   822
  "(P --> ~P) <-> ~P"
wenzelm@26286
   823
  by iprover+
wenzelm@26286
   824
wenzelm@26286
   825
lemma iff_simps:
wenzelm@26286
   826
  "(True <-> P) <-> P"
wenzelm@26286
   827
  "(P <-> True) <-> P"
wenzelm@26286
   828
  "(P <-> P) <-> True"
wenzelm@26286
   829
  "(False <-> P) <-> ~P"
wenzelm@26286
   830
  "(P <-> False) <-> ~P"
wenzelm@26286
   831
  by iprover+
wenzelm@26286
   832
wenzelm@26286
   833
(*The x=t versions are needed for the simplification procedures*)
wenzelm@26286
   834
lemma quant_simps:
wenzelm@26286
   835
  "!!P. (ALL x. P) <-> P"
wenzelm@26286
   836
  "(ALL x. x=t --> P(x)) <-> P(t)"
wenzelm@26286
   837
  "(ALL x. t=x --> P(x)) <-> P(t)"
wenzelm@26286
   838
  "!!P. (EX x. P) <-> P"
wenzelm@26286
   839
  "EX x. x=t"
wenzelm@26286
   840
  "EX x. t=x"
wenzelm@26286
   841
  "(EX x. x=t & P(x)) <-> P(t)"
wenzelm@26286
   842
  "(EX x. t=x & P(x)) <-> P(t)"
wenzelm@26286
   843
  by iprover+
wenzelm@26286
   844
wenzelm@26286
   845
(*These are NOT supplied by default!*)
wenzelm@26286
   846
lemma distrib_simps:
wenzelm@26286
   847
  "P & (Q | R) <-> P&Q | P&R"
wenzelm@26286
   848
  "(Q | R) & P <-> Q&P | R&P"
wenzelm@26286
   849
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"
wenzelm@26286
   850
  by iprover+
wenzelm@26286
   851
wenzelm@26286
   852
wenzelm@26286
   853
text {* Conversion into rewrite rules *}
wenzelm@26286
   854
wenzelm@26286
   855
lemma P_iff_F: "~P ==> (P <-> False)" by iprover
wenzelm@26286
   856
lemma iff_reflection_F: "~P ==> (P == False)" by (rule P_iff_F [THEN iff_reflection])
wenzelm@26286
   857
wenzelm@26286
   858
lemma P_iff_T: "P ==> (P <-> True)" by iprover
wenzelm@26286
   859
lemma iff_reflection_T: "P ==> (P == True)" by (rule P_iff_T [THEN iff_reflection])
wenzelm@26286
   860
wenzelm@26286
   861
wenzelm@26286
   862
text {* More rewrite rules *}
wenzelm@26286
   863
wenzelm@26286
   864
lemma conj_commute: "P&Q <-> Q&P" by iprover
wenzelm@26286
   865
lemma conj_left_commute: "P&(Q&R) <-> Q&(P&R)" by iprover
wenzelm@26286
   866
lemmas conj_comms = conj_commute conj_left_commute
wenzelm@26286
   867
wenzelm@26286
   868
lemma disj_commute: "P|Q <-> Q|P" by iprover
wenzelm@26286
   869
lemma disj_left_commute: "P|(Q|R) <-> Q|(P|R)" by iprover
wenzelm@26286
   870
lemmas disj_comms = disj_commute disj_left_commute
wenzelm@26286
   871
wenzelm@26286
   872
lemma conj_disj_distribL: "P&(Q|R) <-> (P&Q | P&R)" by iprover
wenzelm@26286
   873
lemma conj_disj_distribR: "(P|Q)&R <-> (P&R | Q&R)" by iprover
wenzelm@26286
   874
wenzelm@26286
   875
lemma disj_conj_distribL: "P|(Q&R) <-> (P|Q) & (P|R)" by iprover
wenzelm@26286
   876
lemma disj_conj_distribR: "(P&Q)|R <-> (P|R) & (Q|R)" by iprover
wenzelm@26286
   877
wenzelm@26286
   878
lemma imp_conj_distrib: "(P --> (Q&R)) <-> (P-->Q) & (P-->R)" by iprover
wenzelm@26286
   879
lemma imp_conj: "((P&Q)-->R)   <-> (P --> (Q --> R))" by iprover
wenzelm@26286
   880
lemma imp_disj: "(P|Q --> R)   <-> (P-->R) & (Q-->R)" by iprover
wenzelm@26286
   881
wenzelm@26286
   882
lemma de_Morgan_disj: "(~(P | Q)) <-> (~P & ~Q)" by iprover
wenzelm@26286
   883
wenzelm@26286
   884
lemma not_ex: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover
wenzelm@26286
   885
lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover
wenzelm@26286
   886
wenzelm@26286
   887
lemma ex_disj_distrib:
wenzelm@26286
   888
  "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover
wenzelm@26286
   889
wenzelm@26286
   890
lemma all_conj_distrib:
wenzelm@26286
   891
  "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover
wenzelm@26286
   892
wenzelm@26286
   893
wenzelm@26286
   894
subsection {* Legacy ML bindings *}
paulson@13779
   895
wenzelm@21539
   896
ML {*
wenzelm@22139
   897
val refl = @{thm refl}
wenzelm@22139
   898
val trans = @{thm trans}
wenzelm@22139
   899
val sym = @{thm sym}
wenzelm@22139
   900
val subst = @{thm subst}
wenzelm@22139
   901
val ssubst = @{thm ssubst}
wenzelm@22139
   902
val conjI = @{thm conjI}
wenzelm@22139
   903
val conjE = @{thm conjE}
wenzelm@22139
   904
val conjunct1 = @{thm conjunct1}
wenzelm@22139
   905
val conjunct2 = @{thm conjunct2}
wenzelm@22139
   906
val disjI1 = @{thm disjI1}
wenzelm@22139
   907
val disjI2 = @{thm disjI2}
wenzelm@22139
   908
val disjE = @{thm disjE}
wenzelm@22139
   909
val impI = @{thm impI}
wenzelm@22139
   910
val impE = @{thm impE}
wenzelm@22139
   911
val mp = @{thm mp}
wenzelm@22139
   912
val rev_mp = @{thm rev_mp}
wenzelm@22139
   913
val TrueI = @{thm TrueI}
wenzelm@22139
   914
val FalseE = @{thm FalseE}
wenzelm@22139
   915
val iff_refl = @{thm iff_refl}
wenzelm@22139
   916
val iff_trans = @{thm iff_trans}
wenzelm@22139
   917
val iffI = @{thm iffI}
wenzelm@22139
   918
val iffE = @{thm iffE}
wenzelm@22139
   919
val iffD1 = @{thm iffD1}
wenzelm@22139
   920
val iffD2 = @{thm iffD2}
wenzelm@22139
   921
val notI = @{thm notI}
wenzelm@22139
   922
val notE = @{thm notE}
wenzelm@22139
   923
val allI = @{thm allI}
wenzelm@22139
   924
val allE = @{thm allE}
wenzelm@22139
   925
val spec = @{thm spec}
wenzelm@22139
   926
val exI = @{thm exI}
wenzelm@22139
   927
val exE = @{thm exE}
wenzelm@22139
   928
val eq_reflection = @{thm eq_reflection}
wenzelm@22139
   929
val iff_reflection = @{thm iff_reflection}
wenzelm@22139
   930
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22139
   931
val meta_eq_to_iff = @{thm meta_eq_to_iff}
paulson@13779
   932
*}
paulson@13779
   933
wenzelm@4854
   934
end