354 lines
No EOL
17 KiB
Text
354 lines
No EOL
17 KiB
Text
import kernel
|
||
import macros
|
||
|
||
variable Nat : Type
|
||
alias ℕ : Nat
|
||
|
||
namespace Nat
|
||
builtin numeral
|
||
|
||
builtin add : Nat → Nat → Nat
|
||
infixl 65 + : add
|
||
|
||
builtin mul : Nat → Nat → Nat
|
||
infixl 70 * : mul
|
||
|
||
builtin le : Nat → Nat → Bool
|
||
infix 50 <= : le
|
||
infix 50 ≤ : le
|
||
|
||
definition ge (a b : Nat) := b ≤ a
|
||
infix 50 >= : ge
|
||
infix 50 ≥ : ge
|
||
|
||
definition lt (a b : Nat) := a + 1 ≤ b
|
||
infix 50 < : lt
|
||
|
||
definition gt (a b : Nat) := b < a
|
||
infix 50 > : gt
|
||
|
||
definition id (a : Nat) := a
|
||
notation 55 | _ | : id
|
||
|
||
axiom succ_nz (a : Nat) : a + 1 ≠ 0
|
||
axiom succ_inj {a b : Nat} (H : a + 1 = b + 1) : a = b
|
||
axiom add_zeror (a : Nat) : a + 0 = a
|
||
axiom add_succr (a b : Nat) : a + (b + 1) = (a + b) + 1
|
||
axiom mul_zeror (a : Nat) : a * 0 = 0
|
||
axiom mul_succr (a b : Nat) : a * (b + 1) = a * b + a
|
||
axiom le_def (a b : Nat) : a ≤ b = ∃ c, a + c = b
|
||
axiom induction {P : Nat → Bool} (H1 : P 0) (H2 : ∀ (n : Nat) (iH : P n), P (n + 1)) : ∀ a, P a
|
||
|
||
theorem induction_on {P : Nat → Bool} (a : Nat) (H1 : P 0) (H2 : ∀ (n : Nat) (iH : P n), P (n + 1)) : P a
|
||
:= induction H1 H2 a
|
||
|
||
theorem pred_nz {a : Nat} : a ≠ 0 → ∃ b, b + 1 = a
|
||
:= induction_on a
|
||
(λ H : 0 ≠ 0, false_elim (∃ b, b + 1 = 0) H)
|
||
(λ (n : Nat) (iH : n ≠ 0 → ∃ b, b + 1 = n) (H : n + 1 ≠ 0),
|
||
or_elim (em (n = 0))
|
||
(λ Heq0 : n = 0, exists_intro 0 (calc 0 + 1 = n + 1 : { symm Heq0 }))
|
||
(λ Hne0 : n ≠ 0,
|
||
obtain (w : Nat) (Hw : w + 1 = n), from (iH Hne0),
|
||
exists_intro (w + 1) (calc w + 1 + 1 = n + 1 : { Hw })))
|
||
|
||
theorem discriminate {B : Bool} {a : Nat} (H1: a = 0 → B) (H2 : ∀ n, a = n + 1 → B) : B
|
||
:= or_elim (em (a = 0))
|
||
(λ Heq0 : a = 0, H1 Heq0)
|
||
(λ Hne0 : a ≠ 0, obtain (w : Nat) (Hw : w + 1 = a), from (pred_nz Hne0),
|
||
H2 w (symm Hw))
|
||
|
||
theorem add_zerol (a : Nat) : 0 + a = a
|
||
:= induction_on a
|
||
(have 0 + 0 = 0 : trivial)
|
||
(λ (n : Nat) (iH : 0 + n = n),
|
||
calc 0 + (n + 1) = (0 + n) + 1 : add_succr 0 n
|
||
... = n + 1 : { iH })
|
||
|
||
theorem add_succl (a b : Nat) : (a + 1) + b = (a + b) + 1
|
||
:= induction_on b
|
||
(calc (a + 1) + 0 = a + 1 : add_zeror (a + 1)
|
||
... = (a + 0) + 1 : { symm (add_zeror a) })
|
||
(λ (n : Nat) (iH : (a + 1) + n = (a + n) + 1),
|
||
calc (a + 1) + (n + 1) = ((a + 1) + n) + 1 : add_succr (a + 1) n
|
||
... = ((a + n) + 1) + 1 : { iH }
|
||
... = (a + (n + 1)) + 1 : { have (a + n) + 1 = a + (n + 1) : symm (add_succr a n) })
|
||
|
||
theorem add_comm (a b : Nat) : a + b = b + a
|
||
:= induction_on b
|
||
(calc a + 0 = a : add_zeror a
|
||
... = 0 + a : symm (add_zerol a))
|
||
(λ (n : Nat) (iH : a + n = n + a),
|
||
calc a + (n + 1) = (a + n) + 1 : add_succr a n
|
||
... = (n + a) + 1 : { iH }
|
||
... = (n + 1) + a : symm (add_succl n a))
|
||
|
||
theorem add_assoc (a b c : Nat) : a + (b + c) = (a + b) + c
|
||
:= induction_on a
|
||
(calc 0 + (b + c) = b + c : add_zerol (b + c)
|
||
... = (0 + b) + c : { symm (add_zerol b) })
|
||
(λ (n : Nat) (iH : n + (b + c) = (n + b) + c),
|
||
calc (n + 1) + (b + c) = (n + (b + c)) + 1 : add_succl n (b + c)
|
||
... = ((n + b) + c) + 1 : { iH }
|
||
... = ((n + b) + 1) + c : symm (add_succl (n + b) c)
|
||
... = ((n + 1) + b) + c : { have (n + b) + 1 = (n + 1) + b : symm (add_succl n b) })
|
||
|
||
theorem mul_zerol (a : Nat) : 0 * a = 0
|
||
:= induction_on a
|
||
(have 0 * 0 = 0 : trivial)
|
||
(λ (n : Nat) (iH : 0 * n = 0),
|
||
calc 0 * (n + 1) = (0 * n) + 0 : mul_succr 0 n
|
||
... = 0 + 0 : { iH }
|
||
... = 0 : trivial)
|
||
|
||
theorem mul_succl (a b : Nat) : (a + 1) * b = a * b + b
|
||
:= induction_on b
|
||
(calc (a + 1) * 0 = 0 : mul_zeror (a + 1)
|
||
... = a * 0 : symm (mul_zeror a)
|
||
... = a * 0 + 0 : symm (add_zeror (a * 0)))
|
||
(λ (n : Nat) (iH : (a + 1) * n = a * n + n),
|
||
calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : mul_succr (a + 1) n
|
||
... = a * n + n + (a + 1) : { iH }
|
||
... = a * n + n + a + 1 : add_assoc (a * n + n) a 1
|
||
... = a * n + (n + a) + 1 : { have a * n + n + a = a * n + (n + a) : symm (add_assoc (a * n) n a) }
|
||
... = a * n + (a + n) + 1 : { add_comm n a }
|
||
... = a * n + a + n + 1 : { add_assoc (a * n) a n }
|
||
... = a * (n + 1) + n + 1 : { symm (mul_succr a n) }
|
||
... = a * (n + 1) + (n + 1) : symm (add_assoc (a * (n + 1)) n 1))
|
||
|
||
theorem mul_onel (a : Nat) : 1 * a = a
|
||
:= induction_on a
|
||
(have 1 * 0 = 0 : trivial)
|
||
(λ (n : Nat) (iH : 1 * n = n),
|
||
calc 1 * (n + 1) = 1 * n + 1 : mul_succr 1 n
|
||
... = n + 1 : { iH })
|
||
|
||
theorem mul_oner (a : Nat) : a * 1 = a
|
||
:= induction_on a
|
||
(have 0 * 1 = 0 : trivial)
|
||
(λ (n : Nat) (iH : n * 1 = n),
|
||
calc (n + 1) * 1 = n * 1 + 1 : mul_succl n 1
|
||
... = n + 1 : { iH })
|
||
|
||
theorem mul_comm (a b : Nat) : a * b = b * a
|
||
:= induction_on b
|
||
(calc a * 0 = 0 : mul_zeror a
|
||
... = 0 * a : symm (mul_zerol a))
|
||
(λ (n : Nat) (iH : a * n = n * a),
|
||
calc a * (n + 1) = a * n + a : mul_succr a n
|
||
... = n * a + a : { iH }
|
||
... = (n + 1) * a : symm (mul_succl n a))
|
||
|
||
theorem distributer (a b c : Nat) : a * (b + c) = a * b + a * c
|
||
:= induction_on a
|
||
(calc 0 * (b + c) = 0 : mul_zerol (b + c)
|
||
... = 0 + 0 : trivial
|
||
... = 0 * b + 0 : { symm (mul_zerol b) }
|
||
... = 0 * b + 0 * c : { symm (mul_zerol c) })
|
||
(λ (n : Nat) (iH : n * (b + c) = n * b + n * c),
|
||
calc (n + 1) * (b + c) = n * (b + c) + (b + c) : mul_succl n (b + c)
|
||
... = n * b + n * c + (b + c) : { iH }
|
||
... = n * b + n * c + b + c : add_assoc (n * b + n * c) b c
|
||
... = n * b + (n * c + b) + c : { symm (add_assoc (n * b) (n * c) b) }
|
||
... = n * b + (b + n * c) + c : { add_comm (n * c) b }
|
||
... = n * b + b + n * c + c : { add_assoc (n * b) b (n * c) }
|
||
... = (n + 1) * b + n * c + c : { symm (mul_succl n b) }
|
||
... = (n + 1) * b + (n * c + c) : symm (add_assoc ((n + 1) * b) (n * c) c)
|
||
... = (n + 1) * b + (n + 1) * c : { symm (mul_succl n c) })
|
||
|
||
theorem distributel (a b c : Nat) : (a + b) * c = a * c + b * c
|
||
:= calc (a + b) * c = c * (a + b) : mul_comm (a + b) c
|
||
... = c * a + c * b : distributer c a b
|
||
... = a * c + c * b : { mul_comm c a }
|
||
... = a * c + b * c : { mul_comm c b }
|
||
|
||
theorem mul_assoc (a b c : Nat) : a * (b * c) = a * b * c
|
||
:= induction_on a
|
||
(calc 0 * (b * c) = 0 : mul_zerol (b * c)
|
||
... = 0 * c : symm (mul_zerol c)
|
||
... = (0 * b) * c : { symm (mul_zerol b) })
|
||
(λ (n : Nat) (iH : n * (b * c) = n * b * c),
|
||
calc (n + 1) * (b * c) = n * (b * c) + (b * c) : mul_succl n (b * c)
|
||
... = n * b * c + (b * c) : { iH }
|
||
... = (n * b + b) * c : symm (distributel (n * b) b c)
|
||
... = (n + 1) * b * c : { symm (mul_succl n b) })
|
||
|
||
theorem add_injr {a b c : Nat} : a + b = a + c → b = c
|
||
:= induction_on a
|
||
(λ H : 0 + b = 0 + c,
|
||
calc b = 0 + b : symm (add_zerol b)
|
||
... = 0 + c : H
|
||
... = c : add_zerol c)
|
||
(λ (n : Nat) (iH : n + b = n + c → b = c) (H : n + 1 + b = n + 1 + c),
|
||
let L1 : n + b + 1 = n + c + 1
|
||
:= (calc n + b + 1 = n + (b + 1) : symm (add_assoc n b 1)
|
||
... = n + (1 + b) : { add_comm b 1 }
|
||
... = n + 1 + b : add_assoc n 1 b
|
||
... = n + 1 + c : H
|
||
... = n + (1 + c) : symm (add_assoc n 1 c)
|
||
... = n + (c + 1) : { add_comm 1 c }
|
||
... = n + c + 1 : add_assoc n c 1),
|
||
L2 : n + b = n + c := succ_inj L1
|
||
in iH L2)
|
||
|
||
theorem add_injl {a b c : Nat} (H : a + b = c + b) : a = c
|
||
:= add_injr (calc b + a = a + b : add_comm _ _
|
||
... = c + b : H
|
||
... = b + c : add_comm _ _)
|
||
|
||
theorem add_eqz {a b : Nat} (H : a + b = 0) : a = 0
|
||
:= discriminate
|
||
(λ H1 : a = 0, H1)
|
||
(λ (n : Nat) (H1 : a = n + 1),
|
||
absurd_elim (a = 0)
|
||
H (calc a + b = n + 1 + b : { H1 }
|
||
... = n + (1 + b) : symm (add_assoc n 1 b)
|
||
... = n + (b + 1) : { add_comm 1 b }
|
||
... = n + b + 1 : add_assoc n b 1
|
||
... ≠ 0 : succ_nz (n + b)))
|
||
|
||
theorem le_intro {a b c : Nat} (H : a + c = b) : a ≤ b
|
||
:= (symm (le_def a b)) ◂ (have (∃ x, a + x = b) : exists_intro c H)
|
||
|
||
theorem le_elim {a b : Nat} (H : a ≤ b) : ∃ x, a + x = b
|
||
:= (le_def a b) ◂ H
|
||
|
||
theorem le_refl (a : Nat) : a ≤ a := le_intro (add_zeror a)
|
||
|
||
theorem le_zero (a : Nat) : 0 ≤ a := le_intro (add_zerol a)
|
||
|
||
|
||
theorem le_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c
|
||
:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
|
||
obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2),
|
||
le_intro (calc a + (w1 + w2) = a + w1 + w2 : add_assoc a w1 w2
|
||
... = b + w2 : { Hw1 }
|
||
... = c : Hw2)
|
||
|
||
theorem le_add {a b : Nat} (H : a ≤ b) (c : Nat) : a + c ≤ b + c
|
||
:= obtain (w : Nat) (Hw : a + w = b), from (le_elim H),
|
||
le_intro (calc a + c + w = a + (c + w) : symm (add_assoc a c w)
|
||
... = a + (w + c) : { add_comm c w }
|
||
... = a + w + c : add_assoc a w c
|
||
... = b + c : { Hw })
|
||
|
||
theorem le_antisym {a b : Nat} (H1 : a ≤ b) (H2 : b ≤ a) : a = b
|
||
:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
|
||
obtain (w2 : Nat) (Hw2 : b + w2 = a), from (le_elim H2),
|
||
let L1 : w1 + w2 = 0
|
||
:= add_injr (calc a + (w1 + w2) = a + w1 + w2 : { add_assoc a w1 w2 }
|
||
... = b + w2 : { Hw1 }
|
||
... = a : Hw2
|
||
... = a + 0 : symm (add_zeror a)),
|
||
L2 : w1 = 0 := add_eqz L1
|
||
in calc a = a + 0 : symm (add_zeror a)
|
||
... = a + w1 : { symm L2 }
|
||
... = b : Hw1
|
||
|
||
theorem not_lt_0 (a : Nat) : ¬ a < 0
|
||
:= not_intro (λ H : a + 1 ≤ 0,
|
||
obtain (w : Nat) (Hw1 : a + 1 + w = 0), from (le_elim H),
|
||
absurd
|
||
(calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _)
|
||
... = a + (1 + w) : { add_comm _ _ }
|
||
... = a + 1 + w : add_assoc _ _ _
|
||
... = 0 : Hw1)
|
||
(succ_nz (a + w)))
|
||
|
||
theorem lt_intro {a b c : Nat} (H : a + 1 + c = b) : a < b
|
||
:= le_intro H
|
||
|
||
theorem lt_elim {a b : Nat} (H : a < b) : ∃ x, a + 1 + x = b
|
||
:= le_elim H
|
||
|
||
theorem lt_le {a b : Nat} (H : a < b) : a ≤ b
|
||
:= obtain (w : Nat) (Hw : a + 1 + w = b), from (le_elim H),
|
||
le_intro (calc a + (1 + w) = a + 1 + w : add_assoc _ _ _
|
||
... = b : Hw)
|
||
|
||
theorem lt_ne {a b : Nat} (H : a < b) : a ≠ b
|
||
:= not_intro (λ H1 : a = b,
|
||
obtain (w : Nat) (Hw : a + 1 + w = b), from (lt_elim H),
|
||
absurd (calc w + 1 = 1 + w : add_comm _ _
|
||
... = 0 :
|
||
add_injr (calc b + (1 + w) = b + 1 + w : add_assoc b 1 w
|
||
... = a + 1 + w : { symm H1 }
|
||
... = b : Hw
|
||
... = b + 0 : symm (add_zeror b)))
|
||
(succ_nz w))
|
||
|
||
theorem lt_nrefl (a : Nat) : ¬ a < a
|
||
:= not_intro (λ H : a < a,
|
||
absurd (refl a) (lt_ne H))
|
||
|
||
theorem lt_trans {a b c : Nat} (H1 : a < b) (H2 : b < c) : a < c
|
||
:= obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1),
|
||
obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2),
|
||
lt_intro (calc a + 1 + (w1 + 1 + w2) = a + 1 + (w1 + (1 + w2)) : { symm (add_assoc w1 1 w2) }
|
||
... = (a + 1 + w1) + (1 + w2) : add_assoc _ _ _
|
||
... = b + (1 + w2) : { Hw1 }
|
||
... = b + 1 + w2 : add_assoc _ _ _
|
||
... = c : Hw2)
|
||
|
||
theorem lt_le_trans {a b c : Nat} (H1 : a < b) (H2 : b ≤ c) : a < c
|
||
:= obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1),
|
||
obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2),
|
||
lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _
|
||
... = b + w2 : { Hw1 }
|
||
... = c : Hw2)
|
||
|
||
theorem le_lt_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b < c) : a < c
|
||
:= obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1),
|
||
obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2),
|
||
lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _
|
||
... = a + (1 + w1) + w2 : { symm (add_assoc a 1 w1) }
|
||
... = a + (w1 + 1) + w2 : { add_comm 1 w1 }
|
||
... = a + w1 + 1 + w2 : { add_assoc a w1 1 }
|
||
... = b + 1 + w2 : { Hw1 }
|
||
... = c : Hw2)
|
||
|
||
theorem ne_lt_succ {a b : Nat} (H1 : a ≠ b) (H2 : a < b + 1) : a < b
|
||
:= obtain (w : Nat) (Hw : a + 1 + w = b + 1), from (lt_elim H2),
|
||
let L : a + w = b := add_injl (calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _)
|
||
... = a + (1 + w) : { add_comm _ _ }
|
||
... = a + 1 + w : add_assoc _ _ _
|
||
... = b + 1 : Hw)
|
||
in discriminate (λ Hz : w = 0, absurd_elim (a < b) (calc a = a + 0 : symm (add_zeror _)
|
||
... = a + w : { symm Hz }
|
||
... = b : L)
|
||
H1)
|
||
(λ (p : Nat) (Hp : w = p + 1), lt_intro (calc a + 1 + p = a + (1 + p) : symm (add_assoc _ _ _)
|
||
... = a + (p + 1) : { add_comm _ _ }
|
||
... = a + w : { symm Hp }
|
||
... = b : L))
|
||
|
||
theorem strong_induction {P : Nat → Bool} (H : ∀ n, (∀ m, m < n → P m) → P n) : ∀ a, P a
|
||
:= take a,
|
||
let stronger : P a ∧ ∀ m, m < a → P m :=
|
||
-- we prove a stronger result by regular induction on a
|
||
induction_on a
|
||
(have P 0 ∧ ∀ m, m < 0 → P m :
|
||
let c2 : ∀ m, m < 0 → P m := λ (m : Nat) (Hlt : m < 0), absurd_elim (P m) Hlt (not_lt_0 m),
|
||
c1 : P 0 := H 0 c2
|
||
in and_intro c1 c2)
|
||
(λ (n : Nat) (iH : P n ∧ ∀ m, m < n → P m),
|
||
have P (n + 1) ∧ ∀ m, m < n + 1 → P m :
|
||
let iH1 : P n := and_eliml iH,
|
||
iH2 : ∀ m, m < n → P m := and_elimr iH,
|
||
c2 : ∀ m, m < n + 1 → P m := λ (m : Nat) (Hlt : m < n + 1),
|
||
or_elim (em (m = n))
|
||
(λ Heq : m = n, subst iH1 (symm Heq))
|
||
(λ Hne : m ≠ n, iH2 m (ne_lt_succ Hne Hlt)),
|
||
c1 : P (n + 1) := H (n + 1) c2
|
||
in and_intro c1 c2)
|
||
in and_eliml stronger
|
||
|
||
set_opaque add true
|
||
set_opaque mul true
|
||
set_opaque le true
|
||
set_opaque id true
|
||
set_opaque ge true
|
||
set_opaque lt true
|
||
set_opaque gt true
|
||
set_opaque id true
|
||
end |