(* Title: HOL/HOL.thy
ID: $Id$
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
header {* The basis of Higher-Order Logic *}
theory HOL = CPure
files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
subsection {* Primitive logic *}
subsubsection {* Core syntax *}
global
classes "term" < logic
defaultsort "term"
typedecl bool
arities
bool :: "term"
fun :: ("term", "term") "term"
judgment
Trueprop :: "bool => prop" ("(_)" 5)
consts
Not :: "bool => bool" ("~ _" [40] 40)
True :: bool
False :: bool
If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
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
subsubsection {* Additional concrete syntax *}
nonterminals
letbinds letbind
case_syn cases_syn
syntax
~= :: "['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)"
syntax ("" output)
"=" :: "['a, 'a] => bool" (infix 50)
"~=" :: "['a, 'a] => bool" (infix 50)
syntax (symbols)
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 "\<midarrow>\<rightarrow>" 25)
"op ~=" :: "['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 (symbols output)
"op ~=" :: "['a, 'a] => bool" (infix "\<noteq>" 50)
syntax (xsymbols)
"op -->" :: "[bool, bool] => bool" (infixr "\<longrightarrow>" 25)
syntax (HTML output)
Not :: "bool => bool" ("\<not> _" [40] 40)
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)"
subst: "[| s = t; P(s) |] ==> P(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"
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)"
arbitrary_def: "False ==> arbitrary == (THE x. False)"
-- {* @{term arbitrary} is completely unspecified, but is made to appear as a
definition syntactically *}
subsubsection {* Generic algebraic operations *}
axclass zero < "term"
axclass one < "term"
axclass plus < "term"
axclass minus < "term"
axclass times < "term"
axclass inverse < "term"
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)
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>")
axclass plus_ac0 < plus, zero
commute: "x + y = y + x"
assoc: "(x + y) + z = x + (y + z)"
zero: "0 + x = x"
subsection {* Theory and package setup *}
subsubsection {* Basic lemmas *}
use "HOL_lemmas.ML"
theorems case_split = case_split_thm [case_names True False]
declare trans [trans]
declare impE [CPure.elim] iffD1 [CPure.elim] iffD2 [CPure.elim]
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_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
subsubsection {* Classical Reasoner setup *}
use "cladata.ML"
setup hypsubst_setup
declare atomize_all [symmetric, rulify] atomize_imp [symmetric, rulify]
setup Classical.setup
setup clasetup
declare ext [intro?]
declare disjI1 [elim?] disjI2 [elim?] ex1_implies_ex [elim?] sym [elim?]
use "blastdata.ML"
setup Blast.setup
subsubsection {* Simplifier setup *}
use "simpdata.ML"
setup Simplifier.setup
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
setup Splitter.setup setup Clasimp.setup
subsubsection {* 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) blast
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) blast
lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
by (simp only: atomize_imp atomize_eq induct_conj_def) (rule equal_intr_rule, blast+)
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
by (simp add: induct_implies_def)
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq
lemmas induct_rulify1 = induct_atomize [symmetric, standard]
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";
end);
*}
setup InductMethod.setup
subsection {* Order signatures and orders *}
axclass
ord < "term"
syntax
"op <" :: "['a::ord, 'a] => bool" ("op <")
"op <=" :: "['a::ord, 'a] => bool" ("op <=")
global
consts
"op <" :: "['a::ord, 'a] => bool" ("(_/ < _)" [50, 51] 50)
"op <=" :: "['a::ord, 'a] => bool" ("(_/ <= _)" [50, 51] 50)
local
syntax (symbols)
"op <=" :: "['a::ord, 'a] => bool" ("op \<le>")
"op <=" :: "['a::ord, 'a] => bool" ("(_/ \<le> _)" [50, 51] 50)
(*Tell blast about overloading of < and <= to reduce the risk of
its applying a rule for the wrong type*)
ML {*
Blast.overloaded ("op <" , domain_type);
Blast.overloaded ("op <=", domain_type);
*}
subsubsection {* Monotonicity *}
constdefs
mono :: "['a::ord => 'b::ord] => bool"
"mono f == ALL A B. A <= B --> f A <= f B"
lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
by (unfold mono_def) blast
lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
by (unfold mono_def) blast
constdefs
min :: "['a::ord, 'a] => 'a"
"min a b == (if a <= b then a else b)"
max :: "['a::ord, 'a] => 'a"
"max a b == (if a <= b then b else a)"
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
by (simp add: min_def)
lemma min_of_mono:
"ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
by (simp add: min_def)
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
by (simp add: max_def)
lemma max_of_mono:
"ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
by (simp add: max_def)
subsubsection "Orders"
axclass order < ord
order_refl [iff]: "x <= x"
order_trans: "x <= y ==> y <= z ==> x <= z"
order_antisym: "x <= y ==> y <= x ==> x = y"
order_less_le: "(x < y) = (x <= y & x ~= y)"
text {* Reflexivity. *}
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
-- {* This form is useful with the classical reasoner. *}
apply (erule ssubst)
apply (rule order_refl)
done
lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
by (simp add: order_less_le)
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
apply (simp add: order_less_le)
apply (blast intro!: order_refl)
done
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
by (simp add: order_less_le)
text {* Asymmetry. *}
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
by (simp add: order_less_le order_antisym)
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
apply (drule order_less_not_sym)
apply (erule contrapos_np)
apply simp
done
text {* Transitivity. *}
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
apply (simp add: order_less_le)
apply (blast intro: order_trans order_antisym)
done
text {* Useful for simplification, but too risky to include by default. *}
lemma order_less_imp_not_less: "(x::'a::order) < y ==> (~ y < x) = True"
by (blast elim: order_less_asym)
lemma order_less_imp_triv: "(x::'a::order) < y ==> (y < x --> P) = True"
by (blast elim: order_less_asym)
lemma order_less_imp_not_eq: "(x::'a::order) < y ==> (x = y) = False"
by auto
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==> (y = x) = False"
by auto
text {* Other operators. *}
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
apply (simp add: min_def)
apply (blast intro: order_antisym)
done
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
apply (simp add: max_def)
apply (blast intro: order_antisym)
done
subsubsection {* Least value operator *}
constdefs
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10)
"Least P == THE x. P x & (ALL y. P y --> x <= y)"
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
lemma LeastI2:
"[| P (x::'a::order);
!!y. P y ==> x <= y;
!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
==> Q (Least P)";
apply (unfold Least_def)
apply (rule theI2)
apply (blast intro: order_antisym)+
done
lemma Least_equality:
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k";
apply (simp add: Least_def)
apply (rule the_equality)
apply (auto intro!: order_antisym)
done
subsubsection "Linear / total orders"
axclass linorder < order
linorder_linear: "x <= y | y <= x"
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
apply (simp add: order_less_le)
apply (insert linorder_linear)
apply blast
done
lemma linorder_cases [case_names less equal greater]:
"((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
apply (insert linorder_less_linear)
apply blast
done
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
apply (simp add: order_less_le)
apply (insert linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
apply (simp add: order_less_le)
apply (insert linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
apply (cut_tac x = x and y = y in linorder_less_linear)
apply auto
done
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
apply (simp add: linorder_neq_iff)
apply blast
done
subsubsection "Min and max on (linear) orders"
lemma min_same [simp]: "min (x::'a::order) x = x"
by (simp add: min_def)
lemma max_same [simp]: "max (x::'a::order) x = x"
by (simp add: max_def)
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
apply (simp add: max_def)
apply (insert linorder_linear)
apply (blast intro: order_trans)
done
lemma le_maxI1: "(x::'a::linorder) <= max x y"
by (simp add: le_max_iff_disj)
lemma le_maxI2: "(y::'a::linorder) <= max x y"
-- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
by (simp add: le_max_iff_disj)
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
apply (simp add: max_def order_le_less)
apply (insert linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma max_le_iff_conj [simp]:
"!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
apply (simp add: max_def)
apply (insert linorder_linear)
apply (blast intro: order_trans)
done
lemma max_less_iff_conj [simp]:
"!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
apply (simp add: order_le_less max_def)
apply (insert linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma le_min_iff_conj [simp]:
"!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
-- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
apply (simp add: min_def)
apply (insert linorder_linear)
apply (blast intro: order_trans)
done
lemma min_less_iff_conj [simp]:
"!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
apply (simp add: order_le_less min_def)
apply (insert linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
apply (simp add: min_def)
apply (insert linorder_linear)
apply (blast intro: order_trans)
done
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
apply (simp add: min_def order_le_less)
apply (insert linorder_less_linear)
apply (blast intro: order_less_trans)
done
lemma split_min:
"P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
by (simp add: min_def)
lemma split_max:
"P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
by (simp add: max_def)
subsubsection "Bounded quantifiers"
syntax
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
syntax (symbols)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
syntax (HOL)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
translations
"ALL x<y. P" => "ALL x. x < y --> P"
"EX x<y. P" => "EX x. x < y & P"
"ALL x<=y. P" => "ALL x. x <= y --> P"
"EX x<=y. P" => "EX x. x <= y & P"
end