(* Title: HOL/HOL.thy
ID: $Id$
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
header {* The basis of Higher-Order Logic *}
theory HOL
imports CPure
uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML") ("eqrule_HOL_data.ML")
("~~/src/Provers/eqsubst.ML")
begin
subsection {* Primitive logic *}
subsubsection {* Core syntax *}
classes type
defaultsort type
global
typedecl bool
arities
bool :: type
fun :: (type, type) type
judgment
Trueprop :: "bool => prop" ("(_)" 5)
consts
Not :: "bool => bool" ("~ _" [40] 40)
True :: bool
False :: bool
arbitrary :: 'a
The :: "('a => bool) => 'a"
All :: "('a => bool) => bool" (binder "ALL " 10)
Ex :: "('a => bool) => bool" (binder "EX " 10)
Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
Let :: "['a, 'a => 'b] => 'b"
"=" :: "['a, 'a] => bool" (infixl 50)
& :: "[bool, bool] => bool" (infixr 35)
"|" :: "[bool, bool] => bool" (infixr 30)
--> :: "[bool, bool] => bool" (infixr 25)
local
consts
If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
subsubsection {* Additional concrete syntax *}
nonterminals
letbinds letbind
case_syn cases_syn
syntax
"_not_equal" :: "['a, 'a] => bool" (infixl "~=" 50)
"_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
"_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
"" :: "letbind => letbinds" ("_")
"_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
"_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
"_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
"" :: "case_syn => cases_syn" ("_")
"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
translations
"x ~= y" == "~ (x = y)"
"THE x. P" == "The (%x. P)"
"_Let (_binds b bs) e" == "_Let b (_Let bs e)"
"let x = a in e" == "Let a (%x. e)"
print_translation {*
(* To avoid eta-contraction of body: *)
[("The", fn [Abs abs] =>
let val (x,t) = atomic_abs_tr' abs
in Syntax.const "_The" $ x $ t end)]
*}
syntax (output)
"=" :: "['a, 'a] => bool" (infix 50)
"_not_equal" :: "['a, 'a] => bool" (infix "~=" 50)
syntax (xsymbols)
Not :: "bool => bool" ("\<not> _" [40] 40)
"op &" :: "[bool, bool] => bool" (infixr "\<and>" 35)
"op |" :: "[bool, bool] => bool" (infixr "\<or>" 30)
"op -->" :: "[bool, bool] => bool" (infixr "\<longrightarrow>" 25)
"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50)
"ALL " :: "[idts, bool] => bool" ("(3\<forall>_./ _)" [0, 10] 10)
"EX " :: "[idts, bool] => bool" ("(3\<exists>_./ _)" [0, 10] 10)
"EX! " :: "[idts, bool] => bool" ("(3\<exists>!_./ _)" [0, 10] 10)
"_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
(*"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ \<orelse> _")*)
syntax (xsymbols output)
"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50)
syntax (HTML output)
"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50)
Not :: "bool => bool" ("\<not> _" [40] 40)
"op &" :: "[bool, bool] => bool" (infixr "\<and>" 35)
"op |" :: "[bool, bool] => bool" (infixr "\<or>" 30)
"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50)
"ALL " :: "[idts, bool] => bool" ("(3\<forall>_./ _)" [0, 10] 10)
"EX " :: "[idts, bool] => bool" ("(3\<exists>_./ _)" [0, 10] 10)
"EX! " :: "[idts, bool] => bool" ("(3\<exists>!_./ _)" [0, 10] 10)
syntax (HOL)
"ALL " :: "[idts, bool] => bool" ("(3! _./ _)" [0, 10] 10)
"EX " :: "[idts, bool] => bool" ("(3? _./ _)" [0, 10] 10)
"EX! " :: "[idts, bool] => bool" ("(3?! _./ _)" [0, 10] 10)
subsubsection {* Axioms and basic definitions *}
axioms
eq_reflection: "(x=y) ==> (x==y)"
refl: "t = (t::'a)"
ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
-- {*Extensionality is built into the meta-logic, and this rule expresses
a related property. It is an eta-expanded version of the traditional
rule, and similar to the ABS rule of HOL*}
the_eq_trivial: "(THE x. x = a) = (a::'a)"
impI: "(P ==> Q) ==> P-->Q"
mp: "[| P-->Q; P |] ==> Q"
text{*Thanks to Stephan Merz*}
theorem subst:
assumes eq: "s = t" and p: "P(s)"
shows "P(t::'a)"
proof -
from eq have meta: "s \<equiv> t"
by (rule eq_reflection)
from p show ?thesis
by (unfold meta)
qed
defs
True_def: "True == ((%x::bool. x) = (%x. x))"
All_def: "All(P) == (P = (%x. True))"
Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
False_def: "False == (!P. P)"
not_def: "~ P == P-->False"
and_def: "P & Q == !R. (P-->Q-->R) --> R"
or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
axioms
iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
True_or_False: "(P=True) | (P=False)"
defs
Let_def: "Let s f == f(s)"
if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
finalconsts
"op ="
"op -->"
The
arbitrary
subsubsection {* Generic algebraic operations *}
axclass zero < type
axclass one < type
axclass plus < type
axclass minus < type
axclass times < type
axclass inverse < type
global
consts
"0" :: "'a::zero" ("0")
"1" :: "'a::one" ("1")
"+" :: "['a::plus, 'a] => 'a" (infixl 65)
- :: "['a::minus, 'a] => 'a" (infixl 65)
uminus :: "['a::minus] => 'a" ("- _" [81] 80)
* :: "['a::times, 'a] => 'a" (infixl 70)
syntax
"_index1" :: index ("\<^sub>1")
translations
(index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
local
typed_print_translation {*
let
fun tr' c = (c, fn show_sorts => fn T => fn ts =>
if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
in [tr' "0", tr' "1"] end;
*} -- {* show types that are presumably too general *}
consts
abs :: "'a::minus => 'a"
inverse :: "'a::inverse => 'a"
divide :: "['a::inverse, 'a] => 'a" (infixl "'/" 70)
syntax (xsymbols)
abs :: "'a::minus => 'a" ("\<bar>_\<bar>")
syntax (HTML output)
abs :: "'a::minus => 'a" ("\<bar>_\<bar>")
subsection {*Equality*}
lemma sym: "s=t ==> t=s"
apply (erule subst)
apply (rule refl)
done
(*calling "standard" reduces maxidx to 0*)
lemmas ssubst = sym [THEN subst, standard]
lemma trans: "[| r=s; s=t |] ==> r=t"
apply (erule subst , assumption)
done
lemma def_imp_eq: assumes meq: "A == B" shows "A = B"
apply (unfold meq)
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
text {* For calculational reasoning: *}
lemma forw_subst: "a = b ==> P b ==> P a"
by (rule ssubst)
lemma back_subst: "P a ==> a = b ==> P b"
by (rule subst)
subsection {*Congruence rules for application*}
(*similar to AP_THM in Gordon's HOL*)
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
apply (erule subst)
apply (rule refl)
done
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
lemma arg_cong: "x=y ==> f(x)=f(y)"
apply (erule subst)
apply (rule refl)
done
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
apply (erule ssubst)+
apply (rule refl)
done
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
apply (erule subst)+
apply (rule refl)
done
subsection {*Equality of booleans -- iff*}
lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"
apply (rules intro: iff [THEN mp, THEN mp] impI prems)
done
lemma iffD2: "[| P=Q; Q |] ==> P"
apply (erule ssubst)
apply assumption
done
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
apply (erule iffD2)
apply assumption
done
lemmas iffD1 = sym [THEN iffD2, standard]
lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard]
lemma iffE:
assumes major: "P=Q"
and minor: "[| P --> Q; Q --> P |] ==> R"
shows "R"
by (rules intro: minor impI major [THEN iffD2] major [THEN iffD1])
subsection {*True*}
lemma TrueI: "True"
apply (unfold True_def)
apply (rule refl)
done
lemma eqTrueI: "P ==> P=True"
by (rules intro: iffI TrueI)
lemma eqTrueE: "P=True ==> P"
apply (erule iffD2)
apply (rule TrueI)
done
subsection {*Universal quantifier*}
lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
apply (unfold All_def)
apply (rules intro: ext eqTrueI p)
done
lemma spec: "ALL x::'a. P(x) ==> P(x)"
apply (unfold All_def)
apply (rule eqTrueE)
apply (erule fun_cong)
done
lemma allE:
assumes major: "ALL x. P(x)"
and minor: "P(x) ==> R"
shows "R"
by (rules intro: minor major [THEN spec])
lemma all_dupE:
assumes major: "ALL x. P(x)"
and minor: "[| P(x); ALL x. P(x) |] ==> R"
shows "R"
by (rules intro: minor major major [THEN spec])
subsection {*False*}
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
lemma FalseE: "False ==> P"
apply (unfold False_def)
apply (erule spec)
done
lemma False_neq_True: "False=True ==> P"
by (erule eqTrueE [THEN FalseE])
subsection {*Negation*}
lemma notI:
assumes p: "P ==> False"
shows "~P"
apply (unfold not_def)
apply (rules intro: impI p)
done
lemma False_not_True: "False ~= True"
apply (rule notI)
apply (erule False_neq_True)
done
lemma True_not_False: "True ~= False"
apply (rule notI)
apply (drule sym)
apply (erule False_neq_True)
done
lemma notE: "[| ~P; P |] ==> R"
apply (unfold not_def)
apply (erule mp [THEN FalseE])
apply assumption
done
(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
lemmas notI2 = notE [THEN notI, standard]
subsection {*Implication*}
lemma impE:
assumes "P-->Q" "P" "Q ==> R"
shows "R"
by (rules intro: prems mp)
(* Reduces Q to P-->Q, allowing substitution in P. *)
lemma rev_mp: "[| P; P --> Q |] ==> Q"
by (rules intro: mp)
lemma contrapos_nn:
assumes major: "~Q"
and minor: "P==>Q"
shows "~P"
by (rules intro: notI minor major [THEN notE])
(*not used at all, but we already have the other 3 combinations *)
lemma contrapos_pn:
assumes major: "Q"
and minor: "P ==> ~Q"
shows "~P"
by (rules intro: notI minor major notE)
lemma not_sym: "t ~= s ==> s ~= t"
apply (erule contrapos_nn)
apply (erule sym)
done
(*still used in HOLCF*)
lemma rev_contrapos:
assumes pq: "P ==> Q"
and nq: "~Q"
shows "~P"
apply (rule nq [THEN contrapos_nn])
apply (erule pq)
done
subsection {*Existential quantifier*}
lemma exI: "P x ==> EX x::'a. P x"
apply (unfold Ex_def)
apply (rules intro: allI allE impI mp)
done
lemma exE:
assumes major: "EX x::'a. P(x)"
and minor: "!!x. P(x) ==> Q"
shows "Q"
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
apply (rules intro: impI [THEN allI] minor)
done
subsection {*Conjunction*}
lemma conjI: "[| P; Q |] ==> P&Q"
apply (unfold and_def)
apply (rules intro: impI [THEN allI] mp)
done
lemma conjunct1: "[| P & Q |] ==> P"
apply (unfold and_def)
apply (rules intro: impI dest: spec mp)
done
lemma conjunct2: "[| P & Q |] ==> Q"
apply (unfold and_def)
apply (rules intro: impI dest: spec mp)
done
lemma conjE:
assumes major: "P&Q"
and minor: "[| P; Q |] ==> R"
shows "R"
apply (rule minor)
apply (rule major [THEN conjunct1])
apply (rule major [THEN conjunct2])
done
lemma context_conjI:
assumes prems: "P" "P ==> Q" shows "P & Q"
by (rules intro: conjI prems)
subsection {*Disjunction*}
lemma disjI1: "P ==> P|Q"
apply (unfold or_def)
apply (rules intro: allI impI mp)
done
lemma disjI2: "Q ==> P|Q"
apply (unfold or_def)
apply (rules intro: allI impI mp)
done
lemma disjE:
assumes major: "P|Q"
and minorP: "P ==> R"
and minorQ: "Q ==> R"
shows "R"
by (rules intro: minorP minorQ impI
major [unfolded or_def, THEN spec, THEN mp, THEN mp])
subsection {*Classical logic*}
lemma classical:
assumes prem: "~P ==> P"
shows "P"
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
apply assumption
apply (rule notI [THEN prem, THEN eqTrueI])
apply (erule subst)
apply assumption
done
lemmas ccontr = FalseE [THEN classical, standard]
(*notE with premises exchanged; it discharges ~R so that it can be used to
make elimination rules*)
lemma rev_notE:
assumes premp: "P"
and premnot: "~R ==> ~P"
shows "R"
apply (rule ccontr)
apply (erule notE [OF premnot premp])
done
(*Double negation law*)
lemma notnotD: "~~P ==> P"
apply (rule classical)
apply (erule notE)
apply assumption
done
lemma contrapos_pp:
assumes p1: "Q"
and p2: "~P ==> ~Q"
shows "P"
by (rules intro: classical p1 p2 notE)
subsection {*Unique existence*}
lemma ex1I:
assumes prems: "P a" "!!x. P(x) ==> x=a"
shows "EX! x. P(x)"
by (unfold Ex1_def, rules intro: prems exI conjI allI impI)
text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
lemma ex_ex1I:
assumes ex_prem: "EX x. P(x)"
and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
shows "EX! x. P(x)"
by (rules intro: ex_prem [THEN exE] ex1I eq)
lemma ex1E:
assumes major: "EX! x. P(x)"
and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
shows "R"
apply (rule major [unfolded Ex1_def, THEN exE])
apply (erule conjE)
apply (rules intro: minor)
done
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
apply (erule ex1E)
apply (rule exI)
apply assumption
done
subsection {*THE: definite description operator*}
lemma the_equality:
assumes prema: "P a"
and premx: "!!x. P x ==> x=a"
shows "(THE x. P x) = a"
apply (rule trans [OF _ the_eq_trivial])
apply (rule_tac f = "The" in arg_cong)
apply (rule ext)
apply (rule iffI)
apply (erule premx)
apply (erule ssubst, rule prema)
done
lemma theI:
assumes "P a" and "!!x. P x ==> x=a"
shows "P (THE x. P x)"
by (rules intro: prems the_equality [THEN ssubst])
lemma theI': "EX! x. P x ==> P (THE x. P x)"
apply (erule ex1E)
apply (erule theI)
apply (erule allE)
apply (erule mp)
apply assumption
done
(*Easier to apply than theI: only one occurrence of P*)
lemma theI2:
assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
shows "Q (THE x. P x)"
by (rules intro: prems theI)
lemma the1_equality: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
apply (rule the_equality)
apply assumption
apply (erule ex1E)
apply (erule all_dupE)
apply (drule mp)
apply assumption
apply (erule ssubst)
apply (erule allE)
apply (erule mp)
apply assumption
done
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
apply (rule the_equality)
apply (rule refl)
apply (erule sym)
done
subsection {*Classical intro rules for disjunction and existential quantifiers*}
lemma disjCI:
assumes "~Q ==> P" shows "P|Q"
apply (rule classical)
apply (rules intro: prems disjI1 disjI2 notI elim: notE)
done
lemma excluded_middle: "~P | P"
by (rules intro: disjCI)
text{*case distinction as a natural deduction rule. Note that @{term "~P"}
is the second case, not the first.*}
lemma case_split_thm:
assumes prem1: "P ==> Q"
and prem2: "~P ==> Q"
shows "Q"
apply (rule excluded_middle [THEN disjE])
apply (erule prem2)
apply (erule prem1)
done
(*Classical implies (-->) elimination. *)
lemma impCE:
assumes major: "P-->Q"
and minor: "~P ==> R" "Q ==> R"
shows "R"
apply (rule excluded_middle [of P, THEN disjE])
apply (rules intro: minor major [THEN mp])+
done
(*This version of --> elimination works on Q before P. It works best for
those cases in which P holds "almost everywhere". Can't install as
default: would break old proofs.*)
lemma impCE':
assumes major: "P-->Q"
and minor: "Q ==> R" "~P ==> R"
shows "R"
apply (rule excluded_middle [of P, THEN disjE])
apply (rules intro: minor major [THEN mp])+
done
(*Classical <-> elimination. *)
lemma iffCE:
assumes major: "P=Q"
and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
shows "R"
apply (rule major [THEN iffE])
apply (rules intro: minor elim: impCE notE)
done
lemma exCI:
assumes "ALL x. ~P(x) ==> P(a)"
shows "EX x. P(x)"
apply (rule ccontr)
apply (rules intro: prems exI allI notI notE [of "\<exists>x. P x"])
done
subsection {* Theory and package setup *}
ML
{*
val plusI = thm "plusI"
val minusI = thm "minusI"
val timesI = thm "timesI"
val eq_reflection = thm "eq_reflection"
val refl = thm "refl"
val subst = thm "subst"
val ext = thm "ext"
val impI = thm "impI"
val mp = thm "mp"
val True_def = thm "True_def"
val All_def = thm "All_def"
val Ex_def = thm "Ex_def"
val False_def = thm "False_def"
val not_def = thm "not_def"
val and_def = thm "and_def"
val or_def = thm "or_def"
val Ex1_def = thm "Ex1_def"
val iff = thm "iff"
val True_or_False = thm "True_or_False"
val Let_def = thm "Let_def"
val if_def = thm "if_def"
val sym = thm "sym"
val ssubst = thm "ssubst"
val trans = thm "trans"
val def_imp_eq = thm "def_imp_eq"
val box_equals = thm "box_equals"
val fun_cong = thm "fun_cong"
val arg_cong = thm "arg_cong"
val cong = thm "cong"
val iffI = thm "iffI"
val iffD2 = thm "iffD2"
val rev_iffD2 = thm "rev_iffD2"
val iffD1 = thm "iffD1"
val rev_iffD1 = thm "rev_iffD1"
val iffE = thm "iffE"
val TrueI = thm "TrueI"
val eqTrueI = thm "eqTrueI"
val eqTrueE = thm "eqTrueE"
val allI = thm "allI"
val spec = thm "spec"
val allE = thm "allE"
val all_dupE = thm "all_dupE"
val FalseE = thm "FalseE"
val False_neq_True = thm "False_neq_True"
val notI = thm "notI"
val False_not_True = thm "False_not_True"
val True_not_False = thm "True_not_False"
val notE = thm "notE"
val notI2 = thm "notI2"
val impE = thm "impE"
val rev_mp = thm "rev_mp"
val contrapos_nn = thm "contrapos_nn"
val contrapos_pn = thm "contrapos_pn"
val not_sym = thm "not_sym"
val rev_contrapos = thm "rev_contrapos"
val exI = thm "exI"
val exE = thm "exE"
val conjI = thm "conjI"
val conjunct1 = thm "conjunct1"
val conjunct2 = thm "conjunct2"
val conjE = thm "conjE"
val context_conjI = thm "context_conjI"
val disjI1 = thm "disjI1"
val disjI2 = thm "disjI2"
val disjE = thm "disjE"
val classical = thm "classical"
val ccontr = thm "ccontr"
val rev_notE = thm "rev_notE"
val notnotD = thm "notnotD"
val contrapos_pp = thm "contrapos_pp"
val ex1I = thm "ex1I"
val ex_ex1I = thm "ex_ex1I"
val ex1E = thm "ex1E"
val ex1_implies_ex = thm "ex1_implies_ex"
val the_equality = thm "the_equality"
val theI = thm "theI"
val theI' = thm "theI'"
val theI2 = thm "theI2"
val the1_equality = thm "the1_equality"
val the_sym_eq_trivial = thm "the_sym_eq_trivial"
val disjCI = thm "disjCI"
val excluded_middle = thm "excluded_middle"
val case_split_thm = thm "case_split_thm"
val impCE = thm "impCE"
val impCE = thm "impCE"
val iffCE = thm "iffCE"
val exCI = thm "exCI"
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
local
fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
| wrong_prem (Bound _) = true
| wrong_prem _ = false
val filter_right = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))
in
fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp])
fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]
end
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i)
(*Obsolete form of disjunctive case analysis*)
fun excluded_middle_tac sP =
res_inst_tac [("Q",sP)] (excluded_middle RS disjE)
fun case_tac a = res_inst_tac [("P",a)] case_split_thm
*}
theorems case_split = case_split_thm [case_names True False]
subsubsection {* 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
lemmas [trans] = trans
and [sym] = sym not_sym
and [Pure.elim?] = iffD1 iffD2 impE
subsubsection {* Atomizing meta-level connectives *}
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
proof
assume "!!x. P x"
show "ALL x. P x" by (rule allI)
next
assume "ALL x. P x"
thus "!!x. P x" by (rule allE)
qed
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
proof
assume r: "A ==> B"
show "A --> B" by (rule impI) (rule r)
next
assume "A --> B" and A
thus B by (rule mp)
qed
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
proof
assume r: "A ==> False"
show "~A" by (rule notI) (rule r)
next
assume "~A" and A
thus False by (rule notE)
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
subsubsection {* Classical Reasoner setup *}
use "cladata.ML"
setup hypsubst_setup
setup {*
[ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)]
*}
setup Classical.setup
setup clasetup
lemmas [intro?] = ext
and [elim?] = ex1_implies_ex
use "blastdata.ML"
setup Blast.setup
subsection {* Simplifier setup *}
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
proof -
assume r: "x == y"
show "x = y" by (unfold r) (rule refl)
qed
lemma eta_contract_eq: "(%s. f s) = f" ..
lemma simp_thms:
shows not_not: "(~ ~ P) = P"
and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
and
"(P ~= Q) = (P = (~Q))"
"(P | ~P) = True" "(~P | P) = True"
"(x = x) = True"
"(~True) = False" "(~False) = True"
"(~P) ~= P" "P ~= (~P)"
"(True=P) = P" "(P=True) = P" "(False=P) = (~P)" "(P=False) = (~P)"
"(True --> P) = P" "(False --> P) = True"
"(P --> True) = True" "(P --> P) = True"
"(P --> False) = (~P)" "(P --> ~P) = (~P)"
"(P & True) = P" "(True & P) = P"
"(P & False) = False" "(False & P) = False"
"(P & P) = P" "(P & (P & Q)) = (P & Q)"
"(P & ~P) = False" "(~P & P) = False"
"(P | True) = True" "(True | P) = True"
"(P | False) = P" "(False | P) = P"
"(P | P) = P" "(P | (P | Q)) = (P | Q)" and
"(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
-- {* needed for the one-point-rule quantifier simplification procs *}
-- {* essential for termination!! *} and
"!!P. (EX x. x=t & P(x)) = P(t)"
"!!P. (EX x. t=x & P(x)) = P(t)"
"!!P. (ALL x. x=t --> P(x)) = P(t)"
"!!P. (ALL x. t=x --> P(x)) = P(t)"
by (blast, blast, blast, blast, blast, rules+)
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
by rules
lemma ex_simps:
"!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
"!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
"!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
"!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
"!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
"!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
-- {* Miniscoping: pushing in existential quantifiers. *}
by (rules | blast)+
lemma all_simps:
"!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
"!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
"!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
"!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
"!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
"!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
-- {* Miniscoping: pushing in universal quantifiers. *}
by (rules | blast)+
lemma disj_absorb: "(A | A) = A"
by blast
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
by blast
lemma conj_absorb: "(A & A) = A"
by blast
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
by blast
lemma eq_ac:
shows eq_commute: "(a=b) = (b=a)"
and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
lemma neq_commute: "(a~=b) = (b~=a)" by rules
lemma conj_comms:
shows conj_commute: "(P&Q) = (Q&P)"
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
lemma disj_comms:
shows disj_commute: "(P|Q) = (Q|P)"
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by rules
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
by blast
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
-- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
-- {* cases boil down to the same thing. *}
by blast
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
text {*
\medskip The @{text "&"} congruence rule: not included by default!
May slow rewrite proofs down by as much as 50\% *}
lemma conj_cong:
"(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
by rules
lemma rev_conj_cong:
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
by rules
text {* The @{text "|"} congruence rule: not included by default! *}
lemma disj_cong:
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
by blast
lemma eq_sym_conv: "(x = y) = (y = x)"
by rules
text {* \medskip if-then-else rules *}
lemma if_True: "(if True then x else y) = x"
by (unfold if_def) blast
lemma if_False: "(if False then x else y) = y"
by (unfold if_def) blast
lemma if_P: "P ==> (if P then x else y) = x"
by (unfold if_def) blast
lemma if_not_P: "~P ==> (if P then x else y) = y"
by (unfold if_def) blast
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
apply (rule case_split [of Q])
apply (simplesubst if_P)
prefer 3 apply (simplesubst if_not_P, blast+)
done
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
by (simplesubst split_if, blast)
lemmas if_splits = split_if split_if_asm
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
by (rule split_if)
lemma if_cancel: "(if c then x else x) = x"
by (simplesubst split_if, blast)
lemma if_eq_cancel: "(if x = y then y else x) = x"
by (simplesubst split_if, blast)
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
-- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
by (rule split_if)
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
-- {* And this form is useful for expanding @{text if}s on the LEFT. *}
apply (simplesubst split_if, blast)
done
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
text {* \medskip let rules for simproc *}
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
by (unfold Let_def)
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
by (unfold Let_def)
text {*
The following copy of the implication operator is useful for
fine-tuning congruence rules. It instructs the simplifier to simplify
its premise.
*}
constdefs
simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
"simp_implies \<equiv> op ==>"
lemma simp_impliesI:
assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
shows "PROP P =simp=> PROP Q"
apply (unfold simp_implies_def)
apply (rule PQ)
apply assumption
done
lemma simp_impliesE:
assumes PQ:"PROP P =simp=> PROP Q"
and P: "PROP P"
and QR: "PROP Q \<Longrightarrow> PROP R"
shows "PROP R"
apply (rule QR)
apply (rule PQ [unfolded simp_implies_def])
apply (rule P)
done
lemma simp_implies_cong:
assumes PP' :"PROP P == PROP P'"
and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
proof (unfold simp_implies_def, rule equal_intr_rule)
assume PQ: "PROP P \<Longrightarrow> PROP Q"
and P': "PROP P'"
from PP' [symmetric] and P' have "PROP P"
by (rule equal_elim_rule1)
hence "PROP Q" by (rule PQ)
with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
next
assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
and P: "PROP P"
from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
hence "PROP Q'" by (rule P'Q')
with P'QQ' [OF P', symmetric] show "PROP Q"
by (rule equal_elim_rule1)
qed
subsubsection {* Actual Installation of the Simplifier *}
use "simpdata.ML"
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
setup Splitter.setup setup Clasimp.setup
subsubsection {* Lucas Dixon's eqstep tactic *}
use "~~/src/Provers/eqsubst.ML";
use "eqrule_HOL_data.ML";
setup EQSubstTac.setup
subsection {* Other simple lemmas *}
declare disj_absorb [simp] conj_absorb [simp]
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
by blast+
theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
apply (rule iffI)
apply (rule_tac a = "%x. THE y. P x y" in ex1I)
apply (fast dest!: theI')
apply (fast intro: ext the1_equality [symmetric])
apply (erule ex1E)
apply (rule allI)
apply (rule ex1I)
apply (erule spec)
apply (erule_tac x = "%z. if z = x then y else f z" in allE)
apply (erule impE)
apply (rule allI)
apply (rule_tac P = "xa = x" in case_split_thm)
apply (drule_tac [3] x = x in fun_cong, simp_all)
done
text{*Needs only HOL-lemmas:*}
lemma mk_left_commute:
assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
c: "\<And>x y. f x y = f y x"
shows "f x (f y z) = f y (f x z)"
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
subsection {* Generic cases and induction *}
constdefs
induct_forall :: "('a => bool) => bool"
"induct_forall P == \<forall>x. P x"
induct_implies :: "bool => bool => bool"
"induct_implies A B == A --> B"
induct_equal :: "'a => 'a => bool"
"induct_equal x y == x = y"
induct_conj :: "bool => bool => bool"
"induct_conj A B == A & B"
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
by (simp only: atomize_all induct_forall_def)
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
by (simp only: atomize_imp induct_implies_def)
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
by (simp only: atomize_eq induct_equal_def)
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
induct_conj (induct_forall A) (induct_forall B)"
by (unfold induct_forall_def induct_conj_def) rules
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
induct_conj (induct_implies C A) (induct_implies C B)"
by (unfold induct_implies_def induct_conj_def) rules
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
proof
assume r: "induct_conj A B ==> PROP C" and A B
show "PROP C" by (rule r) (simp! add: induct_conj_def)
next
assume r: "A ==> B ==> PROP C" and "induct_conj A B"
show "PROP C" by (rule r) (simp! add: induct_conj_def)+
qed
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
by (simp add: induct_implies_def)
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
hide const induct_forall induct_implies induct_equal induct_conj
text {* Method setup. *}
ML {*
structure InductMethod = InductMethodFun
(struct
val dest_concls = HOLogic.dest_concls
val cases_default = thm "case_split"
val local_impI = thm "induct_impliesI"
val conjI = thm "conjI"
val atomize = thms "induct_atomize"
val rulify1 = thms "induct_rulify1"
val rulify2 = thms "induct_rulify2"
val localize = [Thm.symmetric (thm "induct_implies_def")]
end);
*}
setup InductMethod.setup
end