src/HOL/HOL.thy
 author paulson Tue, 23 Sep 2003 15:42:01 +0200 changeset 14201 7ad7ab89c402 parent 13764 3e180bf68496 child 14208 144f45277d5a permissions -rw-r--r--
some basic new lemmas
```
(*  Title:      HOL/HOL.thy
ID:         \$Id\$
Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)

header {* The basis of Higher-Order Logic *}

theory HOL = CPure

subsection {* Primitive logic *}

subsubsection {* Core syntax *}

classes type < logic
defaultsort type

global

typedecl bool

arities
bool :: type
fun :: (type, type) type

judgment
Trueprop      :: "bool => prop"                   ("(_)" 5)

consts
Not           :: "bool => bool"                   ("~ _"  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
"_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)
"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           :: "bool => bool"                        ("\<not> _"  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 < 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"              ("- _"  80)
*             :: "['a::times, 'a] => 'a"          (infixl 70)

syntax
"_index1"  :: index    ("\<^sub>1")
translations
(index) "\<^sub>1" == "_index 1"

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]

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 [CPure.elim!] = disjE iffE FalseE conjE exE
and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
and [CPure.elim 2] = allE notE' impE'
and [CPure.intro] = exI disjI2 disjI1

lemmas [trans] = trans
and [sym] = sym not_sym
and [CPure.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_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 *}

setup hypsubst_setup

ML_setup {*
Context.>> (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

subsubsection {* 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
"(P ~= Q) = (P = (~Q))"
"(P | ~P) = True"    "(~P | P) = True"
"((~P) = (~Q)) = (P=Q)"
"(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 (subst if_P)
prefer 3 apply (subst if_not_P)
apply blast+
done

lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
apply (subst split_if)
apply blast
done

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"
apply (subst split_if)
apply blast
done

lemma if_eq_cancel: "(if x = y then y else x) = x"
apply (subst split_if)
apply blast
done

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 (subst split_if)
apply blast
done

lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules

subsubsection {* Actual Installation of the Simplifier *}

use "simpdata.ML"
setup Simplifier.setup
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
setup Splitter.setup setup Clasimp.setup

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 (rule ccontr)
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  x = x in fun_cong)
apply 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"]])

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) 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"

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

subsection {* Order signatures and orders *}

axclass
ord < type

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 (xsymbols)
"op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
"op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)

subsubsection {* Monotonicity *}

locale mono =
fixes f
assumes mono: "A <= B ==> f A <= f B"

lemmas monoI [intro?] = mono.intro
and monoD [dest?] = mono.mono

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"

lemma min_of_mono:
"ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"

lemma max_leastL: "(!!x. least <= x) ==> max least x = x"

lemma max_of_mono:
"ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"

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 [iff]: "~ x < (x::'a::order)"

lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
apply blast
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"

text {* Asymmetry. *}

lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"

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 (blast intro: order_trans order_antisym)
done

lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
apply (blast intro: order_trans order_antisym)
done

lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
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 (blast intro: order_antisym)
done

lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
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 (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 (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 (insert linorder_linear)
apply (blast intro: order_antisym)
done

lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
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 blast
done

subsubsection "Min and max on (linear) orders"

lemma min_same [simp]: "min (x::'a::order) x = x"

lemma max_same [simp]: "max (x::'a::order) x = x"

lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
apply (insert linorder_linear)
apply (blast intro: order_trans)
done

lemma le_maxI1: "(x::'a::linorder) <= max x y"

lemma le_maxI2: "(y::'a::linorder) <= max x y"
-- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}

lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
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 (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 (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 @{text blast} in MiniML *}
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 (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 (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 (insert linorder_less_linear)
apply (blast intro: order_less_trans)
done

lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
apply(rule conjI)
apply(blast intro:order_trans)
apply(blast dest: order_less_trans order_le_less_trans)
done

lemma max_commute: "!!x::'a::linorder. max x y = max y x"
apply(rule conjI)
apply(blast intro:order_antisym)
apply(blast dest: order_less_trans)
done

lemmas max_ac = max_assoc max_commute
mk_left_commute[of max,OF max_assoc max_commute]

lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
apply(rule conjI)
apply(blast intro:order_trans)
apply(blast dest: order_less_trans order_le_less_trans)
done

lemma min_commute: "!!x::'a::linorder. min x y = min y x"
apply(rule conjI)
apply(blast intro:order_antisym)
apply(blast dest: order_less_trans)
done

lemmas min_ac = min_assoc min_commute
mk_left_commute[of min,OF min_assoc min_commute]

lemma split_min:
"P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"

lemma split_max:
"P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"

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 (xsymbols)
"_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
```