(* Title: HOL/HOL.thy
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
header {* The basis of Higher-Order Logic *}
theory HOL
imports Pure "~~/src/Tools/Code_Generator"
keywords
"try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
"print_induct_rules" :: diag and
"quickcheck_params" :: thy_decl
begin
ML_file "~~/src/Tools/misc_legacy.ML"
ML_file "~~/src/Tools/try.ML"
ML_file "~~/src/Tools/quickcheck.ML"
ML_file "~~/src/Tools/solve_direct.ML"
ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
ML_file "~~/src/Tools/IsaPlanner/isand.ML"
ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
ML_file "~~/src/Provers/hypsubst.ML"
ML_file "~~/src/Provers/splitter.ML"
ML_file "~~/src/Provers/classical.ML"
ML_file "~~/src/Provers/blast.ML"
ML_file "~~/src/Provers/clasimp.ML"
ML_file "~~/src/Tools/eqsubst.ML"
ML_file "~~/src/Provers/quantifier1.ML"
ML_file "~~/src/Tools/atomize_elim.ML"
ML_file "~~/src/Tools/induct.ML"
ML_file "~~/src/Tools/cong_tac.ML"
ML_file "~~/src/Tools/intuitionistic.ML"
ML_file "~~/src/Tools/project_rule.ML"
ML_file "~~/src/Tools/subtyping.ML"
ML_file "~~/src/Tools/case_product.ML"
setup {*
Intuitionistic.method_setup @{binding iprover}
#> Subtyping.setup
#> Case_Product.setup
*}
subsection {* Primitive logic *}
subsubsection {* Core syntax *}
setup {* Axclass.class_axiomatization (@{binding type}, []) *}
default_sort type
setup {* Object_Logic.add_base_sort @{sort type} *}
axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
instance "fun" :: (type, type) type by (rule fun_arity)
axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
instance itself :: (type) type by (rule itself_arity)
typedecl bool
judgment
Trueprop :: "bool => prop" ("(_)" 5)
axiomatization
implies :: "[bool, bool] => bool" (infixr "-->" 25) and
eq :: "['a, 'a] => bool" (infixl "=" 50) and
The :: "('a => bool) => 'a"
consts
True :: bool
False :: bool
Not :: "bool => bool" ("~ _" [40] 40)
conj :: "[bool, bool] => bool" (infixr "&" 35)
disj :: "[bool, bool] => bool" (infixr "|" 30)
All :: "('a => bool) => bool" (binder "ALL " 10)
Ex :: "('a => bool) => bool" (binder "EX " 10)
Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
subsubsection {* Additional concrete syntax *}
notation (output)
eq (infix "=" 50)
abbreviation
not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
"x ~= y == ~ (x = y)"
notation (output)
not_equal (infix "~=" 50)
notation (xsymbols)
Not ("\<not> _" [40] 40) and
conj (infixr "\<and>" 35) and
disj (infixr "\<or>" 30) and
implies (infixr "\<longrightarrow>" 25) and
not_equal (infixl "\<noteq>" 50)
notation (xsymbols output)
not_equal (infix "\<noteq>" 50)
notation (HTML output)
Not ("\<not> _" [40] 40) and
conj (infixr "\<and>" 35) and
disj (infixr "\<or>" 30) and
not_equal (infix "\<noteq>" 50)
abbreviation (iff)
iff :: "[bool, bool] => bool" (infixr "<->" 25) where
"A <-> B == A = B"
notation (xsymbols)
iff (infixr "\<longleftrightarrow>" 25)
syntax "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
translations "THE x. P" == "CONST The (%x. P)"
print_translation {*
[(@{const_syntax The}, fn _ => fn [Abs abs] =>
let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
in Syntax.const @{syntax_const "_The"} $ x $ t end)]
*} -- {* To avoid eta-contraction of body *}
nonterminal letbinds and letbind
syntax
"_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
"" :: "letbind => letbinds" ("_")
"_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" [0, 10] 10)
nonterminal case_syn and cases_syn
syntax
"_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" ("_/ | _")
syntax (xsymbols)
"_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
notation (xsymbols)
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
Ex1 (binder "\<exists>!" 10)
notation (HTML output)
All (binder "\<forall>" 10) and
Ex (binder "\<exists>" 10) and
Ex1 (binder "\<exists>!" 10)
notation (HOL)
All (binder "! " 10) and
Ex (binder "? " 10) and
Ex1 (binder "?! " 10)
subsubsection {* Axioms and basic definitions *}
axiomatization where
refl: "t = (t::'a)" and
subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
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*} and
the_eq_trivial: "(THE x. x = a) = (a::'a)"
axiomatization where
impI: "(P ==> Q) ==> P-->Q" and
mp: "[| P-->Q; P |] ==> Q" and
iff: "(P-->Q) --> (Q-->P) --> (P=Q)" and
True_or_False: "(P=True) | (P=False)"
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)"
definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
where "If P x y \<equiv> (THE z::'a. (P=True --> z=x) & (P=False --> z=y))"
definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
where "Let s f \<equiv> f s"
translations
"_Let (_binds b bs) e" == "_Let b (_Let bs e)"
"let x = a in e" == "CONST Let a (%x. e)"
axiomatization undefined :: 'a
class default = fixes default :: 'a
subsection {* Fundamental rules *}
subsubsection {* Equality *}
lemma sym: "s = t ==> t = s"
by (erule subst) (rule refl)
lemma ssubst: "t = s ==> P s ==> P t"
by (drule sym) (erule subst)
lemma trans: "[| r=s; s=t |] ==> r=t"
by (erule subst)
lemma trans_sym [Pure.elim?]: "r = s ==> t = s ==> r = t"
by (rule trans [OF _ sym])
lemma meta_eq_to_obj_eq:
assumes meq: "A == B"
shows "A = B"
by (unfold meq) (rule refl)
text {* Useful with @{text erule} for proving equalities 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)
subsubsection {* Congruence rules for application *}
text {* Similar to @{text AP_THM} in Gordon's HOL. *}
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
apply (erule subst)
apply (rule refl)
done
text {* Similar to @{text AP_TERM} in Gordon's HOL and FOL's @{text 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
ML {* val cong_tac = Cong_Tac.cong_tac @{thm cong} *}
subsubsection {* Equality of booleans -- iff *}
lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
by (iprover intro: iff [THEN mp, THEN mp] impI assms)
lemma iffD2: "[| P=Q; Q |] ==> P"
by (erule ssubst)
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
by (erule iffD2)
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
by (drule sym) (rule iffD2)
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
by (drule sym) (rule rev_iffD2)
lemma iffE:
assumes major: "P=Q"
and minor: "[| P --> Q; Q --> P |] ==> R"
shows R
by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
subsubsection {*True*}
lemma TrueI: "True"
unfolding True_def by (rule refl)
lemma eqTrueI: "P ==> P = True"
by (iprover intro: iffI TrueI)
lemma eqTrueE: "P = True ==> P"
by (erule iffD2) (rule TrueI)
subsubsection {*Universal quantifier*}
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
unfolding All_def by (iprover intro: ext eqTrueI assms)
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 (iprover 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 (iprover intro: minor major major [THEN spec])
subsubsection {* False *}
text {*
Depends upon @{text 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])
subsubsection {* Negation *}
lemma notI:
assumes "P ==> False"
shows "~P"
apply (unfold not_def)
apply (iprover intro: impI assms)
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
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
by (erule notE [THEN notI]) (erule meta_mp)
subsubsection {*Implication*}
lemma impE:
assumes "P-->Q" "P" "Q ==> R"
shows "R"
by (iprover intro: assms mp)
(* Reduces Q to P-->Q, allowing substitution in P. *)
lemma rev_mp: "[| P; P --> Q |] ==> Q"
by (iprover intro: mp)
lemma contrapos_nn:
assumes major: "~Q"
and minor: "P==>Q"
shows "~P"
by (iprover 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 (iprover intro: notI minor major notE)
lemma not_sym: "t ~= s ==> s ~= t"
by (erule contrapos_nn) (erule sym)
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
by (erule subst, erule ssubst, assumption)
subsubsection {*Existential quantifier*}
lemma exI: "P x ==> EX x::'a. P x"
apply (unfold Ex_def)
apply (iprover 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 (iprover intro: impI [THEN allI] minor)
done
subsubsection {*Conjunction*}
lemma conjI: "[| P; Q |] ==> P&Q"
apply (unfold and_def)
apply (iprover intro: impI [THEN allI] mp)
done
lemma conjunct1: "[| P & Q |] ==> P"
apply (unfold and_def)
apply (iprover intro: impI dest: spec mp)
done
lemma conjunct2: "[| P & Q |] ==> Q"
apply (unfold and_def)
apply (iprover 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 "P" "P ==> Q" shows "P & Q"
by (iprover intro: conjI assms)
subsubsection {*Disjunction*}
lemma disjI1: "P ==> P|Q"
apply (unfold or_def)
apply (iprover intro: allI impI mp)
done
lemma disjI2: "Q ==> P|Q"
apply (unfold or_def)
apply (iprover intro: allI impI mp)
done
lemma disjE:
assumes major: "P|Q"
and minorP: "P ==> R"
and minorQ: "Q ==> R"
shows "R"
by (iprover intro: minorP minorQ impI
major [unfolded or_def, THEN spec, THEN mp, THEN mp])
subsubsection {*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]
(*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 (iprover intro: classical p1 p2 notE)
subsubsection {*Unique existence*}
lemma ex1I:
assumes "P a" "!!x. P(x) ==> x=a"
shows "EX! x. P(x)"
by (unfold Ex1_def, iprover intro: assms 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 (iprover 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 (iprover intro: minor)
done
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
apply (erule ex1E)
apply (rule exI)
apply assumption
done
subsubsection {*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 (iprover intro: assms 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 (iprover intro: assms theI)
lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
elim:allE impE)
lemma the1_equality [elim?]: "[| 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
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
lemma disjCI:
assumes "~Q ==> P" shows "P|Q"
apply (rule classical)
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
done
lemma excluded_middle: "~P | P"
by (iprover intro: disjCI)
text {*
case distinction as a natural deduction rule.
Note that @{term "~P"} is the second case, not the first
*}
lemma case_split [case_names True False]:
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 (iprover 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 (iprover 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 (iprover intro: minor elim: impCE notE)
done
lemma exCI:
assumes "ALL x. ~P(x) ==> P(a)"
shows "EX x. P(x)"
apply (rule ccontr)
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
done
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
lemma TrueE: "True ==> P ==> P" .
lemma notFalseE: "~ False ==> P ==> P" .
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
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 *}
axiomatization where
eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
proof
assume "!!x. P x"
then show "ALL x. P x" ..
next
assume "ALL x. P x"
then show "!!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
then show 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
then show False by (rule notE)
qed
lemma atomize_eq [atomize, code]: "(x == y) == Trueprop (x = y)"
proof
assume "x == y"
show "x = y" by (unfold `x == y`) (rule refl)
next
assume "x = y"
then show "x == y" by (rule eq_reflection)
qed
lemma atomize_conj [atomize]: "(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
subsubsection {* Atomizing elimination rules *}
lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
by rule iprover+
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
by rule iprover+
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
by rule iprover+
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
subsection {* Package setup *}
ML_file "Tools/hologic.ML"
subsubsection {* Sledgehammer setup *}
text {*
Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
that are prolific (match too many equality or membership literals) and relate to
seldom-used facts. Some duplicate other rules.
*}
ML {*
structure No_ATPs = Named_Thms
(
val name = @{binding no_atp}
val description = "theorems that should be filtered out by Sledgehammer"
)
*}
setup {* No_ATPs.setup *}
subsubsection {* Classical Reasoner setup *}
lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
by (rule classical) iprover
lemma swap: "~ P ==> (~ R ==> P) ==> R"
by (rule classical) iprover
lemma thin_refl:
"\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
ML {*
structure Hypsubst = Hypsubst
(
val dest_eq = HOLogic.dest_eq
val dest_Trueprop = HOLogic.dest_Trueprop
val dest_imp = HOLogic.dest_imp
val eq_reflection = @{thm eq_reflection}
val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
val imp_intr = @{thm impI}
val rev_mp = @{thm rev_mp}
val subst = @{thm subst}
val sym = @{thm sym}
val thin_refl = @{thm thin_refl};
);
open Hypsubst;
structure Classical = Classical
(
val imp_elim = @{thm imp_elim}
val not_elim = @{thm notE}
val swap = @{thm swap}
val classical = @{thm classical}
val sizef = Drule.size_of_thm
val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
);
structure Basic_Classical: BASIC_CLASSICAL = Classical;
open Basic_Classical;
*}
setup Classical.setup
setup {*
let
fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
| non_bool_eq _ = false;
fun hyp_subst_tac' ctxt =
SUBGOAL (fn (goal, i) =>
if Term.exists_Const non_bool_eq goal
then Hypsubst.hyp_subst_tac ctxt i
else no_tac);
in
Hypsubst.hypsubst_setup
(*prevent substitution on bool*)
#> Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
end
*}
declare iffI [intro!]
and notI [intro!]
and impI [intro!]
and disjCI [intro!]
and conjI [intro!]
and TrueI [intro!]
and refl [intro!]
declare iffCE [elim!]
and FalseE [elim!]
and impCE [elim!]
and disjE [elim!]
and conjE [elim!]
declare ex_ex1I [intro!]
and allI [intro!]
and the_equality [intro]
and exI [intro]
declare exE [elim!]
allE [elim]
ML {* val HOL_cs = claset_of @{context} *}
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
apply (erule swap)
apply (erule (1) meta_mp)
done
declare ex_ex1I [rule del, intro! 2]
and ex1I [intro]
declare ext [intro]
lemmas [intro?] = ext
and [elim?] = ex1_implies_ex
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
lemma alt_ex1E [elim!]:
assumes major: "\<exists>!x. P x"
and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
shows R
apply (rule ex1E [OF major])
apply (rule prem)
apply (tactic {* ares_tac @{thms allI} 1 *})+
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
apply iprover
done
ML {*
structure Blast = Blast
(
structure Classical = Classical
val Trueprop_const = dest_Const @{const Trueprop}
val equality_name = @{const_name HOL.eq}
val not_name = @{const_name Not}
val notE = @{thm notE}
val ccontr = @{thm ccontr}
val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
);
val blast_tac = Blast.blast_tac;
*}
setup Blast.setup
subsubsection {* Simplifier *}
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"
and not_True_eq_False [code]: "(\<not> True) = False"
and not_False_eq_True [code]: "(\<not> False) = True"
and
"(~P) ~= P" "P ~= (~P)"
"(True=P) = P"
and eq_True: "(P = True) = P"
and "(False=P) = (~P)"
and eq_False: "(P = False) = (\<not> P)"
and
"(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"
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, iprover+)
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 \<longleftrightarrow> b = a"
and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
lemma conj_comms:
shows conj_commute: "(P&Q) = (Q&P)"
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
lemma disj_comms:
shows disj_commute: "(P|Q) = (Q|P)"
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
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 imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
by iprover
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
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 iprover
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 iprover
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
declare All_def [no_atp]
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
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 iprover
lemma rev_conj_cong:
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
by iprover
text {* The @{text "|"} congruence rule: not included by default! *}
lemma disj_cong:
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
by blast
text {* \medskip if-then-else rules *}
lemma if_True [code]: "(if True then x else y) = x"
by (unfold If_def) blast
lemma if_False [code]: "(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 [no_atp] = split_if split_if_asm
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) iprover
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
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.
*}
definition simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1) where
"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)
then have "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)
then have "PROP Q'" by (rule P'Q')
with P'QQ' [OF P', symmetric] show "PROP Q"
by (rule equal_elim_rule1)
qed
lemma uncurry:
assumes "P \<longrightarrow> Q \<longrightarrow> R"
shows "P \<and> Q \<longrightarrow> R"
using assms by blast
lemma iff_allI:
assumes "\<And>x. P x = Q x"
shows "(\<forall>x. P x) = (\<forall>x. Q x)"
using assms by blast
lemma iff_exI:
assumes "\<And>x. P x = Q x"
shows "(\<exists>x. P x) = (\<exists>x. Q x)"
using assms by blast
lemma all_comm:
"(\<forall>x y. P x y) = (\<forall>y x. P x y)"
by blast
lemma ex_comm:
"(\<exists>x y. P x y) = (\<exists>y x. P x y)"
by blast
ML_file "Tools/simpdata.ML"
ML {* open Simpdata *}
setup {* map_theory_simpset (put_simpset HOL_basic_ss) *}
simproc_setup defined_Ex ("EX x. P x") = {* fn _ => Quantifier1.rearrange_ex *}
simproc_setup defined_All ("ALL x. P x") = {* fn _ => Quantifier1.rearrange_all *}
setup {*
Simplifier.method_setup Splitter.split_modifiers
#> Splitter.setup
#> clasimp_setup
#> EqSubst.setup
*}
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
simproc_setup neq ("x = y") = {* fn _ =>
let
val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
fun is_neq eq lhs rhs thm =
(case Thm.prop_of thm of
_ $ (Not $ (eq' $ l' $ r')) =>
Not = HOLogic.Not andalso eq' = eq andalso
r' aconv lhs andalso l' aconv rhs
| _ => false);
fun proc ss ct =
(case Thm.term_of ct of
eq $ lhs $ rhs =>
(case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
SOME thm => SOME (thm RS neq_to_EQ_False)
| NONE => NONE)
| _ => NONE);
in proc end;
*}
simproc_setup let_simp ("Let x f") = {*
let
val (f_Let_unfold, x_Let_unfold) =
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}
in (cterm_of @{theory} f, cterm_of @{theory} x) end
val (f_Let_folded, x_Let_folded) =
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}
in (cterm_of @{theory} f, cterm_of @{theory} x) end;
val g_Let_folded =
let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}
in cterm_of @{theory} g end;
fun count_loose (Bound i) k = if i >= k then 1 else 0
| count_loose (s $ t) k = count_loose s k + count_loose t k
| count_loose (Abs (_, _, t)) k = count_loose t (k + 1)
| count_loose _ _ = 0;
fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
case t
of Abs (_, _, t') => count_loose t' 0 <= 1
| _ => true;
in fn _ => fn ctxt => fn ct => if is_trivial_let (Thm.term_of ct)
then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
else let (*Norbert Schirmer's case*)
val thy = Proof_Context.theory_of ctxt;
val t = Thm.term_of ct;
val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
in Option.map (hd o Variable.export ctxt' ctxt o single)
(case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
if is_Free x orelse is_Bound x orelse is_Const x
then SOME @{thm Let_def}
else
let
val n = case f of (Abs (x, _, _)) => x | _ => "x";
val cx = cterm_of thy x;
val {T = xT, ...} = rep_cterm cx;
val cf = cterm_of thy f;
val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
val (_ $ _ $ g) = prop_of fx_g;
val g' = abstract_over (x,g);
val abs_g'= Abs (n,xT,g');
in (if (g aconv g')
then
let
val rl =
cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
in SOME (rl OF [fx_g]) end
else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g') then NONE (*avoid identity conversion*)
else let
val g'x = abs_g'$x;
val g_g'x = Thm.symmetric (Thm.beta_conversion false (cterm_of thy g'x));
val rl = cterm_instantiate
[(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),
(g_Let_folded, cterm_of thy abs_g')]
@{thm Let_folded};
in SOME (rl OF [Thm.transitive fx_g g_g'x])
end)
end
| _ => NONE)
end
end *}
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
proof
assume "True \<Longrightarrow> PROP P"
from this [OF TrueI] show "PROP P" .
next
assume "PROP P"
then show "PROP P" .
qed
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 (iprover | 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 (iprover | blast)+
lemmas [simp] =
triv_forall_equality (*prunes params*)
True_implies_equals (*prune asms `True'*)
if_True
if_False
if_cancel
if_eq_cancel
imp_disjL
(*In general it seems wrong to add distributive laws by default: they
might cause exponential blow-up. But imp_disjL has been in for a while
and cannot be removed without affecting existing proofs. Moreover,
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
grounds that it allows simplification of R in the two cases.*)
conj_assoc
disj_assoc
de_Morgan_conj
de_Morgan_disj
imp_disj1
imp_disj2
not_imp
disj_not1
not_all
not_ex
cases_simp
the_eq_trivial
the_sym_eq_trivial
ex_simps
all_simps
simp_thms
lemmas [cong] = imp_cong simp_implies_cong
lemmas [split] = split_if
ML {* val HOL_ss = simpset_of @{context} *}
text {* Simplifies x assuming c and y assuming ~c *}
lemma if_cong:
assumes "b = c"
and "c \<Longrightarrow> x = u"
and "\<not> c \<Longrightarrow> y = v"
shows "(if b then x else y) = (if c then u else v)"
using assms by simp
text {* Prevents simplification of x and y:
faster and allows the execution of functional programs. *}
lemma if_weak_cong [cong]:
assumes "b = c"
shows "(if b then x else y) = (if c then x else y)"
using assms by (rule arg_cong)
text {* Prevents simplification of t: much faster *}
lemma let_weak_cong:
assumes "a = b"
shows "(let x = a in t x) = (let x = b in t x)"
using assms by (rule arg_cong)
text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
lemma eq_cong2:
assumes "u = u'"
shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
using assms by simp
lemma if_distrib:
"f (if c then x else y) = (if c then f x else f y)"
by simp
text{*As a simplification rule, it replaces all function equalities by
first-order equalities.*}
lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
by auto
subsubsection {* Generic cases and induction *}
text {* Rule projections: *}
ML {*
structure Project_Rule = Project_Rule
(
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val mp = @{thm mp}
)
*}
definition induct_forall where
"induct_forall P == \<forall>x. P x"
definition induct_implies where
"induct_implies A B == A \<longrightarrow> B"
definition induct_equal where
"induct_equal x y == x = y"
definition induct_conj where
"induct_conj A B == A \<and> B"
definition induct_true where
"induct_true == True"
definition induct_false where
"induct_false == False"
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
by (unfold atomize_all induct_forall_def)
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
by (unfold atomize_imp induct_implies_def)
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
by (unfold atomize_eq induct_equal_def)
lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"
by (unfold atomize_conj induct_conj_def)
lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
lemmas induct_atomize = induct_atomize' induct_equal_eq
lemmas induct_rulify' [symmetric] = induct_atomize'
lemmas induct_rulify [symmetric] = induct_atomize
lemmas induct_rulify_fallback =
induct_forall_def induct_implies_def induct_equal_def induct_conj_def
induct_true_def induct_false_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) iprover
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) iprover
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 `A` `B`)
next
assume r: "A ==> B ==> PROP C" and "induct_conj A B"
show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
qed
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
lemma induct_trueI: "induct_true"
by (simp add: induct_true_def)
text {* Method setup. *}
ML {*
structure Induct = Induct
(
val cases_default = @{thm case_split}
val atomize = @{thms induct_atomize}
val rulify = @{thms induct_rulify'}
val rulify_fallback = @{thms induct_rulify_fallback}
val equal_def = @{thm induct_equal_def}
fun dest_def (Const (@{const_name induct_equal}, _) $ t $ u) = SOME (t, u)
| dest_def _ = NONE
val trivial_tac = match_tac @{thms induct_trueI}
)
*}
ML_file "~~/src/Tools/induction.ML"
setup {*
Induct.setup #> Induction.setup #>
Context.theory_map (Induct.map_simpset (fn ss => ss
addsimprocs
[Simplifier.simproc_global @{theory} "swap_induct_false"
["induct_false ==> PROP P ==> PROP Q"]
(fn _ =>
(fn _ $ (P as _ $ @{const induct_false}) $ (_ $ Q $ _) =>
if P <> Q then SOME Drule.swap_prems_eq else NONE
| _ => NONE)),
Simplifier.simproc_global @{theory} "induct_equal_conj_curry"
["induct_conj P Q ==> PROP R"]
(fn _ =>
(fn _ $ (_ $ P) $ _ =>
let
fun is_conj (@{const induct_conj} $ P $ Q) =
is_conj P andalso is_conj Q
| is_conj (Const (@{const_name induct_equal}, _) $ _ $ _) = true
| is_conj @{const induct_true} = true
| is_conj @{const induct_false} = true
| is_conj _ = false
in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
| _ => NONE))]
|> Simplifier.set_mksimps (fn ctxt =>
Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback})))))
*}
text {* Pre-simplification of induction and cases rules *}
lemma [induct_simp]: "(!!x. induct_equal x t ==> PROP P x) == PROP P t"
unfolding induct_equal_def
proof
assume R: "!!x. x = t ==> PROP P x"
show "PROP P t" by (rule R [OF refl])
next
fix x assume "PROP P t" "x = t"
then show "PROP P x" by simp
qed
lemma [induct_simp]: "(!!x. induct_equal t x ==> PROP P x) == PROP P t"
unfolding induct_equal_def
proof
assume R: "!!x. t = x ==> PROP P x"
show "PROP P t" by (rule R [OF refl])
next
fix x assume "PROP P t" "t = x"
then show "PROP P x" by simp
qed
lemma [induct_simp]: "(induct_false ==> P) == Trueprop induct_true"
unfolding induct_false_def induct_true_def
by (iprover intro: equal_intr_rule)
lemma [induct_simp]: "(induct_true ==> PROP P) == PROP P"
unfolding induct_true_def
proof
assume R: "True \<Longrightarrow> PROP P"
from TrueI show "PROP P" by (rule R)
next
assume "PROP P"
then show "PROP P" .
qed
lemma [induct_simp]: "(PROP P ==> induct_true) == Trueprop induct_true"
unfolding induct_true_def
by (iprover intro: equal_intr_rule)
lemma [induct_simp]: "(!!x. induct_true) == Trueprop induct_true"
unfolding induct_true_def
by (iprover intro: equal_intr_rule)
lemma [induct_simp]: "induct_implies induct_true P == P"
by (simp add: induct_implies_def induct_true_def)
lemma [induct_simp]: "(x = x) = True"
by (rule simp_thms)
hide_const induct_forall induct_implies induct_equal induct_conj induct_true induct_false
ML_file "~~/src/Tools/induct_tacs.ML"
setup Induct_Tacs.setup
subsubsection {* Coherent logic *}
ML_file "~~/src/Tools/coherent.ML"
ML {*
structure Coherent = Coherent
(
val atomize_elimL = @{thm atomize_elimL};
val atomize_exL = @{thm atomize_exL};
val atomize_conjL = @{thm atomize_conjL};
val atomize_disjL = @{thm atomize_disjL};
val operator_names = [@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}];
);
*}
subsubsection {* Reorienting equalities *}
ML {*
signature REORIENT_PROC =
sig
val add : (term -> bool) -> theory -> theory
val proc : morphism -> Proof.context -> cterm -> thm option
end;
structure Reorient_Proc : REORIENT_PROC =
struct
structure Data = Theory_Data
(
type T = ((term -> bool) * stamp) list;
val empty = [];
val extend = I;
fun merge data : T = Library.merge (eq_snd op =) data;
);
fun add m = Data.map (cons (m, stamp ()));
fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
fun proc phi ctxt ct =
let
val thy = Proof_Context.theory_of ctxt;
in
case Thm.term_of ct of
(_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
| _ => NONE
end;
end;
*}
subsection {* Other simple lemmas and lemma duplicates *}
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
by blast+
lemma 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: 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 (case_tac "xa = x")
apply (drule_tac [3] x = x in fun_cong, simp_all)
done
lemmas eq_sym_conv = eq_commute
lemma nnf_simps:
"(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
"(\<not> \<not>(P)) = P"
by blast+
subsection {* Basic ML bindings *}
ML {*
val FalseE = @{thm FalseE}
val Let_def = @{thm Let_def}
val TrueI = @{thm TrueI}
val allE = @{thm allE}
val allI = @{thm allI}
val all_dupE = @{thm all_dupE}
val arg_cong = @{thm arg_cong}
val box_equals = @{thm box_equals}
val ccontr = @{thm ccontr}
val classical = @{thm classical}
val conjE = @{thm conjE}
val conjI = @{thm conjI}
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val disjCI = @{thm disjCI}
val disjE = @{thm disjE}
val disjI1 = @{thm disjI1}
val disjI2 = @{thm disjI2}
val eq_reflection = @{thm eq_reflection}
val ex1E = @{thm ex1E}
val ex1I = @{thm ex1I}
val ex1_implies_ex = @{thm ex1_implies_ex}
val exE = @{thm exE}
val exI = @{thm exI}
val excluded_middle = @{thm excluded_middle}
val ext = @{thm ext}
val fun_cong = @{thm fun_cong}
val iffD1 = @{thm iffD1}
val iffD2 = @{thm iffD2}
val iffI = @{thm iffI}
val impE = @{thm impE}
val impI = @{thm impI}
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
val mp = @{thm mp}
val notE = @{thm notE}
val notI = @{thm notI}
val not_all = @{thm not_all}
val not_ex = @{thm not_ex}
val not_iff = @{thm not_iff}
val not_not = @{thm not_not}
val not_sym = @{thm not_sym}
val refl = @{thm refl}
val rev_mp = @{thm rev_mp}
val spec = @{thm spec}
val ssubst = @{thm ssubst}
val subst = @{thm subst}
val sym = @{thm sym}
val trans = @{thm trans}
*}
ML_file "Tools/cnf.ML"
subsection {* Code generator setup *}
subsubsection {* Generic code generator preprocessor setup *}
lemma conj_left_cong:
"P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
by (fact arg_cong)
lemma disj_left_cong:
"P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
by (fact arg_cong)
setup {*
Code_Preproc.map_pre (put_simpset HOL_basic_ss)
#> Code_Preproc.map_post (put_simpset HOL_basic_ss)
#> Code_Simp.map_ss (put_simpset HOL_basic_ss
#> Simplifier.add_cong @{thm conj_left_cong} #> Simplifier.add_cong @{thm disj_left_cong})
*}
subsubsection {* Equality *}
class equal =
fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
begin
lemma equal: "equal = (op =)"
by (rule ext equal_eq)+
lemma equal_refl: "equal x x \<longleftrightarrow> True"
unfolding equal by rule+
lemma eq_equal: "(op =) \<equiv> equal"
by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
end
declare eq_equal [symmetric, code_post]
declare eq_equal [code]
setup {*
Code_Preproc.map_pre (fn ctxt =>
ctxt addsimprocs [Simplifier.simproc_global_i @{theory} "equal" [@{term HOL.eq}]
(fn _ => fn Const (_, Type ("fun", [Type _, _])) => SOME @{thm eq_equal} | _ => NONE)])
*}
subsubsection {* Generic code generator foundation *}
text {* Datatype @{typ bool} *}
code_datatype True False
lemma [code]:
shows "False \<and> P \<longleftrightarrow> False"
and "True \<and> P \<longleftrightarrow> P"
and "P \<and> False \<longleftrightarrow> False"
and "P \<and> True \<longleftrightarrow> P" by simp_all
lemma [code]:
shows "False \<or> P \<longleftrightarrow> P"
and "True \<or> P \<longleftrightarrow> True"
and "P \<or> False \<longleftrightarrow> P"
and "P \<or> True \<longleftrightarrow> True" by simp_all
lemma [code]:
shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
and "(True \<longrightarrow> P) \<longleftrightarrow> P"
and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
and "(P \<longrightarrow> True) \<longleftrightarrow> True" by simp_all
text {* More about @{typ prop} *}
lemma [code nbe]:
shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)" by (auto intro!: equal_intr_rule)
lemma Trueprop_code [code]:
"Trueprop True \<equiv> Code_Generator.holds"
by (auto intro!: equal_intr_rule holds)
declare Trueprop_code [symmetric, code_post]
text {* Equality *}
declare simp_thms(6) [code nbe]
instantiation itself :: (type) equal
begin
definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where
"equal_itself x y \<longleftrightarrow> x = y"
instance proof
qed (fact equal_itself_def)
end
lemma equal_itself_code [code]:
"equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
by (simp add: equal)
setup {*
Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> bool"})
*}
lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)" (is "?ofclass \<equiv> ?equal")
proof
assume "PROP ?ofclass"
show "PROP ?equal"
by (tactic {* ALLGOALS (rtac (Thm.unconstrainT @{thm eq_equal})) *})
(fact `PROP ?ofclass`)
next
assume "PROP ?equal"
show "PROP ?ofclass" proof
qed (simp add: `PROP ?equal`)
qed
setup {*
Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>equal \<Rightarrow> 'a \<Rightarrow> bool"})
*}
setup {*
Nbe.add_const_alias @{thm equal_alias_cert}
*}
text {* Cases *}
lemma Let_case_cert:
assumes "CASE \<equiv> (\<lambda>x. Let x f)"
shows "CASE x \<equiv> f x"
using assms by simp_all
setup {*
Code.add_case @{thm Let_case_cert}
#> Code.add_undefined @{const_name undefined}
*}
declare [[code abort: undefined]]
subsubsection {* Generic code generator target languages *}
text {* type @{typ bool} *}
code_printing
type_constructor bool \<rightharpoonup>
(SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
| constant True \<rightharpoonup>
(SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
| constant False \<rightharpoonup>
(SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"
code_reserved SML
bool true false
code_reserved OCaml
bool
code_reserved Scala
Boolean
code_printing
constant Not \<rightharpoonup>
(SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
| constant HOL.conj \<rightharpoonup>
(SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
| constant HOL.disj \<rightharpoonup>
(SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
| constant HOL.implies \<rightharpoonup>
(SML) "!(if (_)/ then (_)/ else true)"
and (OCaml) "!(if (_)/ then (_)/ else true)"
and (Haskell) "!(if (_)/ then (_)/ else True)"
and (Scala) "!(if ((_))/ (_)/ else true)"
| constant If \<rightharpoonup>
(SML) "!(if (_)/ then (_)/ else (_))"
and (OCaml) "!(if (_)/ then (_)/ else (_))"
and (Haskell) "!(if (_)/ then (_)/ else (_))"
and (Scala) "!(if ((_))/ (_)/ else (_))"
code_reserved SML
not
code_reserved OCaml
not
code_identifier
code_module Pure \<rightharpoonup>
(SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL
text {* using built-in Haskell equality *}
code_printing
type_class equal \<rightharpoonup> (Haskell) "Eq"
| constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
| constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="
text {* undefined *}
code_printing
constant undefined \<rightharpoonup>
(SML) "!(raise/ Fail/ \"undefined\")"
and (OCaml) "failwith/ \"undefined\""
and (Haskell) "error/ \"undefined\""
and (Scala) "!sys.error(\"undefined\")"
subsubsection {* Evaluation and normalization by evaluation *}
method_setup eval = {*
let
fun eval_tac ctxt =
let val conv = Code_Runtime.dynamic_holds_conv ctxt
in CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 conv)) ctxt) THEN' rtac TrueI end
in
Scan.succeed (SIMPLE_METHOD' o eval_tac)
end
*} "solve goal by evaluation"
method_setup normalization = {*
Scan.succeed (fn ctxt =>
SIMPLE_METHOD'
(CHANGED_PROP o
(CONVERSION (Nbe.dynamic_conv ctxt)
THEN_ALL_NEW (TRY o rtac TrueI))))
*} "solve goal by normalization"
subsection {* Counterexample Search Units *}
subsubsection {* Quickcheck *}
quickcheck_params [size = 5, iterations = 50]
subsubsection {* Nitpick setup *}
ML {*
structure Nitpick_Unfolds = Named_Thms
(
val name = @{binding nitpick_unfold}
val description = "alternative definitions of constants as needed by Nitpick"
)
structure Nitpick_Simps = Named_Thms
(
val name = @{binding nitpick_simp}
val description = "equational specification of constants as needed by Nitpick"
)
structure Nitpick_Psimps = Named_Thms
(
val name = @{binding nitpick_psimp}
val description = "partial equational specification of constants as needed by Nitpick"
)
structure Nitpick_Choice_Specs = Named_Thms
(
val name = @{binding nitpick_choice_spec}
val description = "choice specification of constants as needed by Nitpick"
)
*}
setup {*
Nitpick_Unfolds.setup
#> Nitpick_Simps.setup
#> Nitpick_Psimps.setup
#> Nitpick_Choice_Specs.setup
*}
declare if_bool_eq_conj [nitpick_unfold, no_atp]
if_bool_eq_disj [no_atp]
subsection {* Preprocessing for the predicate compiler *}
ML {*
structure Predicate_Compile_Alternative_Defs = Named_Thms
(
val name = @{binding code_pred_def}
val description = "alternative definitions of constants for the Predicate Compiler"
)
structure Predicate_Compile_Inline_Defs = Named_Thms
(
val name = @{binding code_pred_inline}
val description = "inlining definitions for the Predicate Compiler"
)
structure Predicate_Compile_Simps = Named_Thms
(
val name = @{binding code_pred_simp}
val description = "simplification rules for the optimisations in the Predicate Compiler"
)
*}
setup {*
Predicate_Compile_Alternative_Defs.setup
#> Predicate_Compile_Inline_Defs.setup
#> Predicate_Compile_Simps.setup
*}
subsection {* Legacy tactics and ML bindings *}
ML {*
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
local
fun wrong_prem (Const (@{const_name All}, _) $ Abs (_, _, t)) = wrong_prem t
| wrong_prem (Bound _) = true
| wrong_prem _ = false;
val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
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;
local
val nnf_ss =
simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms simp_thms nnf_simps});
in
fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
end
*}
hide_const (open) eq equal
end