src/FOL/IFOL.thy
 author paulson Wed, 31 Jul 2002 14:34:08 +0200 changeset 13435 05631e8f0258 parent 12937 0c4fd7529467 child 13779 2a34dc5cf79e permissions -rw-r--r--
new theorem eq_commute
```
(*  Title:      FOL/IFOL.thy
ID:         \$Id\$
Author:     Lawrence C Paulson and Markus Wenzel
*)

header {* Intuitionistic first-order logic *}

theory IFOL = Pure
files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML"):

subsection {* Syntax and axiomatic basis *}

global

classes "term" < logic
defaultsort "term"

typedecl o

judgment
Trueprop      :: "o => prop"                  ("(_)" 5)

consts
True          :: o
False         :: o

(* Connectives *)

"="           :: "['a, 'a] => o"              (infixl 50)

Not           :: "o => o"                     ("~ _" [40] 40)
&             :: "[o, o] => o"                (infixr 35)
"|"           :: "[o, o] => o"                (infixr 30)
-->           :: "[o, o] => o"                (infixr 25)
<->           :: "[o, o] => o"                (infixr 25)

(* Quantifiers *)

All           :: "('a => o) => o"             (binder "ALL " 10)
Ex            :: "('a => o) => o"             (binder "EX " 10)
Ex1           :: "('a => o) => o"             (binder "EX! " 10)

syntax
"_not_equal"  :: "['a, 'a] => o"              (infixl "~=" 50)
translations
"x ~= y"      == "~ (x = y)"

syntax (xsymbols)
Not           :: "o => o"                     ("\<not> _" [40] 40)
"op &"        :: "[o, o] => o"                (infixr "\<and>" 35)
"op |"        :: "[o, o] => o"                (infixr "\<or>" 30)
"ALL "        :: "[idts, o] => o"             ("(3\<forall>_./ _)" [0, 10] 10)
"EX "         :: "[idts, o] => o"             ("(3\<exists>_./ _)" [0, 10] 10)
"EX! "        :: "[idts, o] => o"             ("(3\<exists>!_./ _)" [0, 10] 10)
"_not_equal"  :: "['a, 'a] => o"              (infixl "\<noteq>" 50)
"op -->"      :: "[o, o] => o"                (infixr "\<longrightarrow>" 25)
"op <->"      :: "[o, o] => o"                (infixr "\<longleftrightarrow>" 25)

syntax (HTML output)
Not           :: "o => o"                     ("\<not> _" [40] 40)

local

axioms

(* Equality *)

refl:         "a=a"
subst:        "[| a=b;  P(a) |] ==> P(b)"

(* Propositional logic *)

conjI:        "[| P;  Q |] ==> P&Q"
conjunct1:    "P&Q ==> P"
conjunct2:    "P&Q ==> Q"

disjI1:       "P ==> P|Q"
disjI2:       "Q ==> P|Q"
disjE:        "[| P|Q;  P ==> R;  Q ==> R |] ==> R"

impI:         "(P ==> Q) ==> P-->Q"
mp:           "[| P-->Q;  P |] ==> Q"

FalseE:       "False ==> P"

(* Definitions *)

True_def:     "True  == False-->False"
not_def:      "~P    == P-->False"
iff_def:      "P<->Q == (P-->Q) & (Q-->P)"

(* Unique existence *)

ex1_def:      "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"

(* Quantifiers *)

allI:         "(!!x. P(x)) ==> (ALL x. P(x))"
spec:         "(ALL x. P(x)) ==> P(x)"

exI:          "P(x) ==> (EX x. P(x))"
exE:          "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"

(* Reflection *)

eq_reflection:  "(x=y)   ==> (x==y)"
iff_reflection: "(P<->Q) ==> (P==Q)"

subsection {* Lemmas and proof tools *}

setup Simplifier.setup
use "IFOL_lemmas.ML"

use "fologic.ML"
use "hypsubstdata.ML"
setup hypsubst_setup
use "intprover.ML"

subsection {* Intuitionistic Reasoning *}

lemma impE':
assumes 1: "P --> Q"
and 2: "Q ==> R"
and 3: "P --> Q ==> P"
shows R
proof -
from 3 and 1 have P .
with 1 have Q by (rule impE)
with 2 show R .
qed

lemma allE':
assumes 1: "ALL x. P(x)"
and 2: "P(x) ==> ALL x. P(x) ==> Q"
shows Q
proof -
from 1 have "P(x)" by (rule spec)
from this and 1 show Q by (rule 2)
qed

lemma notE':
assumes 1: "~ P"
and 2: "~ P ==> P"
shows R
proof -
from 2 and 1 have P .
with 1 show R by (rule notE)
qed

lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
and [Pure.elim 2] = allE notE' impE'
and [Pure.intro] = exI disjI2 disjI1

ML_setup {*
Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac));
*}

lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
by rules

lemmas [sym] = sym iff_sym not_sym iff_not_sym
and [Pure.elim?] = iffD1 iffD2 impE

lemma eq_commute: "a=b <-> b=a"
apply (rule iffI)
apply (erule sym)+
done

subsection {* Atomizing meta-level rules *}

lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
proof
assume "!!x. P(x)"
show "ALL x. P(x)" ..
next
assume "ALL x. P(x)"
thus "!!x. P(x)" ..
qed

lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
proof
assume "A ==> B"
thus "A --> B" ..
next
assume "A --> B" and A
thus B by (rule mp)
qed

lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
proof
assume "x == y"
show "x = y" by (unfold prems) (rule refl)
next
assume "x = y"
thus "x == y" by (rule eq_reflection)
qed

lemma atomize_conj [atomize]:
"(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
proof
assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
show "A & B" by (rule conjI)
next
fix C
assume "A & B"
assume "A ==> B ==> PROP C"
thus "PROP C"
proof this
show A by (rule conjunct1)
show B by (rule conjunct2)
qed
qed

lemmas [symmetric, rulify] = atomize_all atomize_imp

subsection {* Calculational rules *}

lemma forw_subst: "a = b ==> P(b) ==> P(a)"
by (rule ssubst)

lemma back_subst: "P(a) ==> a = b ==> P(b)"
by (rule subst)

text {*
Note that this list of rules is in reverse order of priorities.
*}

lemmas basic_trans_rules [trans] =
forw_subst
back_subst
rev_mp
mp
trans

end
```