(* Title: FOL/IFOL.thy
ID: $Id$
Author: Lawrence C Paulson and Markus Wenzel
*)
header {* Intuitionistic first-order logic *}
theory IFOL
imports Pure
files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML")
begin
subsection {* Syntax and axiomatic basis *}
global
classes "term"
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)
"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)
local
finalconsts
False All Ex
"op ="
"op &"
"op |"
"op -->"
axioms
(* Equality *)
refl: "a=a"
(* 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"
(* 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)"
text{*Thanks to Stephan Merz*}
theorem subst:
assumes eq: "a = b" and p: "P(a)"
shows "P(b)"
proof -
from eq have meta: "a \<equiv> b"
by (rule eq_reflection)
from p show ?thesis
by (unfold meta)
qed
defs
(* Definitions *)
True_def: "True == False-->False"
not_def: "~P == P-->False"
iff_def: "P<->Q == (P-->Q) & (Q-->P)"
(* Unique existence *)
ex1_def: "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)"
subsection {* Lemmas and proof tools *}
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
setup {*
[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
subsection {* ``Let'' declarations *}
nonterminals letbinds letbind
constdefs
Let :: "['a::{}, 'a => 'b] => ('b::{})"
"Let(s, f) == f(s)"
syntax
"_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
"" :: "letbind => letbinds" ("_")
"_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
translations
"_Let(_binds(b, bs), e)" == "_Let(b, _Let(bs, e))"
"let x = a in e" == "Let(a, %x. e)"
lemma LetI:
assumes prem: "(!!x. x=t ==> P(u(x)))"
shows "P(let x=t in u(x))"
apply (unfold Let_def)
apply (rule refl [THEN prem])
done
ML
{*
val Let_def = thm "Let_def";
val LetI = thm "LetI";
*}
end