Natural Number Game
因为用 Verilog 会有颜色显示，所以用的 Verilog。
记录一下 Natural Number Game 的答案，好久以前做的，但没做完。
目录
Tutorial world
\[\text{example1}:xy+z=xy+z
\]
lemma example1 (x y z : mynat) : x * y + z = x * y + z :=
begin
refl,
end
\[\text{example2}:y=x+1\to2y=2(x+7)
\]
lemma example2 (x y : mynat) (h : y = x + 7) : 2 * y = 2 * (x + 7) :=
begin
rw h,
refl,
end
\[\text{example3}:\operatorname{succ}(a)=b\to\operatorname{succ}(\operatorname{succ}(a))=\operatorname{succ}(b)
\]
lemma example3 (a b : mynat) (h : succ a = b) : succ(succ(a)) = succ(b) :=
begin
rw h,
refl,
end
\[\text{add_succ_zero}:a+\operatorname{succ}(0)=\operatorname{succ}(a)
\]
lemma add_succ_zero (a : mynat) : a + succ(0) = succ(a) :=
begin
rw add_succ,
rw add_zero,
refl,
end
Addition world
\[\text{zero_add}:0+n=n
\]
lemma zero_add (n : mynat) : 0 + n = n :=
begin
induction n with d m,
rw add_zero,
refl,
rw add_succ,
rw m,
refl,
end
\[\text{add_assoc}:(a+b)+c=a+(b+c)
\]
lemma add_assoc (a b c : mynat) : (a + b) + c = a + (b + c) :=
begin
induction c with d hd,
rw add_zero,
rw add_zero,
refl,
rw add_succ,
rw add_succ,
rw add_succ,
rw hd,
refl,
end
\[\text{succ_add}:\operatorname{succ}(a)+b=\operatorname{succ}(a+b)
\]
lemma succ_add (a b : mynat) : succ a + b = succ (a + b) :=
begin
induction b with d hd,
rw add_zero,
rw add_zero,
refl,
rw add_succ,
rw add_succ,
rw hd,
refl,
end
\[\text{add_comm}:a+b=b+a
\]
lemma add_comm (a b : mynat) : a + b = b + a :=
begin
induction b with d hd,
rw add_zero,
rw zero_add,
refl,
rw succ_add,
rw add_succ,
rw hd,
refl,
end
\[\text{succ_eq_add_one}:\operatorname{succ}(n)=n+1
\]
theorem succ_eq_add_one (n : mynat) : succ n = n + 1 :=
begin
induction n with d hd,
rw one_eq_succ_zero,
rw zero_add,
refl,
rw succ_add,
rw hd,
refl,
end
\[\text{add_right_comm}:a+b+c=a+c+b
\]
lemma add_right_comm (a b c : mynat) : a + b + c = a + c + b :=
begin
rw add_assoc,
rw add_assoc,
induction a with d hd,
rw zero_add,
rw zero_add,
rw add_comm,
refl,
rw succ_add,
rw succ_add,
rw hd,
refl,
end
Multiplication world
\[\text{zero_mul}:0\times m=0
\]
lemma zero_mul (m : mynat) : 0 * m = 0 :=
begin
induction m with d hd,
rw mul_zero,
refl,
rw mul_succ,
rw add_zero,
rw nd,
refl,
end
\[\text{mul_one}:m\times1=m
\]
lemma mul_one (m : mynat) : m * 1 = m :=
begin
induction m with d hd,
rw zero_mul,
refl,
rw one_eq_succ_zero,
rw mul_succ,
rw mul_zero,
rw zero_add,
refl,
end
\[\text{one_mul}:1\times m=m
\]
lemma one_mul (m : mynat) : 1 * m = m :=
begin
induction m with d hd,
rw mul_zero,
refl,
rw mul_succ,
rw hd,
rw succ_eq_add_one,
refl,
end
\[\text{mul_add}:t(a+b)=ta+tb
\]
lemma mul_add (t a b : mynat) : t * (a + b) = t * a + t * b :=
begin
induction b with d hd,
rw mul_zero,
repeat {rw add_zero},
rw add_succ,
rw mul_succ,
rw mul_succ,
rw hd,
rw add_assoc,
refl,
end
\[\text{mul_assoc}:(ab)c=a(bc)
\]
lemma mul_assoc (a b c : mynat) : (a * b) * c = a * (b * c) :=
begin
induction c with d hd,
repeat {rw mul_zero},
repeat {rw mul_succ},
repeat {rw mul_add},
rw hd,
refl,
end
\[\text{succ_mul}:\operatorname{succ}(a)\times b=ab+b
\]
lemma succ_mul (a b : mynat) : succ a * b = a * b + b :=
begin
induction b with d hd,
repeat {rw mul_zero},
repeat {rw zero_add},
repeat {rw mul_succ},
rw hd,
repeat {rw add_succ},
rw add_right_comm,
refl,
end
\[\text{add_mul}:(a+b)\times t=at+bt
\]
lemma add_mul (a b t : mynat) : (a + b) * t = a * t + b * t :=
begin
induction t with d hd,
repeat {rw mul_zero},
repeat {rw add_zero},
repeat {rw mul_succ},
rw hd,
simp,
end
\[\text{mul_comm}:a\times b=b\times a
\]
lemma mul_comm (a b : mynat) : a * b = b * a :=
begin
induction b with d hd,
rw mul_zero,
rw zero_mul,
refl,
rw mul_succ,
rw succ_mul,
rw hd,
refl,
end
\[\text{mul_left_comm}:a(bc)=b(ac)
\]
lemma mul_left_comm (a b c : mynat) : a * (b * c) = b * (a * c) :=
begin
induction b with d hd,
repeat {rw zero_mul},
rw mul_zero,
refl,
repeat {rw succ_mul},
rw mul_add,
rw hd,
simp,
end
\[\text{zero_pow_zero}:0^0=1
\]
lemma zero_pow_zero : (0 : mynat) ^ (0 : mynat) = 1 :=
begin
rw pow_zero,
refl,
end
\[\text{zero_pow_succ}:0^{\operatorname{succ}(m)}=0
\]
lemma zero_pow_succ (m : mynat) : (0 : mynat) ^ (succ m) = 0 :=
begin
rw pow_succ,
rw mul_zero,
refl,
end
\[\text{pow_one}:a^1=a
\]
lemma pow_one (a : mynat) : a ^ (1 : mynat) = a :=
begin
rw one_eq_succ_zero,
rw pow_succ,
rw pow_zero,
rw one_mul,
refl,
end
\[\text{one_pow}:1^m=1
\]
lemma one_pow (m : mynat) : (1 : mynat) ^ m = 1 :=
begin
induction m with d hd,
rw pow_zero,
refl,
rw pow_succ,
rw hd,
rw one_mul,
refl,
end
\[\text{pow_add}:a^{m+n}=a^ma^n
\]
lemma pow_add (a m n : mynat) : a ^ (m + n) = a ^ m * a ^ n :=
begin
induction n with d hd,
rw add_zero,
rw pow_zero,
rw mul_one,
refl,
rw add_succ,
repeat {rw pow_succ},
rw hd,
simp,
end
\[\text{mul_pow}:(ab)^n=a^nb^n
\]
lemma mul_pow (a b n : mynat) : (a * b) ^ n = a ^ n * b ^ n :=
begin
induction n with d hd,
repeat {rw pow_zero},
rw mul_one,
refl,
repeat {rw pow_succ},
rw hd,
simp,
end
\[\text{pow_pow}:(a^m)^n=a^{mn}
\]
lemma pow_pow (a m n : mynat) : (a ^ m) ^ n = a ^ (m * n) :=
begin
induction n with d hd,
rw mul_zero,
repeat {rw pow_zero},
rw mul_succ,
repeat {rw pow_succ},
rw hd,
rw pow_add,
refl,
end
\[\text{add_squared}:(a+b)^2=a^2+b^2+2ab
\]
lemma add_squared (a b : mynat) :
(a + b) ^ (2 : mynat) = a ^ (2 : mynat) + b ^ (2 : mynat) + 2 * a * b :=
begin
rw two_eq_succ_one,
rw one_eq_succ_zero,
repeat {rw pow_succ},
repeat {rw pow_zero},
repeat {rw one_mul},
repeat {rw succ_mul},
rw zero_mul,
rw zero_add,
repeat {rw add_mul},
repeat {rw mul_add},
simp,
end
Function world
\[\text{example}:\operatorname{exact}
\]
example (P Q : Type) (p : P) (h : P → Q) : Q :=
begin
exact h p,
end
\[\text{example}:\operatorname{intro}
\]
example : mynat → mynat :=
begin
intro n,
exact n,
end
\[\text{example}:\operatorname{have}
\]
example (P Q R S T U: Type)
(p : P)
(h : P → Q)
(i : Q → R)
(j : Q → T)
(k : S → T)
(l : T → U)
: U :=
begin
have t : T := j (h p),
exact l t ,
end
\[\text{example}:\operatorname{apply}
\]
example (P Q R S T U: Type)
(p : P)
(h : P → Q)
(i : Q → R)
(j : Q → T)
(k : S → T)
(l : T → U)
: U :=
begin
have t : T := j (h p),
apply l,
apply t,
end
\[\text{example}:P\to(Q\to P)
\]
example (P Q : Type) : P → (Q → P) :=
begin
intro p,
intro q,
exact p,
end
\[\text{example}:(P\to(Q\to R))\to((P\to Q)\to(P\to R))
\]
example (P Q R : Type) : (P → (Q → R)) → ((P → Q) → (P → R)) :=
begin
intros f1 f2 p,
have q := f2 p,
apply f1,
apply p,
apply q,
end
\[\text{example}:(P\to Q)\to((Q\to F)\to(P\to F))
\]
example (P Q F : Type) : (P → Q) → ((Q → F) → (P → F)) :=
begin
intros f1 f2 p,
exact f2 (f1 p),
end
\[\text{example}:(P\to Q)\to((Q\to \varnothing)\to(P\to\varnothing))
\]
example (P Q : Type) : (P → Q) → ((Q → empty) → (P → empty)) :=
begin
intros f1 f2 p,
exact f2 (f1 p),
end
\[\text{example}:\text{a big maze}
\]
example (A B C D E F G H I J K L : Type)
(f1 : A → B) (f2 : B → E) (f3 : E → D) (f4 : D → A) (f5 : E → F)
(f6 : F → C) (f7 : B → C) (f8 : F → G) (f9 : G → J) (f10 : I → J)
(f11 : J → I) (f12 : I → H) (f13 : E → H) (f14 : H → K) (f15 : I → L)
: A → L :=
begin
intro a,
apply f15,
apply f11,
apply f9,
exact f8 (f5 (f2 (f1 a))),
end
Proposition world
\[\text{example}:\begin{cases}
P\text{ is true}\\
P\to Q\text{ is true}
\end{cases}\to Q\text{ is true}
\]
example (P Q : Prop) (p : P) (h : P → Q) : Q :=
begin
exact h p,
end
\[\text{imp_self}:P\to P
\]
lemma imp_self (P : Prop) : P → P :=
begin
intro p,
exact p,
end
\[\text{maze}:
\]
lemma maze (P Q R S T U: Prop)
(p : P)
(h : P → Q)
(i : Q → R)
(j : Q → T)
(k : S → T)
(l : T → U)
: U :=
begin
exact l (j (h p)),
end
\[\text{example}:P\to(Q\to P)
\]
example (P Q : Prop) : P → (Q → P) :=
begin
intros p q,
exact p,
end
\[\text{example}:(P\to(Q\to R))\to((P\to Q)\to(P\to R))
\]
example (P Q R : Prop) : (P → (Q → R)) → ((P → Q) → (P → R)) :=
begin
intros f1 f2 p,
have q := f2 p,
exact f1 p q,
end
\[\text{imp_trans}:\begin{cases}
P\to Q\\
Q\to R\\
\end{cases}\to(P\to R)\\
\]
lemma imp_trans (P Q R : Prop) : (P → Q) → ((Q → R) → (P → R)) :=
begin
intros f1 f2 p,
exact f2 (f1 p),
end
\[\text{contrapositive}:(P\to Q)\to(\neg Q\to\neg P)
\]
lemma contrapositive (P Q : Prop) : (P → Q) → (¬ Q → ¬ P) :=
begin
repeat {rw not_iff_imp_false},
intros f nq p,
exact nq (f p),
end
\[\text{example}:\text{a big maze}
\]
example (A B C D E F G H I J K L : Prop)
(f1 : A → B) (f2 : B → E) (f3 : E → D) (f4 : D → A) (f5 : E → F)
(f6 : F → C) (f7 : B → C) (f8 : F → G) (f9 : G → J) (f10 : I → J)
(f11 : J → I) (f12 : I → H) (f13 : E → H) (f14 : H → K) (f15 : I → L)
: A → L :=
begin
intro a,
exact f15 (f11 (f9 (f8 (f5 (f2 (f1 a)))))),
end
Advanced proposition world
\[\text{example}:\operatorname{split}
\]
example (P Q : Prop) (p : P) (q : Q) : P ∧ Q :=
begin
split,
exact p,
exact q,
end
\[\text{and_symm}:P\wedge Q\to Q\wedge P
\]
lemma and_symm (P Q : Prop) : P ∧ Q → Q ∧ P :=
begin
intro h,
cases h with p q,
split,
exact q,
exact p,
end
\[\text{and_trans}:\begin{cases}
P\wedge Q\\
Q\wedge R
\end{cases}\to P\wedge R
\]
lemma and_trans (P Q R : Prop) : P ∧ Q → Q ∧ R → P ∧ R :=
begin
intros h1 h2,
cases h1 with p q,
cases h2 with q r,
split,
exact p,
exact r,
end
\[\text{iff_trans}:\begin{cases}
P\leftrightarrow Q\\
Q\leftrightarrow R
\end{cases}\to(P\leftrightarrow R)
\]
lemma iff_trans (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) :=
begin
intros h1 h2,
cases h1 with hpq hqp,
cases h2 with hqr hrq,
split,
intro p,
exact hqr (hpq p),
intro r,
exact hqp (hrq r),
end
lemma iff_trans (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) :=
begin
intros h1 h2,
rw h1,
rw h2,
end
lemma iff_trans (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) :=
begin
intros h1 h2,
split,
intro p,
exact h2.1 (h1.1 p),
intro r,
exact h1.2 (h2.2 r),
end
\[\text{example}:Q\to(P\vee Q)
\]
example (P Q : Prop) : Q → (P ∨ Q) :=
begin
intro q,
right,
exact q,
end
\[\text{or_symm}:P\vee Q\to Q\vee P
\]
lemma or_symm (P Q : Prop) : P ∨ Q → Q ∨ P :=
begin
intro h,
cases h with p q,
right,
exact p,
left,
exact q,
end
\[\text{and_or_distrib_left}:P\wedge(Q\vee R)\leftrightarrow(P\wedge Q)\vee(P\wedge R)
\]
lemma and_or_distrib_left (P Q R : Prop) : P ∧ (Q ∨ R) ↔ (P ∧ Q) ∨ (P ∧ R) :=
begin
split,
intro h,
cases h with p or_qr,
cases or_qr with q r,
left,
split,
exact p,
exact q,
right,
split,
exact p,
exact r,
intro h,
cases h with and_pq and_pr,
cases and_pq with p q,
split,
exact p,
left,
exact q,
cases and_pr with p r,
split,
exact p,
right,
exact r,
end
\[\text{contra}:(P\wedge(\neg P))\to Q
\]
lemma contra (P Q : Prop) : (P ∧ ¬ P) → Q :=
begin
intro h,
rw not_iff_imp_false at h,
exfalso,
exact h.2 h.1,
end
\[\text{contrapositive2}:(\neg Q\to\neg P)\to(P\to Q)
\]
lemma contrapositive2 (P Q : Prop) : (¬ Q → ¬ P) → (P → Q) :=
begin
by_cases p: P; by_cases q: Q,
repeat {cc},
end
Advanced Addition World
\[\text{succ_inj'}:\operatorname{succ}(a)=\operatorname{succ}(b)\to a=b
\]
theorem succ_inj' {a b : mynat} (hs : succ(a) = succ(b)) : a = b :=
begin
apply succ_inj,
rw hs,
refl,
-- exact succ_inj hs,
end
\[\text{succ_succ_inj}:\operatorname{succ}(\operatorname{succ}(a))=\operatorname{succ}(\operatorname{succ}(b))\to\operatorname{succ}(a)=\operatorname{succ}(b)
\]
theorem succ_succ_inj {a b : mynat} (h : succ(succ(a)) = succ(succ(b))) : a = b :=
begin
exact succ_inj (succ_inj h),
end
\[\text{succ_eq_succ_of_eq}:a=b\to\operatorname{succ}(a)=\operatorname{succ}(b)
\]
theorem succ_eq_succ_of_eq {a b : mynat} : a = b → succ(a) = succ(b) :=
begin
intro eq,
induction b with d hd,
rw eq,
refl,
rw eq,
refl,
end
\[\text{eq_iff_succ_eq_succ}:a=b\leftrightarrow\operatorname{succ}(a)=\operatorname{succ}(b)
\]
theorem succ_eq_succ_iff (a b : mynat) : succ a = succ b ↔ a = b :=
begin
split,
exact succ_inj,
exact succ_eq_succ_of_eq,
end
\[\text{add_right_cancel}:a+t=b+t\to a=b
\]
theorem add_right_cancel (a t b : mynat) : a + t = b + t → a = b :=
begin
intro h,
induction t with d hd,
repeat {rw add_zero at h},
exact h,
repeat {rw add_succ at h},
exact hd (succ_inj h),
end
\[\text{add_left_cancel}:t+a=t+b\to a=b
\]
theorem add_left_cancel (t a b : mynat) : t + a = t + b → a = b :=
begin
intro h,
induction t with d hd,
repeat {rw zero_add at h},
exact h,
repeat {rw succ_add at h},
exact hd (succ_inj h),
end
theorem add_left_cancel (t a b : mynat) : t + a = t + b → a = b :=
begin
intro h,
apply add_right_cancel a t b,
rw add_comm at h,
rw h,
rw add_comm,
refl,
end
theorem add_left_cancel (t a b : mynat) : t + a = t + b → a = b :=
begin
intro h,
apply add_right_cancel a t b,
simp,
exact h,
end
theorem add_left_cancel (t a b : mynat) : t + a = t + b → a = b :=
begin
rw add_comm t a,
rw add_comm t b,
exact add_right_cancel a t b,
end
\[\text{add_right_cancel_iff}:a+t=b+t\leftrightarrow a=b
\]
theorem add_right_cancel_iff (t a b : mynat) : a + t = b + t ↔ a = b :=
begin
split,
exact add_right_cancel _ _ _,
intro h,
rw h,
refl,
end
\[\text{eq_zero_of_add_right_eq_self}:a+b=a\to b=0
\]
lemma eq_zero_of_add_right_eq_self {a b : mynat} : a + b = a → b = 0 :=
begin
rw ← add_zero a,
rw add_assoc,
rw zero_add b,
exact add_left_cancel a b 0,
end
\[\text{succ_ne_zero}:\operatorname{succ}(a)\ne0
\]
theorem succ_ne_zero (a : mynat) : succ a ≠ 0 :=
begin
symmetry,
exact zero_ne_succ a,
end
\[\text{add_left_eq_zero}:a+b=0\to b=0
\]
lemma add_left_eq_zero {{a b : mynat}} (H : a + b = 0) : b = 0 :=
begin
cases b with d,
refl,
rw add_succ at H,
exfalso,
exact succ_ne_zero (a + d) H,
end
\[\text{add_right_eq_zero}:a+b=0\to a=0.
\]
lemma add_right_eq_zero {a b : mynat} : a + b = 0 → a = 0 :=
begin
rw add_comm,
apply add_left_eq_zero,
end
\[\text{add_one_eq_succ}:d+1=\operatorname{succ}(d)
\]
theorem add_one_eq_succ (d : mynat) : d + 1 = succ d :=
begin
symmetry,
apply succ_eq_add_one,
end
\[\text{ne_succ_self}:n\ne\operatorname{succ}(n)
\]
lemma ne_succ_self (n : mynat) : n ≠ succ n :=
begin
symmetry,
rw succ_eq_add_one,
rw one_eq_succ_zero,
intro h,
exact succ_ne_zero 0 (eq_zero_of_add_right_eq_self h),
end
Advanced Multiplication World
\[\text{mul_pos}:a\ne0\to b\ne0\to ab\ne0
\]
theorem mul_pos (a b : mynat) : a ≠ 0 → b ≠ 0 → a * b ≠ 0 :=
begin
intros h1 h2,
cases a with c,
simp,
apply h1,
refl,
rw succ_mul,
intro h3,
apply h2,
exact add_left_eq_zero h3,
end
theorem mul_pos (a b : mynat) : a ≠ 0 → b ≠ 0 → a * b ≠ 0 :=
begin
cases a with c; cases b with d,
simp, simp, simp,
intros h1 h2 h3,
rw succ_mul at h3,
rw add_succ at h3,
exact succ_ne_zero _ h3,
end
\[\text{eq_zero_or_eq_zero_of_mul_eq_zero}:ab=0\to a=0\vee b=0
\]
theorem eq_zero_or_eq_zero_of_mul_eq_zero (a b : mynat) (h : a * b = 0) :
a = 0 ∨ b = 0 :=
begin
cases a with c; cases b with d,
simp, simp, simp,
exfalso,
rw succ_mul at h,
rw add_succ at h,
exact succ_ne_zero _ h,
end
\[\text{mul_eq_zero_iff}:ab=0\leftrightarrow a=0\vee b=0
\]
theorem mul_eq_zero_iff (a b : mynat): a * b = 0 ↔ a = 0 ∨ b = 0 :=
begin
split,
apply eq_zero_or_eq_zero_of_mul_eq_zero,
intro h,
cases h,
rw h, simp,
rw h, simp,
end
