# Theory Orderings

```(*  Title:      HOL/Orderings.thy
Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)

section ‹Abstract orderings›

theory Orderings
imports HOL
keywords "print_orders" :: diag
begin

subsection ‹Abstract ordering›

locale partial_preordering =
fixes less_eq :: ‹'a ⇒ 'a ⇒ bool› (infix ‹❙≤› 50)
assumes refl: ‹a ❙≤ a› ― ‹not ‹iff›: makes problems due to multiple (dual) interpretations›
and trans: ‹a ❙≤ b ⟹ b ❙≤ c ⟹ a ❙≤ c›

locale preordering = partial_preordering +
fixes less :: ‹'a ⇒ 'a ⇒ bool› (infix ‹❙<› 50)
assumes strict_iff_not: ‹a ❙< b ⟷ a ❙≤ b ∧ ¬ b ❙≤ a›
begin

lemma strict_implies_order:
‹a ❙< b ⟹ a ❙≤ b›

lemma irrefl: ― ‹not ‹iff›: makes problems due to multiple (dual) interpretations›
‹¬ a ❙< a›

lemma asym:
‹a ❙< b ⟹ b ❙< a ⟹ False›

lemma strict_trans1:
‹a ❙≤ b ⟹ b ❙< c ⟹ a ❙< c›
by (auto simp add: strict_iff_not intro: trans)

lemma strict_trans2:
‹a ❙< b ⟹ b ❙≤ c ⟹ a ❙< c›
by (auto simp add: strict_iff_not intro: trans)

lemma strict_trans:
‹a ❙< b ⟹ b ❙< c ⟹ a ❙< c›
by (auto intro: strict_trans1 strict_implies_order)

end

lemma preordering_strictI: ― ‹Alternative introduction rule with bias towards strict order›
fixes less_eq (infix ‹❙≤› 50)
and less (infix ‹❙<› 50)
assumes less_eq_less: ‹⋀a b. a ❙≤ b ⟷ a ❙< b ∨ a = b›
assumes asym: ‹⋀a b. a ❙< b ⟹ ¬ b ❙< a›
assumes irrefl: ‹⋀a. ¬ a ❙< a›
assumes trans: ‹⋀a b c. a ❙< b ⟹ b ❙< c ⟹ a ❙< c›
shows ‹preordering (❙≤) (❙<)›
proof
fix a b
show ‹a ❙< b ⟷ a ❙≤ b ∧ ¬ b ❙≤ a›
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show ‹a ❙≤ a›
next
fix a b c
assume ‹a ❙≤ b› and ‹b ❙≤ c› then show ‹a ❙≤ c›
by (auto simp add: less_eq_less intro: trans)
qed

lemma preordering_dualI:
fixes less_eq (infix ‹❙≤› 50)
and less (infix ‹❙<› 50)
assumes ‹preordering (λa b. b ❙≤ a) (λa b. b ❙< a)›
shows ‹preordering (❙≤) (❙<)›
proof -
from assms interpret preordering ‹λa b. b ❙≤ a› ‹λa b. b ❙< a› .
show ?thesis
by standard (auto simp: strict_iff_not refl intro: trans)
qed

locale ordering = partial_preordering +
fixes less :: ‹'a ⇒ 'a ⇒ bool› (infix ‹❙<› 50)
assumes strict_iff_order: ‹a ❙< b ⟷ a ❙≤ b ∧ a ≠ b›
assumes antisym: ‹a ❙≤ b ⟹ b ❙≤ a ⟹ a = b›
begin

sublocale preordering ‹(❙≤)› ‹(❙<)›
proof
show ‹a ❙< b ⟷ a ❙≤ b ∧ ¬ b ❙≤ a› for a b
by (auto simp add: strict_iff_order intro: antisym)
qed

lemma strict_implies_not_eq:
‹a ❙< b ⟹ a ≠ b›

lemma not_eq_order_implies_strict:
‹a ≠ b ⟹ a ❙≤ b ⟹ a ❙< b›

lemma order_iff_strict:
‹a ❙≤ b ⟷ a ❙< b ∨ a = b›
by (auto simp add: strict_iff_order refl)

lemma eq_iff: ‹a = b ⟷ a ❙≤ b ∧ b ❙≤ a›
by (auto simp add: refl intro: antisym)

end

lemma ordering_strictI: ― ‹Alternative introduction rule with bias towards strict order›
fixes less_eq (infix ‹❙≤› 50)
and less (infix ‹❙<› 50)
assumes less_eq_less: ‹⋀a b. a ❙≤ b ⟷ a ❙< b ∨ a = b›
assumes asym: ‹⋀a b. a ❙< b ⟹ ¬ b ❙< a›
assumes irrefl: ‹⋀a. ¬ a ❙< a›
assumes trans: ‹⋀a b c. a ❙< b ⟹ b ❙< c ⟹ a ❙< c›
shows ‹ordering (❙≤) (❙<)›
proof
fix a b
show ‹a ❙< b ⟷ a ❙≤ b ∧ a ≠ b›
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show ‹a ❙≤ a›
next
fix a b c
assume ‹a ❙≤ b› and ‹b ❙≤ c› then show ‹a ❙≤ c›
by (auto simp add: less_eq_less intro: trans)
next
fix a b
assume ‹a ❙≤ b› and ‹b ❙≤ a› then show ‹a = b›
by (auto simp add: less_eq_less asym)
qed

lemma ordering_dualI:
fixes less_eq (infix ‹❙≤› 50)
and less (infix ‹❙<› 50)
assumes ‹ordering (λa b. b ❙≤ a) (λa b. b ❙< a)›
shows ‹ordering (❙≤) (❙<)›
proof -
from assms interpret ordering ‹λa b. b ❙≤ a› ‹λa b. b ❙< a› .
show ?thesis
by standard (auto simp: strict_iff_order refl intro: antisym trans)
qed

locale ordering_top = ordering +
fixes top :: ‹'a›  (‹❙⊤›)
assumes extremum [simp]: ‹a ❙≤ ❙⊤›
begin

lemma extremum_uniqueI:
‹❙⊤ ❙≤ a ⟹ a = ❙⊤›
by (rule antisym) auto

lemma extremum_unique:
‹❙⊤ ❙≤ a ⟷ a = ❙⊤›
by (auto intro: antisym)

lemma extremum_strict [simp]:
‹¬ (❙⊤ ❙< a)›
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)

lemma not_eq_extremum:
‹a ≠ ❙⊤ ⟷ a ❙< ❙⊤›
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)

end

subsection ‹Syntactic orders›

class ord =
fixes less_eq :: "'a ⇒ 'a ⇒ bool"
and less :: "'a ⇒ 'a ⇒ bool"
begin

notation
less_eq  ("'(≤')") and
less_eq  ("(_/ ≤ _)"  [51, 51] 50) and
less  ("'(<')") and
less  ("(_/ < _)"  [51, 51] 50)

abbreviation (input)
greater_eq  (infix "≥" 50)
where "x ≥ y ≡ y ≤ x"

abbreviation (input)
greater  (infix ">" 50)
where "x > y ≡ y < x"

notation (ASCII)
less_eq  ("'(<=')") and
less_eq  ("(_/ <= _)" [51, 51] 50)

notation (input)
greater_eq  (infix ">=" 50)

end

subsection ‹Quasi orders›

class preorder = ord +
assumes less_le_not_le: "x < y ⟷ x ≤ y ∧ ¬ (y ≤ x)"
and order_refl [iff]: "x ≤ x"
and order_trans: "x ≤ y ⟹ y ≤ z ⟹ x ≤ z"
begin

sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater
proof -
interpret preordering less_eq less
by standard (auto intro: order_trans simp add: less_le_not_le)
show ‹preordering less_eq less›
by (fact preordering_axioms)
then show ‹preordering greater_eq greater›
by (rule preordering_dualI)
qed

text ‹Reflexivity.›

lemma eq_refl: "x = y ⟹ x ≤ y"
― ‹This form is useful with the classical reasoner.›
by (erule ssubst) (rule order_refl)

lemma less_irrefl [iff]: "¬ x < x"

lemma less_imp_le: "x < y ⟹ x ≤ y"

text ‹Asymmetry.›

lemma less_not_sym: "x < y ⟹ ¬ (y < x)"

lemma less_asym: "x < y ⟹ (¬ P ⟹ y < x) ⟹ P"
by (drule less_not_sym, erule contrapos_np) simp

text ‹Transitivity.›

lemma less_trans: "x < y ⟹ y < z ⟹ x < z"
by (auto simp add: less_le_not_le intro: order_trans)

lemma le_less_trans: "x ≤ y ⟹ y < z ⟹ x < z"
by (auto simp add: less_le_not_le intro: order_trans)

lemma less_le_trans: "x < y ⟹ y ≤ z ⟹ x < z"
by (auto simp add: less_le_not_le intro: order_trans)

text ‹Useful for simplification, but too risky to include by default.›

lemma less_imp_not_less: "x < y ⟹ (¬ y < x) ⟷ True"
by (blast elim: less_asym)

lemma less_imp_triv: "x < y ⟹ (y < x ⟶ P) ⟷ True"
by (blast elim: less_asym)

text ‹Transitivity rules for calculational reasoning›

lemma less_asym': "a < b ⟹ b < a ⟹ P"
by (rule less_asym)

text ‹Dual order›

lemma dual_preorder:
‹class.preorder (≥) (>)›
by standard (auto simp add: less_le_not_le intro: order_trans)

end

lemma preordering_preorderI:
‹class.preorder (❙≤) (❙<)› if ‹preordering (❙≤) (❙<)›
for less_eq (infix ‹❙≤› 50) and less (infix ‹❙<› 50)
proof -
from that interpret preordering ‹(❙≤)› ‹(❙<)› .
show ?thesis
by standard (auto simp add: strict_iff_not refl intro: trans)
qed

subsection ‹Partial orders›

class order = preorder +
assumes order_antisym: "x ≤ y ⟹ y ≤ x ⟹ x = y"
begin

lemma less_le: "x < y ⟷ x ≤ y ∧ x ≠ y"
by (auto simp add: less_le_not_le intro: order_antisym)

sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
proof -
interpret ordering less_eq less
by standard (auto intro: order_antisym order_trans simp add: less_le)
show "ordering less_eq less"
by (fact ordering_axioms)
then show "ordering greater_eq greater"
by (rule ordering_dualI)
qed

text ‹Reflexivity.›

lemma le_less: "x ≤ y ⟷ x < y ∨ x = y"
― ‹NOT suitable for iff, since it can cause PROOF FAILED.›
by (fact order.order_iff_strict)

lemma le_imp_less_or_eq: "x ≤ y ⟹ x < y ∨ x = y"

text ‹Useful for simplification, but too risky to include by default.›

lemma less_imp_not_eq: "x < y ⟹ (x = y) ⟷ False"
by auto

lemma less_imp_not_eq2: "x < y ⟹ (y = x) ⟷ False"
by auto

text ‹Transitivity rules for calculational reasoning›

lemma neq_le_trans: "a ≠ b ⟹ a ≤ b ⟹ a < b"
by (fact order.not_eq_order_implies_strict)

lemma le_neq_trans: "a ≤ b ⟹ a ≠ b ⟹ a < b"
by (rule order.not_eq_order_implies_strict)

text ‹Asymmetry.›

lemma order_eq_iff: "x = y ⟷ x ≤ y ∧ y ≤ x"
by (fact order.eq_iff)

lemma antisym_conv: "y ≤ x ⟹ x ≤ y ⟷ x = y"

lemma less_imp_neq: "x < y ⟹ x ≠ y"
by (fact order.strict_implies_not_eq)

lemma antisym_conv1: "¬ x < y ⟹ x ≤ y ⟷ x = y"

lemma antisym_conv2: "x ≤ y ⟹ ¬ x < y ⟷ x = y"

lemma leD: "y ≤ x ⟹ ¬ x < y"
by (auto simp: less_le order.antisym)

text ‹Least value operator›

definition (in ord)
Least :: "('a ⇒ bool) ⇒ 'a" (binder "LEAST " 10) where
"Least P = (THE x. P x ∧ (∀y. P y ⟶ x ≤ y))"

lemma Least_equality:
assumes "P x"
and "⋀y. P y ⟹ x ≤ y"
shows "Least P = x"
unfolding Least_def by (rule the_equality)
(blast intro: assms order.antisym)+

lemma LeastI2_order:
assumes "P x"
and "⋀y. P y ⟹ x ≤ y"
and "⋀x. P x ⟹ ∀y. P y ⟶ x ≤ y ⟹ Q x"
shows "Q (Least P)"
unfolding Least_def by (rule theI2)
(blast intro: assms order.antisym)+

lemma Least_ex1:
assumes   "∃!x. P x ∧ (∀y. P y ⟶ x ≤ y)"
shows     Least1I: "P (Least P)" and Least1_le: "P z ⟹ Least P ≤ z"
using     theI'[OF assms]
unfolding Least_def
by        auto

text ‹Greatest value operator›

definition Greatest :: "('a ⇒ bool) ⇒ 'a" (binder "GREATEST " 10) where
"Greatest P = (THE x. P x ∧ (∀y. P y ⟶ x ≥ y))"

lemma GreatestI2_order:
"⟦ P x;
⋀y. P y ⟹ x ≥ y;
⋀x. ⟦ P x; ∀y. P y ⟶ x ≥ y ⟧ ⟹ Q x ⟧
⟹ Q (Greatest P)"
unfolding Greatest_def
by (rule theI2) (blast intro: order.antisym)+

lemma Greatest_equality:
"⟦ P x;  ⋀y. P y ⟹ x ≥ y ⟧ ⟹ Greatest P = x"
unfolding Greatest_def
by (rule the_equality) (blast intro: order.antisym)+

end

lemma ordering_orderI:
fixes less_eq (infix "❙≤" 50)
and less (infix "❙<" 50)
assumes "ordering less_eq less"
shows "class.order less_eq less"
proof -
from assms interpret ordering less_eq less .
show ?thesis
by standard (auto intro: antisym trans simp add: refl strict_iff_order)
qed

lemma order_strictI:
fixes less (infix "❙<" 50)
and less_eq (infix "❙≤" 50)
assumes "⋀a b. a ❙≤ b ⟷ a ❙< b ∨ a = b"
assumes "⋀a b. a ❙< b ⟹ ¬ b ❙< a"
assumes "⋀a. ¬ a ❙< a"
assumes "⋀a b c. a ❙< b ⟹ b ❙< c ⟹ a ❙< c"
shows "class.order less_eq less"
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)

context order
begin

text ‹Dual order›

lemma dual_order:
"class.order (≥) (>)"
using dual_order.ordering_axioms by (rule ordering_orderI)

end

subsection ‹Linear (total) orders›

class linorder = order +
assumes linear: "x ≤ y ∨ y ≤ x"
begin

lemma less_linear: "x < y ∨ x = y ∨ y < x"
unfolding less_le using less_le linear by blast

lemma le_less_linear: "x ≤ y ∨ y < x"

lemma le_cases [case_names le ge]:
"(x ≤ y ⟹ P) ⟹ (y ≤ x ⟹ P) ⟹ P"
using linear by blast

lemma (in linorder) le_cases3:
"⟦⟦x ≤ y; y ≤ z⟧ ⟹ P; ⟦y ≤ x; x ≤ z⟧ ⟹ P; ⟦x ≤ z; z ≤ y⟧ ⟹ P;
⟦z ≤ y; y ≤ x⟧ ⟹ P; ⟦y ≤ z; z ≤ x⟧ ⟹ P; ⟦z ≤ x; x ≤ y⟧ ⟹ P⟧ ⟹ P"
by (blast intro: le_cases)

lemma linorder_cases [case_names less equal greater]:
"(x < y ⟹ P) ⟹ (x = y ⟹ P) ⟹ (y < x ⟹ P) ⟹ P"
using less_linear by blast

lemma linorder_wlog[case_names le sym]:
"(⋀a b. a ≤ b ⟹ P a b) ⟹ (⋀a b. P b a ⟹ P a b) ⟹ P a b"
by (cases rule: le_cases[of a b]) blast+

lemma not_less: "¬ x < y ⟷ y ≤ x"
unfolding less_le
using linear by (blast intro: order.antisym)

lemma not_less_iff_gr_or_eq: "¬(x < y) ⟷ (x > y ∨ x = y)"

lemma not_le: "¬ x ≤ y ⟷ y < x"
unfolding less_le
using linear by (blast intro: order.antisym)

lemma neq_iff: "x ≠ y ⟷ x < y ∨ y < x"
by (cut_tac x = x and y = y in less_linear, auto)

lemma neqE: "x ≠ y ⟹ (x < y ⟹ R) ⟹ (y < x ⟹ R) ⟹ R"

lemma antisym_conv3: "¬ y < x ⟹ ¬ x < y ⟷ x = y"
by (blast intro: order.antisym dest: not_less [THEN iffD1])

lemma leI: "¬ x < y ⟹ y ≤ x"
unfolding not_less .

lemma not_le_imp_less: "¬ y ≤ x ⟹ x < y"
unfolding not_le .

lemma linorder_less_wlog[case_names less refl sym]:
"⟦⋀a b. a < b ⟹ P a b;  ⋀a. P a a;  ⋀a b. P b a ⟹ P a b⟧ ⟹ P a b"
using antisym_conv3 by blast

text ‹Dual order›

lemma dual_linorder:
"class.linorder (≥) (>)"
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)

end

text ‹Alternative introduction rule with bias towards strict order›

lemma linorder_strictI:
fixes less_eq (infix "❙≤" 50)
and less (infix "❙<" 50)
assumes "class.order less_eq less"
assumes trichotomy: "⋀a b. a ❙< b ∨ a = b ∨ b ❙< a"
shows "class.linorder less_eq less"
proof -
interpret order less_eq less
by (fact ‹class.order less_eq less›)
show ?thesis
proof
fix a b
show "a ❙≤ b ∨ b ❙≤ a"
using trichotomy by (auto simp add: le_less)
qed
qed

subsection ‹Reasoning tools setup›

ML_file ‹~~/src/Provers/order_procedure.ML›
ML_file ‹~~/src/Provers/order_tac.ML›

ML ‹
structure Logic_Signature : LOGIC_SIGNATURE = struct
val mk_Trueprop = HOLogic.mk_Trueprop
val dest_Trueprop = HOLogic.dest_Trueprop
val Trueprop_conv = HOLogic.Trueprop_conv
val Not = HOLogic.Not
val conj = HOLogic.conj
val disj = HOLogic.disj

val notI = @{thm notI}
val ccontr = @{thm ccontr}
val conjI = @{thm conjI}
val conjE = @{thm conjE}
val disjE = @{thm disjE}

val not_not_conv = Conv.rewr_conv @{thm eq_reflection[OF not_not]}
val de_Morgan_conj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]}
val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_disj]}
val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]}
val conj_disj_distribR_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribR]}
end

structure HOL_Base_Order_Tac = Base_Order_Tac(
structure Logic_Sig = Logic_Signature;
(* Exclude types with specialised solvers. *)
val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic.realT]
)

structure HOL_Order_Tac = Order_Tac(structure Base_Tac = HOL_Base_Order_Tac)

fun print_orders ctxt0 =
let
val ctxt = Config.put show_sorts true ctxt0
val orders = HOL_Order_Tac.Data.get (Context.Proof ctxt)
fun pretty_term t = Pretty.block
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
Pretty.str "::", Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt (type_of t)), Pretty.brk 1]
fun pretty_order ({kind = kind, ops = ops, ...}, _) =
Pretty.block ([Pretty.str (@{make_string} kind), Pretty.str ":", Pretty.brk 1]
@ map pretty_term ops)
in
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders))
end

val _ =
Outer_Syntax.command \<^command_keyword>‹print_orders›
"print order structures available to order reasoner"
(Scan.succeed (Toplevel.keep (print_orders o Toplevel.context_of)))

›

method_setup order = ‹
Scan.succeed (fn ctxt => SIMPLE_METHOD' (HOL_Order_Tac.tac [] ctxt))
› "partial and linear order reasoner"

text ‹Declarations to set up transitivity reasoner of partial and linear orders.›

context order
begin

lemma nless_le: "(¬ a < b) ⟷ (¬ a ≤ b) ∨ a = b"
using local.dual_order.order_iff_strict by blast

local_setup ‹
HOL_Order_Tac.declare_order {
ops = {eq = @{term ‹(=) :: 'a ⇒ 'a ⇒ bool›}, le = @{term ‹(≤)›}, lt = @{term ‹(<)›}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF nless_le]}}
}
›

end

context linorder
begin

lemma nle_le: "(¬ a ≤ b) ⟷ b ≤ a ∧ b ≠ a"
using not_le less_le by simp

local_setup ‹
HOL_Order_Tac.declare_linorder {
ops = {eq = @{term ‹(=) :: 'a ⇒ 'a ⇒ bool›}, le = @{term ‹(≤)›}, lt = @{term ‹(<)›}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF not_less]},
nle_le = @{thm eq_reflection[OF nle_le]}}
}
›

end

setup ‹
map_theory_simpset (fn ctxt0 => ctxt0 addSolver
mk_solver "partial and linear orders" (fn ctxt => HOL_Order_Tac.tac (Simplifier.prems_of ctxt) ctxt))
›

ML ‹
local
fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
in

fun antisym_le_simproc ctxt ct =
(case Thm.term_of ct of
(le as Const (_, T)) \$ r \$ s =>
(let
val prems = Simplifier.prems_of ctxt;
val less = Const (\<^const_name>‹less›, T);
val t = HOLogic.mk_Trueprop(le \$ s \$ r);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop(HOLogic.Not \$ (less \$ r \$ s)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
end handle THM _ => NONE)
| _ => NONE);

fun antisym_less_simproc ctxt ct =
(case Thm.term_of ct of
NotC \$ ((less as Const(_,T)) \$ r \$ s) =>
(let
val prems = Simplifier.prems_of ctxt;
val le = Const (\<^const_name>‹less_eq›, T);
val t = HOLogic.mk_Trueprop(le \$ r \$ s);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop (NotC \$ (less \$ s \$ r)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2})))
end handle THM _ => NONE)
| _ => NONE);

end;
›

simproc_setup antisym_le ("(x::'a::order) ≤ y") = "K antisym_le_simproc"
simproc_setup antisym_less ("¬ (x::'a::linorder) < y") = "K antisym_less_simproc"

subsection ‹Bounded quantifiers›

syntax (ASCII)
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)

"_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)

"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3ALL _~=_./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3EX _~=_./ _)"  [0, 0, 10] 10)

syntax
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3∀_<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3∃_<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3∀_≤_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3∃_≤_./ _)" [0, 0, 10] 10)

"_All_greater" :: "[idt, 'a, bool] => bool"    ("(3∀_>_./ _)"  [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3∃_>_./ _)"  [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3∀_≥_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3∃_≥_./ _)" [0, 0, 10] 10)

"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3∀_≠_./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3∃_≠_./ _)"  [0, 0, 10] 10)

syntax (input)
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3! _~=_./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3? _~=_./ _)"  [0, 0, 10] 10)

translations
"∀x<y. P" ⇀ "∀x. x < y ⟶ P"
"∃x<y. P" ⇀ "∃x. x < y ∧ P"
"∀x≤y. P" ⇀ "∀x. x ≤ y ⟶ P"
"∃x≤y. P" ⇀ "∃x. x ≤ y ∧ P"
"∀x>y. P" ⇀ "∀x. x > y ⟶ P"
"∃x>y. P" ⇀ "∃x. x > y ∧ P"
"∀x≥y. P" ⇀ "∀x. x ≥ y ⟶ P"
"∃x≥y. P" ⇀ "∃x. x ≥ y ∧ P"
"∀x≠y. P" ⇀ "∀x. x ≠ y ⟶ P"
"∃x≠y. P" ⇀ "∃x. x ≠ y ∧ P"

print_translation ‹
let
val All_binder = Mixfix.binder_name \<^const_syntax>‹All›;
val Ex_binder = Mixfix.binder_name \<^const_syntax>‹Ex›;
val impl = \<^const_syntax>‹HOL.implies›;
val conj = \<^const_syntax>‹HOL.conj›;
val less = \<^const_syntax>‹less›;
val less_eq = \<^const_syntax>‹less_eq›;

val trans =
[((All_binder, impl, less),
(\<^syntax_const>‹_All_less›, \<^syntax_const>‹_All_greater›)),
((All_binder, impl, less_eq),
(\<^syntax_const>‹_All_less_eq›, \<^syntax_const>‹_All_greater_eq›)),
((Ex_binder, conj, less),
(\<^syntax_const>‹_Ex_less›, \<^syntax_const>‹_Ex_greater›)),
((Ex_binder, conj, less_eq),
(\<^syntax_const>‹_Ex_less_eq›, \<^syntax_const>‹_Ex_greater_eq›))];

fun matches_bound v t =
(case t of
Const (\<^syntax_const>‹_bound›, _) \$ Free (v', _) => v = v'
| _ => false);
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
fun mk x c n P = Syntax.const c \$ Syntax_Trans.mark_bound_body x \$ n \$ P;

fun tr' q = (q, fn _ =>
(fn [Const (\<^syntax_const>‹_bound›, _) \$ Free (v, T),
Const (c, _) \$ (Const (d, _) \$ t \$ u) \$ P] =>
(case AList.lookup (=) trans (q, c, d) of
NONE => raise Match
| SOME (l, g) =>
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
else raise Match)
| _ => raise Match));
in [tr' All_binder, tr' Ex_binder] end
›

subsection ‹Transitivity reasoning›

context ord
begin

lemma ord_le_eq_trans: "a ≤ b ⟹ b = c ⟹ a ≤ c"
by (rule subst)

lemma ord_eq_le_trans: "a = b ⟹ b ≤ c ⟹ a ≤ c"
by (rule ssubst)

lemma ord_less_eq_trans: "a < b ⟹ b = c ⟹ a < c"
by (rule subst)

lemma ord_eq_less_trans: "a = b ⟹ b < c ⟹ a < c"
by (rule ssubst)

end

lemma order_less_subst2: "(a::'a::order) < b ⟹ f b < (c::'c::order) ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ f a < c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b < c"
finally (less_trans) show ?thesis .
qed

lemma order_less_subst1: "(a::'a::order) < f b ⟹ (b::'b::order) < c ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ a < f c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a < f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (less_trans) show ?thesis .
qed

lemma order_le_less_subst2: "(a::'a::order) <= b ⟹ f b < (c::'c::order) ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ f a < c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b < c"
finally (le_less_trans) show ?thesis .
qed

lemma order_le_less_subst1: "(a::'a::order) <= f b ⟹ (b::'b::order) < c ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ a < f c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a <= f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (le_less_trans) show ?thesis .
qed

lemma order_less_le_subst2: "(a::'a::order) < b ⟹ f b <= (c::'c::order) ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ f a < c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b <= c"
finally (less_le_trans) show ?thesis .
qed

lemma order_less_le_subst1: "(a::'a::order) < f b ⟹ (b::'b::order) <= c ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ a < f c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a < f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (less_le_trans) show ?thesis .
qed

lemma order_subst1: "(a::'a::order) <= f b ⟹ (b::'b::order) <= c ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ a <= f c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a <= f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (order_trans) show ?thesis .
qed

lemma order_subst2: "(a::'a::order) <= b ⟹ f b <= (c::'c::order) ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ f a <= c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b <= c"
finally (order_trans) show ?thesis .
qed

lemma ord_le_eq_subst: "a <= b ⟹ f b = c ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ f a <= c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b = c"
finally (ord_le_eq_trans) show ?thesis .
qed

lemma ord_eq_le_subst: "a = f b ⟹ b <= c ⟹
(!!x y. x <= y ⟹ f x <= f y) ⟹ a <= f c"
proof -
assume r: "!!x y. x <= y ⟹ f x <= f y"
assume "a = f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (ord_eq_le_trans) show ?thesis .
qed

lemma ord_less_eq_subst: "a < b ⟹ f b = c ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ f a < c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b = c"
finally (ord_less_eq_trans) show ?thesis .
qed

lemma ord_eq_less_subst: "a = f b ⟹ b < c ⟹
(!!x y. x < y ⟹ f x < f y) ⟹ a < f c"
proof -
assume r: "!!x y. x < y ⟹ f x < f y"
assume "a = f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (ord_eq_less_trans) show ?thesis .
qed

text ‹
Note that this list of rules is in reverse order of priorities.
›

lemmas [trans] =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp

lemmas (in order) [trans] =
neq_le_trans
le_neq_trans

lemmas (in preorder) [trans] =
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans

lemmas (in order) [trans] =
order.antisym

lemmas (in ord) [trans] =
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans

lemmas [trans] =
trans

lemmas order_trans_rules =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
neq_le_trans
le_neq_trans
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
order.antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans

text ‹These support proving chains of decreasing inequalities
a ‹≥› b ‹≥› c ... in Isar proofs.›

lemma xt1 [no_atp]:
"a = b ⟹ b > c ⟹ a > c"
"a > b ⟹ b = c ⟹ a > c"
"a = b ⟹ b ≥ c ⟹ a ≥ c"
"a ≥ b ⟹ b = c ⟹ a ≥ c"
"(x::'a::order) ≥ y ⟹ y ≥ x ⟹ x = y"
"(x::'a::order) ≥ y ⟹ y ≥ z ⟹ x ≥ z"
"(x::'a::order) > y ⟹ y ≥ z ⟹ x > z"
"(x::'a::order) ≥ y ⟹ y > z ⟹ x > z"
"(a::'a::order) > b ⟹ b > a ⟹ P"
"(x::'a::order) > y ⟹ y > z ⟹ x > z"
"(a::'a::order) ≥ b ⟹ a ≠ b ⟹ a > b"
"(a::'a::order) ≠ b ⟹ a ≥ b ⟹ a > b"
"a = f b ⟹ b > c ⟹ (⋀x y. x > y ⟹ f x > f y) ⟹ a > f c"
"a > b ⟹ f b = c ⟹ (⋀x y. x > y ⟹ f x > f y) ⟹ f a > c"
"a = f b ⟹ b ≥ c ⟹ (⋀x y. x ≥ y ⟹ f x ≥ f y) ⟹ a ≥ f c"
"a ≥ b ⟹ f b = c ⟹ (⋀x y. x ≥ y ⟹ f x ≥ f y) ⟹ f a ≥ c"
by auto

lemma xt2 [no_atp]:
assumes "(a::'a::order) ≥ f b"
and "b ≥ c"
and "⋀x y. x ≥ y ⟹ f x ≥ f y"
shows  "a ≥ f c"
using assms by force

lemma xt3 [no_atp]:
assumes "(a::'a::order) ≥ b"
and "(f b::'b::order) ≥ c"
and "⋀x y. x ≥ y ⟹ f x ≥ f y"
shows  "f a ≥ c"
using assms by force

lemma xt4 [no_atp]:
assumes "(a::'a::order) > f b"
and "(b::'b::order) ≥ c"
and "⋀x y. x ≥ y ⟹ f x ≥ f y"
shows  "a > f c"
using assms by force

lemma xt5 [no_atp]:
assumes "(a::'a::order) > b"
and "(f b::'b::order) ≥ c"
and "⋀x y. x > y ⟹ f x > f y"
shows  "f a > c"
using assms by force

lemma xt6 [no_atp]:
assumes "(a::'a::order) ≥ f b"
and "b > c"
and "⋀x y. x > y ⟹ f x > f y"
shows  "a > f c"
using assms by force

lemma xt7 [no_atp]:
assumes "(a::'a::order) ≥ b"
and "(f b::'b::order) > c"
and "⋀x y. x ≥ y ⟹ f x ≥ f y"
shows  "f a > c"
using assms by force

lemma xt8 [no_atp]:
assumes "(a::'a::order) > f b"
and "(b::'b::order) > c"
and "⋀x y. x > y ⟹ f x > f y"
shows  "a > f c"
using assms by force

lemma xt9 [no_atp]:
assumes "(a::'a::order) > b"
and "(f b::'b::order) > c"
and "⋀x y. x > y ⟹ f x > f y"
shows  "f a > c"
using assms by force

lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9

(*
Since "a ≥ b" abbreviates "b ≤ a", the abbreviation "..." stands
for the wrong thing in an Isar proof.

The extra transitivity rules can be used as follows:

lemma "(a::'a::order) > z"
proof -
have "a ≥ b" (is "_ ≥ ?rhs")
sorry
also have "?rhs ≥ c" (is "_ ≥ ?rhs")
sorry
also (xtrans) have "?rhs = d" (is "_ = ?rhs")
sorry
also (xtrans) have "?rhs ≥ e" (is "_ ≥ ?rhs")
sorry
also (xtrans) have "?rhs > f" (is "_ > ?rhs")
sorry
also (xtrans) have "?rhs > z"
sorry
finally (xtrans) show ?thesis .
qed

Alternatively, one can use "declare xtrans [trans]" and then
leave out the "(xtrans)" above.
*)

subsection ‹min and max -- fundamental›

definition (in ord) min :: "'a ⇒ 'a ⇒ 'a" where
"min a b = (if a ≤ b then a else b)"

definition (in ord) max :: "'a ⇒ 'a ⇒ 'a" where
"max a b = (if a ≤ b then b else a)"

lemma min_absorb1: "x ≤ y ⟹ min x y = x"

lemma max_absorb2: "x ≤ y ⟹ max x y = y"

lemma min_absorb2: "(y::'a::order) ≤ x ⟹ min x y = y"

lemma max_absorb1: "(y::'a::order) ≤ x ⟹ max x y = x"

lemma max_min_same [simp]:
fixes x y :: "'a :: linorder"
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"

subsection ‹(Unique) top and bottom elements›

class bot =
fixes bot :: 'a ("⊥")

class order_bot = order + bot +
assumes bot_least: "⊥ ≤ a"
begin

sublocale bot: ordering_top greater_eq greater bot
by standard (fact bot_least)

lemma le_bot:
"a ≤ ⊥ ⟹ a = ⊥"
by (fact bot.extremum_uniqueI)

lemma bot_unique:
"a ≤ ⊥ ⟷ a = ⊥"
by (fact bot.extremum_unique)

lemma not_less_bot:
"¬ a < ⊥"
by (fact bot.extremum_strict)

lemma bot_less:
"a ≠ ⊥ ⟷ ⊥ < a"
by (fact bot.not_eq_extremum)

lemma max_bot[simp]: "max bot x = x"

lemma max_bot2[simp]: "max x bot = x"

lemma min_bot[simp]: "min bot x = bot"

lemma min_bot2[simp]: "min x bot = bot"

end

class top =
fixes top :: 'a ("⊤")

class order_top = order + top +
assumes top_greatest: "a ≤ ⊤"
begin

sublocale top: ordering_top less_eq less top
by standard (fact top_greatest)

lemma top_le:
"⊤ ≤ a ⟹ a = ⊤"
by (fact top.extremum_uniqueI)

lemma top_unique:
"⊤ ≤ a ⟷ a = ⊤"
by (fact top.extremum_unique)

lemma not_top_less:
"¬ ⊤ < a"
by (fact top.extremum_strict)

lemma less_top:
"a ≠ ⊤ ⟷ a < ⊤"
by (fact top.not_eq_extremum)

lemma max_top[simp]: "max top x = top"

lemma max_top2[simp]: "max x top = top"

lemma min_top[simp]: "min top x = x"

lemma min_top2[simp]: "min x top = x"

end

subsection ‹Dense orders›

class dense_order = order +
assumes dense: "x < y ⟹ (∃z. x < z ∧ z < y)"

class dense_linorder = linorder + dense_order
begin

lemma dense_le:
fixes y z :: 'a
assumes "⋀x. x < y ⟹ x ≤ z"
shows "y ≤ z"
proof (rule ccontr)
assume "¬ ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "x ≤ z" using assms[OF ‹x < y›] .
ultimately show False by auto
qed

lemma dense_le_bounded:
fixes x y z :: 'a
assumes "x < y"
assumes *: "⋀w. ⟦ x < w ; w < y ⟧ ⟹ w ≤ z"
shows "y ≤ z"
proof (rule dense_le)
fix w assume "w < y"
from dense[OF ‹x < y›] obtain u where "x < u" "u < y" by safe
from linear[of u w]
show "w ≤ z"
proof (rule disjE)
assume "u ≤ w"
from less_le_trans[OF ‹x < u› ‹u ≤ w›] ‹w < y›
show "w ≤ z" by (rule *)
next
assume "w ≤ u"
from ‹w ≤ u› *[OF ‹x < u› ‹u < y›]
show "w ≤ z" by (rule order_trans)
qed
qed

lemma dense_ge:
fixes y z :: 'a
assumes "⋀x. z < x ⟹ y ≤ x"
shows "y ≤ z"
proof (rule ccontr)
assume "¬ ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "y ≤ x" using assms[OF ‹z < x›] .
ultimately show False by auto
qed

lemma dense_ge_bounded:
fixes x y z :: 'a
assumes "z < x"
assumes *: "⋀w. ⟦ z < w ; w < x ⟧ ⟹ y ≤ w"
shows "y ≤ z"
proof (rule dense_ge)
fix w assume "z < w"
from dense[OF ‹z < x›] obtain u where "z < u" "u < x" by safe
from linear[of u w]
show "y ≤ w"
proof (rule disjE)
assume "w ≤ u"
from ‹z < w› le_less_trans[OF ‹w ≤ u› ‹u < x›]
show "y ≤ w" by (rule *)
next
assume "u ≤ w"
from *[OF ‹z < u› ‹u < x›] ‹u ≤ w›
show "y ≤ w" by (rule order_trans)
qed
qed

end

class no_top = order +
assumes gt_ex: "∃y. x < y"

class no_bot = order +
assumes lt_ex: "∃y. y < x"

class unbounded_dense_linorder = dense_linorder + no_top + no_bot

subsection ‹Wellorders›

class wellorder = linorder +
assumes less_induct [case_names less]: "(⋀x. (⋀y. y < x ⟹ P y) ⟹ P x) ⟹ P a"
begin

lemma wellorder_Least_lemma:
fixes k :: 'a
assumes "P k"
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) ≤ k"
proof -
have "P (LEAST x. P x) ∧ (LEAST x. P x) ≤ k"
using assms proof (induct k rule: less_induct)
case (less x) then have "P x" by simp
show ?case proof (rule classical)
assume assm: "¬ (P (LEAST a. P a) ∧ (LEAST a. P a) ≤ x)"
have "⋀y. P y ⟹ x ≤ y"
proof (rule classical)
fix y
assume "P y" and "¬ x ≤ y"
with less have "P (LEAST a. P a)" and "(LEAST a. P a) ≤ y"
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) ≤ y"
by auto
then show "x ≤ y" by auto
qed
with ‹P x› have Least: "(LEAST a. P a) = x"
by (rule Least_equality)
with ‹P x› show ?thesis by simp
qed
qed
then show "P (LEAST x. P x)" and "(LEAST x. P x) ≤ k" by auto
qed

― ‹The following 3 lemmas are due to Brian Huffman›
lemma LeastI_ex: "∃x. P x ⟹ P (Least P)"
by (erule exE) (erule LeastI)

lemma LeastI2:
"P a ⟹ (⋀x. P x ⟹ Q x) ⟹ Q (Least P)"
by (blast intro: LeastI)

lemma LeastI2_ex:
"∃a. P a ⟹ (⋀x. P x ⟹ Q x) ⟹ Q (Least P)"
by (blast intro: LeastI_ex)

lemma LeastI2_wellorder:
assumes "P a"
and "⋀a. ⟦ P a; ∀b. P b ⟶ a ≤ b ⟧ ⟹ Q a"
shows "Q (Least P)"
proof (rule LeastI2_order)
show "P (Least P)" using ‹P a› by (rule LeastI)
next
fix y assume "P y" thus "Least P ≤ y" by (rule Least_le)
next
fix x assume "P x" "∀y. P y ⟶ x ≤ y" thus "Q x" by (rule assms(2))
qed

lemma LeastI2_wellorder_ex:
assumes "∃x. P x"
and "⋀a. ⟦ P a; ∀b. P b ⟶ a ≤ b ⟧ ⟹ Q a"
shows "Q (Least P)"
using assms by clarify (blast intro!: LeastI2_wellorder)

lemma not_less_Least: "k < (LEAST x. P x) ⟹ ¬ P k"
apply (erule contrapos_nn)
apply (erule Least_le)
done

lemma exists_least_iff: "(∃n. P n) ⟷ (∃n. P n ∧ (∀m < n. ¬ P m))" (is "?lhs ⟷ ?rhs")
proof
assume ?rhs thus ?lhs by blast
next
assume H: ?lhs then obtain n where n: "P n" by blast
let ?x = "Least P"
{ fix m assume m: "m < ?x"
from not_less_Least[OF m] have "¬ P m" . }
with LeastI_ex[OF H] show ?rhs by blast
qed

end

subsection ‹Order on \<^typ>‹bool››

instantiation bool :: "{order_bot, order_top, linorder}"
begin

definition
le_bool_def [simp]: "P ≤ Q ⟷ P ⟶ Q"

definition
[simp]: "(P::bool) < Q ⟷ ¬ P ∧ Q"

definition
[simp]: "⊥ ⟷ False"

definition
[simp]: "⊤ ⟷ True"

instance proof
qed auto

end

lemma le_boolI: "(P ⟹ Q) ⟹ P ≤ Q"
by simp

lemma le_boolI': "P ⟶ Q ⟹ P ≤ Q"
by simp

lemma le_boolE: "P ≤ Q ⟹ P ⟹ (Q ⟹ R) ⟹ R"
by simp

lemma le_boolD: "P ≤ Q ⟹ P ⟶ Q"
by simp

lemma bot_boolE: "⊥ ⟹ P"
by simp

lemma top_boolI: ⊤
by simp

lemma [code]:
"False ≤ b ⟷ True"
"True ≤ b ⟷ b"
"False < b ⟷ b"
"True < b ⟷ False"
by simp_all

subsection ‹Order on \<^typ>‹_ ⇒ _››

instantiation "fun" :: (type, ord) ord
begin

definition
le_fun_def: "f ≤ g ⟷ (∀x. f x ≤ g x)"

definition
"(f::'a ⇒ 'b) < g ⟷ f ≤ g ∧ ¬ (g ≤ f)"

instance ..

end

instance "fun" :: (type, preorder) preorder proof
qed (auto simp add: le_fun_def less_fun_def
intro: order_trans order.antisym)

instance "fun" :: (type, order) order proof
qed (auto simp add: le_fun_def intro: order.antisym)

instantiation "fun" :: (type, bot) bot
begin

definition
"⊥ = (λx. ⊥)"

instance ..

end

instantiation "fun" :: (type, order_bot) order_bot
begin

lemma bot_apply [simp, code]:
"⊥ x = ⊥"

instance proof

end

instantiation "fun" :: (type, top) top
begin

definition
[no_atp]: "⊤ = (λx. ⊤)"

instance ..

end

instantiation "fun" :: (type, order_top) order_top
begin

lemma top_apply [simp, code]:
"⊤ x = ⊤"

instance proof

end

lemma le_funI: "(⋀x. f x ≤ g x) ⟹ f ≤ g"
unfolding le_fun_def by simp

lemma le_funE: "f ≤ g ⟹ (f x ≤ g x ⟹ P) ⟹ P"
unfolding le_fun_def by simp

lemma le_funD: "f ≤ g ⟹ f x ≤ g x"
by (rule le_funE)

subsection ‹Order on unary and binary predicates›

lemma predicate1I:
assumes PQ: "⋀x. P x ⟹ Q x"
shows "P ≤ Q"
apply (rule le_funI)
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done

lemma predicate1D:
"P ≤ Q ⟹ P x ⟹ Q x"
apply (erule le_funE)
apply (erule le_boolE)
apply assumption+
done

lemma rev_predicate1D:
"P x ⟹ P ≤ Q ⟹ Q x"
by (rule predicate1D)

lemma predicate2I:
assumes PQ: "⋀x y. P x y ⟹ Q x y"
shows "P ≤ Q"
apply (rule le_funI)+
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done

lemma predicate2D:
"P ≤ Q ⟹ P x y ⟹ Q x y"
apply (erule le_funE)+
apply (erule le_boolE)
apply assumption+
done

lemma rev_predicate2D:
"P x y ⟹ P ≤ Q ⟹ Q x y"
by (rule predicate2D)

lemma bot1E [no_atp]: "⊥ x ⟹ P"

lemma bot2E: "⊥ x y ⟹ P"

lemma top1I: "⊤ x"

lemma top2I: "⊤ x y"

subsection ‹Name duplicates›

lemmas antisym = order.antisym
lemmas eq_iff = order.eq_iff

lemmas order_eq_refl = preorder_class.eq_refl
lemmas order_less_irrefl = preorder_class.less_irrefl
lemmas order_less_imp_le = preorder_class.less_imp_le
lemmas order_less_not_sym = preorder_class.less_not_sym
lemmas order_less_asym = preorder_class.less_asym
lemmas order_less_trans = preorder_class.less_trans
lemmas order_le_less_trans = preorder_class.le_less_trans
lemmas order_less_le_trans = preorder_class.less_le_trans
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
lemmas order_less_imp_triv = preorder_class.less_imp_triv
lemmas order_less_asym' = preorder_class.less_asym'

lemmas order_less_le = order_class.less_le
lemmas order_le_less = order_class.le_less
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
lemmas order_neq_le_trans = order_class.neq_le_trans
lemmas order_le_neq_trans = order_class.le_neq_trans
lemmas order_eq_iff = order_class.order.eq_iff
lemmas order_antisym_conv = order_class.antisym_conv

lemmas linorder_linear = linorder_class.linear
lemmas linorder_less_linear = linorder_class.less_linear
lemmas linorder_le_less_linear = linorder_class.le_less_linear
lemmas linorder_le_cases = linorder_class.le_cases
lemmas linorder_not_less = linorder_class.not_less
lemmas linorder_not_le = linorder_class.not_le
lemmas linorder_neq_iff = linorder_class.neq_iff
lemmas linorder_neqE = linorder_class.neqE

end
```