src/FOL/IFOL.thy
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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 = Pure
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files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML"):
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subsection {* Syntax and axiomatic basis *}
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global
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classes "term" < logic
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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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  "="           :: "['a, 'a] => o"              (infixl 50)
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  Not           :: "o => o"                     ("~ _" [40] 40)
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  &             :: "[o, o] => o"                (infixr 35)
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  "|"           :: "[o, o] => o"                (infixr 30)
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  -->           :: "[o, o] => o"                (infixr 25)
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  <->           :: "[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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syntax
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  "~="          :: "['a, 'a] => o"              (infixl 50)
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translations
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  "x ~= y"      == "~ (x = y)"
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syntax (xsymbols)
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  Not           :: "o => o"                     ("\<not> _" [40] 40)
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  "op &"        :: "[o, o] => o"                (infixr "\<and>" 35)
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  "op |"        :: "[o, o] => o"                (infixr "\<or>" 30)
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  "ALL "        :: "[idts, o] => o"             ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, o] => o"             ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, o] => o"             ("(3\<exists>!_./ _)" [0, 10] 10)
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  "op ~="       :: "['a, 'a] => o"              (infixl "\<noteq>" 50)
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  "op -->"      :: "[o, o] => o"                (infixr "\<longrightarrow>" 25)
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  "op <->"      :: "[o, o] => o"                (infixr "\<longleftrightarrow>" 25)
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syntax (HTML output)
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  Not           :: "o => o"                     ("\<not> _" [40] 40)
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local
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axioms
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  (* Equality *)
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  refl:         "a=a"
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  subst:        "[| a=b;  P(a) |] ==> 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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  (* 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:      "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
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  (* Quantifiers *)
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  allI:         "(!!x. P(x)) ==> (ALL x. P(x))"
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  spec:         "(ALL x. P(x)) ==> P(x)"
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  exI:          "P(x) ==> (EX x. P(x))"
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  exE:          "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"
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  (* Reflection *)
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  eq_reflection:  "(x=y)   ==> (x==y)"
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  iff_reflection: "(P<->Q) ==> (P==Q)"
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subsection {* Lemmas and proof tools *}
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setup Simplifier.setup
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use "IFOL_lemmas.ML"
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declare impE [Pure.elim?]  iffD1 [Pure.elim?]  iffD2 [Pure.elim?]
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use "fologic.ML"
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use "hypsubstdata.ML"
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setup hypsubst_setup
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use "intprover.ML"
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lemma impE':
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  (assumes 1: "P --> Q" and 2: "Q ==> R" and 3: "P --> Q ==> P") R
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proof -
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  from 3 and 1 have P .
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  with 1 have Q ..
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  with 2 show R .
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qed
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lemma allE':
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  (assumes 1: "ALL x. P(x)" and 2: "P(x) ==> ALL x. P(x) ==> Q") Q
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proof -
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  from 1 have "P(x)" by (rule spec)
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  from this and 1 show Q by (rule 2)
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qed
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lemma notE': (assumes 1: "~ P" and 2: "~ P ==> P") R
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proof -
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  from 2 and 1 have P .
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  with 1 show R by (rule notE)
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qed
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lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
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  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
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  and [Pure.elim 2] = allE notE' impE'
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  and [Pure.intro] = exI disjI2 disjI1
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ML_setup {*
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  Context.>> (RuleContext.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
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*}
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subsection {* Atomizing meta-level rules *}
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lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
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proof
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  assume "!!x. P(x)"
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  show "ALL x. P(x)" by (rule allI)
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next
ee12f18599e5 atomize stuff from theory FOL;
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   166
  assume "ALL x. P(x)"
ee12f18599e5 atomize stuff from theory FOL;
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  thus "!!x. P(x)" by (rule allE)
ee12f18599e5 atomize stuff from theory FOL;
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qed
ee12f18599e5 atomize stuff from theory FOL;
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   169
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17a6dcd6f3cf judgment Trueprop;
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lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
11976
075df6e46cef equal_intr_rule already declared in Pure;
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proof
11677
ee12f18599e5 atomize stuff from theory FOL;
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   172
  assume r: "A ==> B"
ee12f18599e5 atomize stuff from theory FOL;
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   173
  show "A --> B" by (rule impI) (rule r)
ee12f18599e5 atomize stuff from theory FOL;
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   174
next
ee12f18599e5 atomize stuff from theory FOL;
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   175
  assume "A --> B" and A
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  thus B by (rule mp)
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qed
ee12f18599e5 atomize stuff from theory FOL;
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   178
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17a6dcd6f3cf judgment Trueprop;
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lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
11976
075df6e46cef equal_intr_rule already declared in Pure;
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   180
proof
11677
ee12f18599e5 atomize stuff from theory FOL;
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   181
  assume "x == y"
ee12f18599e5 atomize stuff from theory FOL;
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   182
  show "x = y" by (unfold prems) (rule refl)
ee12f18599e5 atomize stuff from theory FOL;
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   183
next
ee12f18599e5 atomize stuff from theory FOL;
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   184
  assume "x = y"
ee12f18599e5 atomize stuff from theory FOL;
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   185
  thus "x == y" by (rule eq_reflection)
ee12f18599e5 atomize stuff from theory FOL;
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qed
ee12f18599e5 atomize stuff from theory FOL;
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11953
f98623fdf6ef atomize_conj;
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lemma atomize_conj [atomize]: "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
11976
075df6e46cef equal_intr_rule already declared in Pure;
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proof
11953
f98623fdf6ef atomize_conj;
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   190
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
f98623fdf6ef atomize_conj;
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   191
  show "A & B" by (rule conjI)
f98623fdf6ef atomize_conj;
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   192
next
f98623fdf6ef atomize_conj;
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   193
  fix C
f98623fdf6ef atomize_conj;
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   194
  assume "A & B"
f98623fdf6ef atomize_conj;
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   195
  assume "A ==> B ==> PROP C"
f98623fdf6ef atomize_conj;
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   196
  thus "PROP C"
f98623fdf6ef atomize_conj;
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   197
  proof this
f98623fdf6ef atomize_conj;
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   198
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
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   199
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
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   200
  qed
f98623fdf6ef atomize_conj;
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   201
qed
f98623fdf6ef atomize_conj;
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   202
11771
b7b100a2de1d moved rulify to ObjectLogic;
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   203
declare atomize_all [symmetric, rulify]  atomize_imp [symmetric, rulify]
b7b100a2de1d moved rulify to ObjectLogic;
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   204
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6e3017adb8c0 calculational rules moved from FOL to IFOL;
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6e3017adb8c0 calculational rules moved from FOL to IFOL;
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subsection {* Calculational rules *}
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6e3017adb8c0 calculational rules moved from FOL to IFOL;
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lemma forw_subst: "a = b ==> P(b) ==> P(a)"
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  by (rule ssubst)
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6e3017adb8c0 calculational rules moved from FOL to IFOL;
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lemma back_subst: "P(a) ==> a = b ==> P(b)"
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   212
  by (rule subst)
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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6e3017adb8c0 calculational rules moved from FOL to IFOL;
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text {*
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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  Note that this list of rules is in reverse order of priorities.
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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*}
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   217
12019
abe9b7c6016e transitive declared in Pure;
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   218
lemmas basic_trans_rules [trans] =
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   219
  forw_subst
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   220
  back_subst
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   221
  rev_mp
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   222
  mp
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   223
  trans
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   224
12349
94e812f9683e setup "rules" method;
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   225
lemmas [Pure.elim?] = sym
11848
6e3017adb8c0 calculational rules moved from FOL to IFOL;
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   226
4854
d1850e0964f2 tuned setup;
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   227
end