56 lines
1.8 KiB
Plaintext
56 lines
1.8 KiB
Plaintext
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namespace Nat
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theorem add_equations: ∀{a b c d: Nat}, a = b → c = d → a + c = b + d := by
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intro a b c d ab cd
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rw [ab, cd]
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theorem add_to_equation_right: ∀{a b c: Nat}, a = b → a + c = b + c := by
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intro a b c ab
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exact add_equations ab rfl
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theorem succ_equation : ∀{a b: Nat}, Nat.succ a = Nat.succ b → a = b := by
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intro a b
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apply Eq.mp -- (a + 1 = b + 1) = a + b
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apply Nat.succ.injEq -- this was already here?
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inductive A : Nat -> Type
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| ooo: (n: Nat) → A n
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theorem subtract_to_equation_right: ∀{a b c: Nat}, a + c = b + c → a = b := by
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intro a b c acbc
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induction c with
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| zero =>
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repeat rw [Nat.add_zero] at acbc
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exact acbc
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| succ pc inner =>
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/-
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a + S pc = b + S pc -- comm
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S pc + a = S pc + b -- addition definition
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S (pc + a) = S (pc + b) -- injective constructor
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pc + a = pc + b
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-/
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rw [Nat.add_comm a (Nat.succ pc), Nat.add_comm b (Nat.succ pc)] at acbc
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simp [Nat.succ_add, Nat.add_comm] at acbc
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exact (inner acbc)
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theorem subtract_to_equation_left: ∀{a b c: Nat}, c + a = c + b → a = b := by
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intro a b c cacb
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rw [Nat.add_comm c a, Nat.add_comm c b] at cacb
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exact (subtract_to_equation_right cacb)
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theorem add_equation_both_sides_right: ∀{a b c: Nat}, (a = b) ↔ (a + c = b + c) := by
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intro a b c
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apply Iff.intro
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. exact add_to_equation_right
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. exact subtract_to_equation_right
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theorem multiply_equation_left: ∀{a b c: Nat}, (a = b) → (c * a = c * b) := by
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intro a b c ab
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rw [ab]
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theorem double.addition_is_multiplication (x: Nat): 2 * x = x + x := by
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induction x with
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| zero => simp
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| succ px ic =>
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simp [<-Nat.add_one, Nat.left_distrib, ic, Nat.add_left_comm, Nat.add_assoc]
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end Nat
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