`mul_def`

proved

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module: IndisputableMonolith.Foundation.ArithmeticFromLogic
domain: Foundation
line: 177 · github
papers citing: none yet

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IndisputableMonolith.Foundation.ArithmeticFromLogic on GitHub at line 177.

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formal source

 174
 175instance : Mul LogicNat := ⟨mul⟩
 176
 177@[simp] theorem mul_def (n m : LogicNat) : n * m = mul n m := rfl
 178
 179@[simp] theorem mul_zero (n : LogicNat) : n * zero = zero := rfl
 180
 181@[simp] theorem mul_succ (n m : LogicNat) : n * succ m = n * m + n := rfl
 182
 183theorem zero_mul (n : LogicNat) : zero * n = zero := by
 184  induction n with
 185  | identity => rfl
 186  | step n ih =>
 187    show zero * succ n = zero
 188    rw [mul_succ, ih, zero_add]
 189
 190theorem mul_one (n : LogicNat) : n * succ zero = n := by
 191  show n * succ zero = n
 192  rw [mul_succ, mul_zero, zero_add]
 193
 194theorem one_mul (n : LogicNat) : succ zero * n = n := by
 195  induction n with
 196  | identity => rfl
 197  | step n ih =>
 198    show succ zero * succ n = succ n
 199    rw [mul_succ, ih]
 200    show n + succ zero = succ n
 201    rw [add_succ, add_zero]
 202
 203theorem mul_add (a b c : LogicNat) : a * (b + c) = a * b + a * c := by
 204  induction c with
 205  | identity =>
 206    show a * (b + zero) = a * b + a * zero
 207    rw [add_zero, mul_zero, add_zero]

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