# Theory Higher_Order_Logic

theory Higher_Order_Logic
imports Pure
```(*  Title:      HOL/Isar_Examples/Higher_Order_Logic.thy
Author:     Makarius
*)

section ‹Foundations of HOL›

theory Higher_Order_Logic
imports Pure
begin

text ‹
The following theory development illustrates the foundations of Higher-Order
Logic. The ``HOL'' logic that is given here resembles @{cite
"Gordon:1985:HOL"} and its predecessor @{cite "church40"}, but the order of
axiomatizations and defined connectives has be adapted to modern
presentations of ‹λ›-calculus and Constructive Type Theory. Thus it fits
nicely to the underlying Natural Deduction framework of Isabelle/Pure and
Isabelle/Isar.
›

section ‹HOL syntax within Pure›

class type
default_sort type

typedecl o
instance o :: type ..
instance "fun" :: (type, type) type ..

judgment Trueprop :: "o ⇒ prop"  ("_" 5)

section ‹Minimal logic (axiomatization)›

axiomatization imp :: "o ⇒ o ⇒ o"  (infixr "⟶" 25)
where impI [intro]: "(A ⟹ B) ⟹ A ⟶ B"
and impE [dest, trans]: "A ⟶ B ⟹ A ⟹ B"

axiomatization All :: "('a ⇒ o) ⇒ o"  (binder "∀" 10)
where allI [intro]: "(⋀x. P x) ⟹ ∀x. P x"
and allE [dest]: "∀x. P x ⟹ P a"

lemma atomize_imp [atomize]: "(A ⟹ B) ≡ Trueprop (A ⟶ B)"
by standard (fact impI, fact impE)

lemma atomize_all [atomize]: "(⋀x. P x) ≡ Trueprop (∀x. P x)"
by standard (fact allI, fact allE)

subsubsection ‹Derived connectives›

definition False :: o
where "False ≡ ∀A. A"

lemma FalseE [elim]:
assumes "False"
shows A
proof -
from ‹False› have "∀A. A" by (simp only: False_def)
then show A ..
qed

definition True :: o
where "True ≡ False ⟶ False"

lemma TrueI [intro]: True
unfolding True_def ..

definition not :: "o ⇒ o"  ("¬ _"  40)
where "not ≡ λA. A ⟶ False"

lemma notI [intro]:
assumes "A ⟹ False"
shows "¬ A"
using assms unfolding not_def ..

lemma notE [elim]:
assumes "¬ A" and A
shows B
proof -
from ‹¬ A› have "A ⟶ False" by (simp only: not_def)
from this and ‹A› have "False" ..
then show B ..
qed

lemma notE': "A ⟹ ¬ A ⟹ B"
by (rule notE)

lemmas contradiction = notE notE'  ― ‹proof by contradiction in any order›

definition conj :: "o ⇒ o ⇒ o"  (infixr "∧" 35)
where "A ∧ B ≡ ∀C. (A ⟶ B ⟶ C) ⟶ C"

lemma conjI [intro]:
assumes A and B
shows "A ∧ B"
unfolding conj_def
proof
fix C
show "(A ⟶ B ⟶ C) ⟶ C"
proof
assume "A ⟶ B ⟶ C"
also note ‹A›
also note ‹B›
finally show C .
qed
qed

lemma conjE [elim]:
assumes "A ∧ B"
obtains A and B
proof
from ‹A ∧ B› have *: "(A ⟶ B ⟶ C) ⟶ C" for C
unfolding conj_def ..
show A
proof -
note * [of A]
also have "A ⟶ B ⟶ A"
proof
assume A
then show "B ⟶ A" ..
qed
finally show ?thesis .
qed
show B
proof -
note * [of B]
also have "A ⟶ B ⟶ B"
proof
show "B ⟶ B" ..
qed
finally show ?thesis .
qed
qed

definition disj :: "o ⇒ o ⇒ o"  (infixr "∨" 30)
where "A ∨ B ≡ ∀C. (A ⟶ C) ⟶ (B ⟶ C) ⟶ C"

lemma disjI1 [intro]:
assumes A
shows "A ∨ B"
unfolding disj_def
proof
fix C
show "(A ⟶ C) ⟶ (B ⟶ C) ⟶ C"
proof
assume "A ⟶ C"
from this and ‹A› have C ..
then show "(B ⟶ C) ⟶ C" ..
qed
qed

lemma disjI2 [intro]:
assumes B
shows "A ∨ B"
unfolding disj_def
proof
fix C
show "(A ⟶ C) ⟶ (B ⟶ C) ⟶ C"
proof
show "(B ⟶ C) ⟶ C"
proof
assume "B ⟶ C"
from this and ‹B› show C ..
qed
qed
qed

lemma disjE [elim]:
assumes "A ∨ B"
obtains (a) A | (b) B
proof -
from ‹A ∨ B› have "(A ⟶ thesis) ⟶ (B ⟶ thesis) ⟶ thesis"
unfolding disj_def ..
also have "A ⟶ thesis"
proof
assume A
then show thesis by (rule a)
qed
also have "B ⟶ thesis"
proof
assume B
then show thesis by (rule b)
qed
finally show thesis .
qed

definition Ex :: "('a ⇒ o) ⇒ o"  (binder "∃" 10)
where "∃x. P x ≡ ∀C. (∀x. P x ⟶ C) ⟶ C"

lemma exI [intro]: "P a ⟹ ∃x. P x"
unfolding Ex_def
proof
fix C
assume "P a"
show "(∀x. P x ⟶ C) ⟶ C"
proof
assume "∀x. P x ⟶ C"
then have "P a ⟶ C" ..
from this and ‹P a› show C ..
qed
qed

lemma exE [elim]:
assumes "∃x. P x"
obtains (that) x where "P x"
proof -
from ‹∃x. P x› have "(∀x. P x ⟶ thesis) ⟶ thesis"
unfolding Ex_def ..
also have "∀x. P x ⟶ thesis"
proof
fix x
show "P x ⟶ thesis"
proof
assume "P x"
then show thesis by (rule that)
qed
qed
finally show thesis .
qed

subsubsection ‹Extensional equality›

axiomatization equal :: "'a ⇒ 'a ⇒ o"  (infixl "=" 50)
where refl [intro]: "x = x"
and subst: "x = y ⟹ P x ⟹ P y"

abbreviation not_equal :: "'a ⇒ 'a ⇒ o"  (infixl "≠" 50)
where "x ≠ y ≡ ¬ (x = y)"

abbreviation iff :: "o ⇒ o ⇒ o"  (infixr "⟷" 25)
where "A ⟷ B ≡ A = B"

axiomatization
where ext [intro]: "(⋀x. f x = g x) ⟹ f = g"
and iff [intro]: "(A ⟹ B) ⟹ (B ⟹ A) ⟹ A ⟷ B"

lemma sym [sym]: "y = x" if "x = y"
using that by (rule subst) (rule refl)

lemma [trans]: "x = y ⟹ P y ⟹ P x"
by (rule subst) (rule sym)

lemma [trans]: "P x ⟹ x = y ⟹ P y"
by (rule subst)

lemma arg_cong: "f x = f y" if "x = y"
using that by (rule subst) (rule refl)

lemma fun_cong: "f x = g x" if "f = g"
using that by (rule subst) (rule refl)

lemma trans [trans]: "x = y ⟹ y = z ⟹ x = z"
by (rule subst)

lemma iff1 [elim]: "A ⟷ B ⟹ A ⟹ B"
by (rule subst)

lemma iff2 [elim]: "A ⟷ B ⟹ B ⟹ A"
by (rule subst) (rule sym)

subsection ‹Cantor's Theorem›

text ‹
Cantor's Theorem states that there is no surjection from a set to its
powerset. The subsequent formulation uses elementary ‹λ›-calculus and
predicate logic, with standard introduction and elimination rules.
›

assumes *: "¬ A ⟷ A"
shows C
proof (rule notE)
show "¬ A"
proof
assume A
with * have "¬ A" ..
from this and ‹A› show False ..
qed
with * show A ..
qed

theorem Cantor: "¬ (∃f :: 'a ⇒ 'a ⇒ o. ∀A. ∃x. A = f x)"
proof
assume "∃f :: 'a ⇒ 'a ⇒ o. ∀A. ∃x. A = f x"
then obtain f :: "'a ⇒ 'a ⇒ o" where *: "∀A. ∃x. A = f x" ..
let ?D = "λx. ¬ f x x"
from * have "∃x. ?D = f x" ..
then obtain a where "?D = f a" ..
then have "?D a ⟷ f a a" using refl by (rule subst)
then have "¬ f a a ⟷ f a a" .
then show False by (rule iff_contradiction)
qed

subsection ‹Characterization of Classical Logic›

text ‹
The subsequent rules of classical reasoning are all equivalent.
›

locale classical =
assumes classical: "(¬ A ⟹ A) ⟹ A"
― ‹predicate definition and hypothetical context›
begin

assumes "¬ A ⟹ False"
shows A
proof (rule classical)
assume "¬ A"
then have False by (rule assms)
then show A ..
qed

lemma double_negation:
assumes "¬ ¬ A"
shows A
assume "¬ A"
with ‹¬ ¬ A› show False by (rule contradiction)
qed

lemma tertium_non_datur: "A ∨ ¬ A"
proof (rule double_negation)
show "¬ ¬ (A ∨ ¬ A)"
proof
assume "¬ (A ∨ ¬ A)"
have "¬ A"
proof
assume A then have "A ∨ ¬ A" ..
with ‹¬ (A ∨ ¬ A)› show False by (rule contradiction)
qed
then have "A ∨ ¬ A" ..
with ‹¬ (A ∨ ¬ A)› show False by (rule contradiction)
qed
qed

lemma classical_cases:
obtains A | "¬ A"
using tertium_non_datur
proof
assume A
then show thesis ..
next
assume "¬ A"
then show thesis ..
qed

end

lemma classical_if_cases: classical
if cases: "⋀A C. (A ⟹ C) ⟹ (¬ A ⟹ C) ⟹ C"
proof
fix A
assume *: "¬ A ⟹ A"
show A
proof (rule cases)
assume A
then show A .
next
assume "¬ A"
then show A by (rule *)
qed
qed

section ‹Peirce's Law›

text ‹
Peirce's Law is another characterization of classical reasoning. Its
statement only requires implication.
›

theorem (in classical) Peirce's_Law: "((A ⟶ B) ⟶ A) ⟶ A"
proof
assume *: "(A ⟶ B) ⟶ A"
show A
proof (rule classical)
assume "¬ A"
have "A ⟶ B"
proof
assume A
with ‹¬ A› show B by (rule contradiction)
qed
with * show A ..
qed
qed

section ‹Hilbert's choice operator (axiomatization)›

axiomatization Eps :: "('a ⇒ o) ⇒ 'a"
where someI: "P x ⟹ P (Eps P)"

syntax "_Eps" :: "pttrn ⇒ o ⇒ 'a"  ("(3SOME _./ _)" [0, 10] 10)
translations "SOME x. P" ⇌ "CONST Eps (λx. P)"

text ‹
┉
It follows a derivation of the classical law of tertium-non-datur by
means of Hilbert's choice operator (due to Berghofer, Beeson, Harrison,
based on a proof by Diaconescu).
┉
›

theorem Diaconescu: "A ∨ ¬ A"
proof -
let ?P = "λx. (A ∧ x) ∨ ¬ x"
let ?Q = "λx. (A ∧ ¬ x) ∨ x"

have a: "?P (Eps ?P)"
proof (rule someI)
have "¬ False" ..
then show "?P False" ..
qed
have b: "?Q (Eps ?Q)"
proof (rule someI)
have True ..
then show "?Q True" ..
qed

from a show ?thesis
proof
assume "A ∧ Eps ?P"
then have A ..
then show ?thesis ..
next
assume "¬ Eps ?P"
from b show ?thesis
proof
assume "A ∧ ¬ Eps ?Q"
then have A ..
then show ?thesis ..
next
assume "Eps ?Q"
have neq: "?P ≠ ?Q"
proof
assume "?P = ?Q"
then have "Eps ?P ⟷ Eps ?Q" by (rule arg_cong)
also note ‹Eps ?Q›
finally have "Eps ?P" .
with ‹¬ Eps ?P› show False by (rule contradiction)
qed
have "¬ A"
proof
assume A
have "?P = ?Q"
proof (rule ext)
show "?P x ⟷ ?Q x" for x
proof
assume "?P x"
then show "?Q x"
proof
assume "¬ x"
with ‹A› have "A ∧ ¬ x" ..
then show ?thesis ..
next
assume "A ∧ x"
then have x ..
then show ?thesis ..
qed
next
assume "?Q x"
then show "?P x"
proof
assume "A ∧ ¬ x"
then have "¬ x" ..
then show ?thesis ..
next
assume x
with ‹A› have "A ∧ x" ..
then show ?thesis ..
qed
qed
qed
with neq show False by (rule contradiction)
qed
then show ?thesis ..
qed
qed
qed

text ‹
This means, the hypothetical predicate \<^const>‹classical› always holds
unconditionally (with all consequences).
›

interpretation classical
proof (rule classical_if_cases)
fix A C
assume *: "A ⟹ C"
and **: "¬ A ⟹ C"
from Diaconescu [of A] show C
proof
assume A
then show C by (rule *)
next
assume "¬ A"
then show C by (rule **)
qed
qed

thm classical