(* Title: FOL/IFOL.thy
ID: $Id$
Author: Lawrence C Paulson and Markus Wenzel
*)
header {* Intuitionistic first-order logic *}
theory IFOL
imports Pure
uses ("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 *)
"op =" :: "['a, 'a] => o" (infixl "=" 50)
Not :: "o => o" ("~ _" [40] 40)
"op &" :: "[o, o] => o" (infixr "&" 35)
"op |" :: "[o, o] => o" (infixr "|" 30)
"op -->" :: "[o, o] => o" (infixr "-->" 25)
"op <->" :: "[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)
abbreviation
not_equal :: "['a, 'a] => o" (infixl "~=" 50) where
"x ~= y == ~ (x = y)"
notation (xsymbols)
not_equal (infixl "\<noteq>" 50)
notation (HTML output)
not_equal (infixl "\<noteq>" 50)
notation (xsymbols)
Not ("\<not> _" [40] 40) and
"op &" (infixr "\<and>" 35) and
"op |" (infixr "\<or>" 30) and
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
Ex1 (binder "\<exists>!" 10) and
"op -->" (infixr "\<longrightarrow>" 25) and
"op <->" (infixr "\<longleftrightarrow>" 25)
notation (HTML output)
Not ("\<not> _" [40] 40) and
"op &" (infixr "\<and>" 35) and
"op |" (infixr "\<or>" 30) and
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
Ex1 (binder "\<exists>!" 10)
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)"
lemmas strip = impI allI
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 *}
lemma TrueI: True
unfolding True_def by (rule impI)
(*** Sequent-style elimination rules for & --> and ALL ***)
lemma conjE:
assumes major: "P & Q"
and r: "[| P; Q |] ==> R"
shows R
apply (rule r)
apply (rule major [THEN conjunct1])
apply (rule major [THEN conjunct2])
done
lemma impE:
assumes major: "P --> Q"
and P
and r: "Q ==> R"
shows R
apply (rule r)
apply (rule major [THEN mp])
apply (rule `P`)
done
lemma allE:
assumes major: "ALL x. P(x)"
and r: "P(x) ==> R"
shows R
apply (rule r)
apply (rule major [THEN spec])
done
(*Duplicates the quantifier; for use with eresolve_tac*)
lemma all_dupE:
assumes major: "ALL x. P(x)"
and r: "[| P(x); ALL x. P(x) |] ==> R"
shows R
apply (rule r)
apply (rule major [THEN spec])
apply (rule major)
done
(*** Negation rules, which translate between ~P and P-->False ***)
lemma notI: "(P ==> False) ==> ~P"
unfolding not_def by (erule impI)
lemma notE: "[| ~P; P |] ==> R"
unfolding not_def by (erule mp [THEN FalseE])
lemma rev_notE: "[| P; ~P |] ==> R"
by (erule notE)
(*This is useful with the special implication rules for each kind of P. *)
lemma not_to_imp:
assumes "~P"
and r: "P --> False ==> Q"
shows Q
apply (rule r)
apply (rule impI)
apply (erule notE [OF `~P`])
done
(* For substitution into an assumption P, reduce Q to P-->Q, substitute into
this implication, then apply impI to move P back into the assumptions.
To specify P use something like
eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *)
lemma rev_mp: "[| P; P --> Q |] ==> Q"
by (erule mp)
(*Contrapositive of an inference rule*)
lemma contrapos:
assumes major: "~Q"
and minor: "P ==> Q"
shows "~P"
apply (rule major [THEN notE, THEN notI])
apply (erule minor)
done
(*** Modus Ponens Tactics ***)
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
ML {*
fun mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i THEN assume_tac i
fun eq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i THEN eq_assume_tac i
*}
(*** If-and-only-if ***)
lemma iffI: "[| P ==> Q; Q ==> P |] ==> P<->Q"
apply (unfold iff_def)
apply (rule conjI)
apply (erule impI)
apply (erule impI)
done
(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
lemma iffE:
assumes major: "P <-> Q"
and r: "P-->Q ==> Q-->P ==> R"
shows R
apply (insert major, unfold iff_def)
apply (erule conjE)
apply (erule r)
apply assumption
done
(* Destruct rules for <-> similar to Modus Ponens *)
lemma iffD1: "[| P <-> Q; P |] ==> Q"
apply (unfold iff_def)
apply (erule conjunct1 [THEN mp])
apply assumption
done
lemma iffD2: "[| P <-> Q; Q |] ==> P"
apply (unfold iff_def)
apply (erule conjunct2 [THEN mp])
apply assumption
done
lemma rev_iffD1: "[| P; P <-> Q |] ==> Q"
apply (erule iffD1)
apply assumption
done
lemma rev_iffD2: "[| Q; P <-> Q |] ==> P"
apply (erule iffD2)
apply assumption
done
lemma iff_refl: "P <-> P"
by (rule iffI)
lemma iff_sym: "Q <-> P ==> P <-> Q"
apply (erule iffE)
apply (rule iffI)
apply (assumption | erule mp)+
done
lemma iff_trans: "[| P <-> Q; Q<-> R |] ==> P <-> R"
apply (rule iffI)
apply (assumption | erule iffE | erule (1) notE impE)+
done
(*** Unique existence. NOTE THAT the following 2 quantifications
EX!x such that [EX!y such that P(x,y)] (sequential)
EX!x,y such that P(x,y) (simultaneous)
do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential.
***)
lemma ex1I:
assumes "P(a)"
and "!!x. P(x) ==> x=a"
shows "EX! x. P(x)"
apply (unfold ex1_def)
apply (assumption | rule assms exI conjI allI impI)+
done
(*Sometimes easier to use: the premises have no shared variables. Safe!*)
lemma ex_ex1I:
assumes ex: "EX x. P(x)"
and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
shows "EX! x. P(x)"
apply (rule ex [THEN exE])
apply (assumption | rule ex1I eq)+
done
lemma ex1E:
assumes ex1: "EX! x. P(x)"
and r: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
shows R
apply (insert ex1, unfold ex1_def)
apply (assumption | erule exE conjE)+
done
(*** <-> congruence rules for simplification ***)
(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*)
ML {*
fun iff_tac prems i =
resolve_tac (prems RL @{thms iffE}) i THEN
REPEAT1 (eresolve_tac [@{thm asm_rl}, @{thm mp}] i)
*}
lemma conj_cong:
assumes "P <-> P'"
and "P' ==> Q <-> Q'"
shows "(P&Q) <-> (P'&Q')"
apply (insert assms)
apply (assumption | rule iffI conjI | erule iffE conjE mp |
tactic {* iff_tac (thms "assms") 1 *})+
done
(*Reversed congruence rule! Used in ZF/Order*)
lemma conj_cong2:
assumes "P <-> P'"
and "P' ==> Q <-> Q'"
shows "(Q&P) <-> (Q'&P')"
apply (insert assms)
apply (assumption | rule iffI conjI | erule iffE conjE mp |
tactic {* iff_tac (thms "assms") 1 *})+
done
lemma disj_cong:
assumes "P <-> P'" and "Q <-> Q'"
shows "(P|Q) <-> (P'|Q')"
apply (insert assms)
apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
done
lemma imp_cong:
assumes "P <-> P'"
and "P' ==> Q <-> Q'"
shows "(P-->Q) <-> (P'-->Q')"
apply (insert assms)
apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
tactic {* iff_tac (thms "assms") 1 *})+
done
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
done
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
done
lemma all_cong:
assumes "!!x. P(x) <-> Q(x)"
shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
tactic {* iff_tac (thms "assms") 1 *})+
done
lemma ex_cong:
assumes "!!x. P(x) <-> Q(x)"
shows "(EX x. P(x)) <-> (EX x. Q(x))"
apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
tactic {* iff_tac (thms "assms") 1 *})+
done
lemma ex1_cong:
assumes "!!x. P(x) <-> Q(x)"
shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
tactic {* iff_tac (thms "assms") 1 *})+
done
(*** Equality rules ***)
lemma sym: "a=b ==> b=a"
apply (erule subst)
apply (rule refl)
done
lemma trans: "[| a=b; b=c |] ==> a=c"
apply (erule subst, assumption)
done
(** **)
lemma not_sym: "b ~= a ==> a ~= b"
apply (erule contrapos)
apply (erule sym)
done
(* Two theorms for rewriting only one instance of a definition:
the first for definitions of formulae and the second for terms *)
lemma def_imp_iff: "(A == B) ==> A <-> B"
apply unfold
apply (rule iff_refl)
done
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
apply unfold
apply (rule refl)
done
lemma meta_eq_to_iff: "x==y ==> x<->y"
by unfold (rule iff_refl)
(*substitution*)
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
apply (drule sym)
apply (erule (1) subst)
done
(*A special case of ex1E that would otherwise need quantifier expansion*)
lemma ex1_equalsE:
"[| EX! x. P(x); P(a); P(b) |] ==> a=b"
apply (erule ex1E)
apply (rule trans)
apply (rule_tac [2] sym)
apply (assumption | erule spec [THEN mp])+
done
(** Polymorphic congruence rules **)
lemma subst_context: "[| a=b |] ==> t(a)=t(b)"
apply (erule ssubst)
apply (rule refl)
done
lemma subst_context2: "[| a=b; c=d |] ==> t(a,c)=t(b,d)"
apply (erule ssubst)+
apply (rule refl)
done
lemma subst_context3: "[| a=b; c=d; e=f |] ==> t(a,c,e)=t(b,d,f)"
apply (erule ssubst)+
apply (rule refl)
done
(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
| |
c = d *)
lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
apply (rule trans)
apply (rule trans)
apply (rule sym)
apply assumption+
done
(*Dual of box_equals: for proving equalities backwards*)
lemma simp_equals: "[| a=c; b=d; c=d |] ==> a=b"
apply (rule trans)
apply (rule trans)
apply assumption+
apply (erule sym)
done
(** Congruence rules for predicate letters **)
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
apply (rule iffI)
apply (erule (1) subst)
apply (erule (1) ssubst)
done
lemma pred2_cong: "[| a=a'; b=b' |] ==> P(a,b) <-> P(a',b')"
apply (rule iffI)
apply (erule subst)+
apply assumption
apply (erule ssubst)+
apply assumption
done
lemma pred3_cong: "[| a=a'; b=b'; c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
apply (rule iffI)
apply (erule subst)+
apply assumption
apply (erule ssubst)+
apply assumption
done
(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
ML {*
bind_thms ("pred_congs",
List.concat (map (fn c =>
map (fn th => read_instantiate [("P",c)] th)
[@{thm pred1_cong}, @{thm pred2_cong}, @{thm pred3_cong}])
(explode"PQRS")))
*}
(*special case for the equality predicate!*)
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
apply (erule (1) pred2_cong)
done
(*** Simplifications of assumed implications.
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
used with mp_tac (restricted to atomic formulae) is COMPLETE for
intuitionistic propositional logic. See
R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
(preprint, University of St Andrews, 1991) ***)
lemma conj_impE:
assumes major: "(P&Q)-->S"
and r: "P-->(Q-->S) ==> R"
shows R
by (assumption | rule conjI impI major [THEN mp] r)+
lemma disj_impE:
assumes major: "(P|Q)-->S"
and r: "[| P-->S; Q-->S |] ==> R"
shows R
by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed. *)
lemma imp_impE:
assumes major: "(P-->Q)-->S"
and r1: "[| P; Q-->S |] ==> Q"
and r2: "S ==> R"
shows R
by (assumption | rule impI major [THEN mp] r1 r2)+
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed. *)
lemma not_impE:
assumes major: "~P --> S"
and r1: "P ==> False"
and r2: "S ==> R"
shows R
apply (assumption | rule notI impI major [THEN mp] r1 r2)+
done
(*Simplifies the implication. UNSAFE. *)
lemma iff_impE:
assumes major: "(P<->Q)-->S"
and r1: "[| P; Q-->S |] ==> Q"
and r2: "[| Q; P-->S |] ==> P"
and r3: "S ==> R"
shows R
apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
done
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
lemma all_impE:
assumes major: "(ALL x. P(x))-->S"
and r1: "!!x. P(x)"
and r2: "S ==> R"
shows R
apply (assumption | rule allI impI major [THEN mp] r1 r2)+
done
(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *)
lemma ex_impE:
assumes major: "(EX x. P(x))-->S"
and r: "P(x)-->S ==> R"
shows R
apply (assumption | rule exI impI major [THEN mp] r)+
done
(*** Courtesy of Krzysztof Grabczewski ***)
lemma disj_imp_disj:
assumes major: "P|Q"
and "P==>R" and "Q==>S"
shows "R|S"
apply (rule disjE [OF major])
apply (rule disjI1) apply assumption
apply (rule disjI2) apply assumption
done
ML {*
structure ProjectRule = ProjectRuleFun
(struct
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val mp = @{thm mp}
end)
*}
use "fologic.ML"
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
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 iprover
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_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
proof
assume "A == B"
show "A <-> B" by (unfold prems) (rule iff_refl)
next
assume "A <-> B"
thus "A == B" by (rule iff_reflection)
qed
lemma atomize_conj [atomize]:
includes meta_conjunction_syntax
shows "(A && B) == Trueprop (A & B)"
proof
assume conj: "A && B"
show "A & B"
proof (rule conjI)
from conj show A by (rule conjunctionD1)
from conj show B by (rule conjunctionD2)
qed
next
assume conj: "A & B"
show "A && B"
proof -
from conj show A ..
from conj show B ..
qed
qed
lemmas [symmetric, rulify] = atomize_all atomize_imp
and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
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 "!!x. x=t ==> P(u(x))"
shows "P(let x=t in u(x))"
apply (unfold Let_def)
apply (rule refl [THEN assms])
done
subsection {* ML bindings *}
ML {*
val refl = @{thm refl}
val trans = @{thm trans}
val sym = @{thm sym}
val subst = @{thm subst}
val ssubst = @{thm ssubst}
val conjI = @{thm conjI}
val conjE = @{thm conjE}
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val disjI1 = @{thm disjI1}
val disjI2 = @{thm disjI2}
val disjE = @{thm disjE}
val impI = @{thm impI}
val impE = @{thm impE}
val mp = @{thm mp}
val rev_mp = @{thm rev_mp}
val TrueI = @{thm TrueI}
val FalseE = @{thm FalseE}
val iff_refl = @{thm iff_refl}
val iff_trans = @{thm iff_trans}
val iffI = @{thm iffI}
val iffE = @{thm iffE}
val iffD1 = @{thm iffD1}
val iffD2 = @{thm iffD2}
val notI = @{thm notI}
val notE = @{thm notE}
val allI = @{thm allI}
val allE = @{thm allE}
val spec = @{thm spec}
val exI = @{thm exI}
val exE = @{thm exE}
val eq_reflection = @{thm eq_reflection}
val iff_reflection = @{thm iff_reflection}
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
val meta_eq_to_iff = @{thm meta_eq_to_iff}
*}
end