2a67a49f31
This PR deprecates the tactics `simp_arith`, `simp_arith!`, `simp_all_arith` and `simp_all_arith!`. Users can just use the `+arith` option.
64 lines
1.8 KiB
Text
64 lines
1.8 KiB
Text
def Nat.bit (b : Bool) (n : Nat) : Nat :=
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cond b (2*n+1) (2*n)
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theorem Nat.bit_div_even (h : n % 2 = 0) : bit false (n / 2) = n := by
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simp [bit]
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have := Nat.div_add_mod n 2
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simp [h] at this
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assumption
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theorem Nat.bit_div_odd (h : n % 2 ≠ 0) : bit true (n / 2) = n := by
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have h : n % 2 = 1 := by
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have := mod_lt n (by decide : 2 > 0)
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revert h this
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generalize n%2 = k
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match k with
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| 0 => decide
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| 1 => decide
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| n+2 => intros; contradiction
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simp [bit]
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have := Nat.div_add_mod n 2
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simp [h] at this
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assumption
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theorem Nat.div2_lt (h : n ≠ 0) : n / 2 < n := by
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match n with
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| 1 => decide
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| 2 => decide
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| 3 => decide
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| n+4 =>
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rw [div_eq, if_pos]
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refine succ_lt_succ (Nat.lt_trans ?_ (lt_succ_self _))
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exact @div2_lt (n+2) (by simp +arith)
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simp +arith
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@[specialize]
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def Nat.binrec
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(motive : Nat → Sort u)
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(base : Unit → motive 0)
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(ind : (b : Bool) → (n : Nat) → (Unit → motive n) → motive (bit b n))
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(n : Nat) : motive n :=
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if h₁ : n = 0 then
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h₁ ▸ base ()
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else if h₂ : n % 2 = 0 then
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bit_div_even h₂ ▸ ind false (n / 2) (fun _ => binrec motive base ind (n / 2))
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else
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bit_div_odd h₂ ▸ ind true (n / 2) (fun _ => binrec motive base ind (n / 2))
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termination_by n
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decreasing_by all_goals exact Nat.div2_lt h₁
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theorem Nat.binind
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(motive : Nat → Prop)
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(base : motive 0)
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(ind : (b : Bool) → (n : Nat) → motive n → motive (bit b n))
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(n : Nat) : motive n :=
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binrec motive (fun _ => base) (fun b n ih => ind b n (ih ())) n
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set_option trace.compiler.ir.result true in
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def Nat.toBit (n : Nat) : List Bool :=
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binrec (fun _ => List Bool)
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(fun _ => [])
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(fun b n ih => b :: ih ())
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n
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#guard Nat.toBit 18 == [false, true, false, false, true]
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