332 lines
8.9 KiB
Text
332 lines
8.9 KiB
Text
open Lean Grind
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/--
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info: Try these:
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[apply] cases #c4b6 <;> cases #4c68 <;> ring
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[apply] finish only [#c4b6, #4c68]
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-/
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#guard_msgs in
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example {α : Type} [CommRing α] (a b c d e : α) :
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(a * a = b * c ∨ a^2 = c * b) →
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(a^2 = c * b ∨ e^2 = d * c) →
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(b^2 = d*c ∨ b^2 = c*d) →
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a*b*(b*a) = c^2*b*d := by
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grind => finish?
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/--
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info: Try these:
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[apply] ⏎
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cases #b0f4
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· cases #50fc
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· cases #50fc <;> lia
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[apply] finish only [#b0f4, #50fc]
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-/
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#guard_msgs in
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example (p : Nat → Prop) (x y z w : Int) :
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(x = 1 ∨ x = 2) →
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(w = 1 ∨ w = 4) →
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(y = 1 ∨ (∃ x : Nat, y = 3 - x ∧ p x)) →
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(z = 1 ∨ z = 0) → x + y ≤ 6 := by
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grind => finish?
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/-- error: tactic is not applicable -/
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#guard_msgs in
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example (a b c : Int) : a + b ≤ 2 → b = c → 2*b - c + a ≤ 3 := by
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grind => cases_next
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example (p : Nat → Prop) (x y z w : Int) :
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(x = 1 ∨ x = 2) →
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(w = 1 ∨ w = 4) →
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(y = 1 ∨ (∃ x : Nat, y = 3 - x ∧ p x)) →
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(z = 1 ∨ z = 0) → x + y ≤ 6 := by
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grind =>
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cases_next <;> cases_next <;> cases_next <;> cases_next <;> lia
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example (p : Nat → Prop) (x y z w : Int) :
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(x = 1 ∨ x = 2) →
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(w = 1 ∨ w = 4) →
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(y = 1 ∨ (∃ x : Nat, y = 3 - x ∧ p x)) →
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(z = 1 ∨ z = 0) → x + y ≤ 6 := by
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grind =>
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repeat (first (lia) (cases_next))
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example (p : Nat → Prop) (x y z w : Int) :
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(x = 1 ∨ x = 2) →
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(w = 1 ∨ w = 4) →
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(y = 1 ∨ (∃ x : Nat, y = 3 - x ∧ p x)) →
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(z = 1 ∨ z = 0) → x + y ≤ 6 := by
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grind =>
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repeat (first (cases_next) (lia))
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example (p : Nat → Prop) (x y z w : Int) :
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(x = 1 ∨ x = 2) →
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(w = 1 ∨ w = 4) →
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(y = 1 ∨ (∃ x : Nat, y = 3 - x ∧ p x)) →
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(z = 1 ∨ z = 0) → x + y ≤ 6 := by
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grind =>
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repeat (first (ring) (cases_next) (lia))
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/--
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info: Try these:
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[apply] cases #5c4b <;> cases #896f <;> ac
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[apply] finish only [#5c4b, #896f]
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-/
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#guard_msgs in
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example {α : Type} (op : α → α → α) [Std.Associative op] [Std.Commutative op] (a b c d e : α) :
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(op a a = op b c ∨ op a a = op c b) →
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(op a a = op c b ∨ op e e = op d c) →
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(op b b = op d c ∨ op b b = op c d) →
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op (op a b) (op b a) = op (op c c) (op b d) := by
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grind => finish?
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/--
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info: Try these:
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[apply] ⏎
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instantiate only [= Array.getElem_set]
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instantiate only [= Array.getElem_set]
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[apply] finish only [= Array.getElem_set]
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-/
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#guard_msgs in
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example (as bs cs : Array α) (v₁ v₂ : α)
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(i₁ i₂ j : Nat)
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(h₁ : i₁ < as.size)
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(h₂ : bs = as.set i₁ v₁)
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(h₃ : i₂ < bs.size)
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(h₃ : cs = bs.set i₂ v₂)
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(h₄ : i₁ ≠ j ∧ i₂ ≠ j)
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(h₅ : j < cs.size)
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(h₆ : j < as.size)
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: cs[j] = as[j] := by
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grind => finish?
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set_option warn.sorry false
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/--
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info: Try this:
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[apply] ⏎
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cases #c4b6
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· cases #8c9f
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· ring
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· sorry
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· cases #8c9f
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· ring
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· sorry
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-/
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#guard_msgs in
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example {α : Type} [CommRing α] (a b c d e : α) :
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(a^2 = c * b ∨ e^2 = d * c) →
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(b^2 = d*c ∨ b^2 = c*d) →
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a*b*(b*a) = c^2*b*d := by
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grind => finish?
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/--
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info: Try this:
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[apply] ⏎
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instantiate only [= Nat.min_def]
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cases #7640
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· sorry
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· lia
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-/
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#guard_msgs in
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example (as : Array α) (lo hi i j : Nat) (h₁ : lo ≤ i) (_ : i < j) (_ : j ≤ hi) (_ : j < as.size)
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(_ : ¬as.size = 0) : min lo (as.size - 1) < i := by
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grind => finish?
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/--
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info: Try these:
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[apply] ⏎
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instantiate only [= getMsbD_setWidth']
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cases #aa9d
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[apply] finish only [= getMsbD_setWidth', #aa9d]
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-/
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#guard_msgs in
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open BitVec in
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example (ge : m ≥ n) (x : BitVec n) (i : Nat) :
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getMsbD (setWidth' ge x) i = (decide (m - n ≤ i) && getMsbD x (i + n - m)) := by
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grind => finish?
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open BitVec in
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example (ge : m ≥ n) (x : BitVec n) (i : Nat) :
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getMsbD (setWidth' ge x) i = (decide (m - n ≤ i) && getMsbD x (i + n - m)) := by
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grind =>
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instantiate only [= getMsbD_setWidth']
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cases #aa9d
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/--
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info: Try these:
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[apply] cases #9942 <;>
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instantiate only [= BitVec.getElem_and] <;> instantiate only [= BitVec.getElem_or] <;> cases #cfbc
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[apply] finish only [= BitVec.getElem_and, = BitVec.getElem_or, #9942, #cfbc]
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-/
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#guard_msgs in
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example (x y : BitVec 64) : (x ||| y) &&& x = x := by
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grind => finish?
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set_option trace.Meta.debug true in
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example (x y : BitVec 64) : (x ||| y) &&& x = x := by
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grind => finish?
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macro_rules | `(tactic| get_elem_tactic_extensible) => `(tactic| grind)
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/--
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info: Try these:
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[apply] ⏎
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instantiate only [= Array.getElem_set]
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ring
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[apply] finish only [= Array.getElem_set]
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-/
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#guard_msgs in
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example (a : Array (BitVec 64)) (i : Nat) (v : BitVec 64)
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: (_ : i < a.size) → (_ : i + 1 < a.size) → (a.set i v)[i+1] + a[i+1] = 2*a[i+1] := by
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grind => finish?
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/--
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info: Try these:
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[apply] ⏎
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mbtc
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cases #a6c8
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[apply] finish only [#a6c8]
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-/
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#guard_msgs in
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example (f : Nat → Nat) (x : Nat)
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: x ≠ 0 → x ≤ 1 → f x = 2 → f 1 = 2 := by
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grind => finish?
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/--
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info: Try these:
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[apply] ⏎
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mbtc
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cases #beb4
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[apply] finish only [#beb4]
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-/
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#guard_msgs in
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example (f : Int → Int → Int) (x y : Int)
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: 0 ≤ x → x ≠ 0 → x ≤ 1 → f x y = 2 → f 1 y = 2 := by
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grind => finish?
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example (f : Int → Int → Int) (x y : Int)
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: 0 ≤ x → x ≠ 0 → x ≤ 1 → f x y = 2 → f 1 y = 2 := by
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grind =>
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-- We can use `have` to golf proofs using `mbtc` and `cases`
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have : x = 1
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example (f : Int → Int) (x y : Int)
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: 0 ≤ x → x ≤ 2 → f 0 = y → f 1 = y → f 2 = y → f x = y := by
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grind
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example (f : Int → Int) (x y : Int)
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: 0 ≤ x → x ≤ 2 → f 0 = y → f 1 = y → f 2 = y → f x = y := by
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grind =>
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mbtc
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cases #23ad <;> mbtc <;> cases #beb4 <;> mbtc <;> cases #beed
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example (f : Int → Int) (x y : Int)
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: 0 ≤ x → x ≤ 2 → f 0 = y → f 1 = y → f 2 = y → f x = y := by
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grind =>
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-- Again, we can use `have` to golf the proof with `mbtc`
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have : x ≠ 0
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have : x ≠ 1
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have : x ≠ 2
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example (f g : Int → Int) (x y z w : Int)
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: 0 ≤ x → x ≤ 1 → 0 ≤ w →
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g 0 = z → g 1 = z → g 2 = z →
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f 0 = y → f 1 = y →
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g w ≠ z → f x = y := by
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set_option trace.grind.split true in
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grind =>
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mbtc
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cases #23ad
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mbtc
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cases #beb4
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/--
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trace: [grind.split] w = 0, generation: 0
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[grind.split] x = 0, generation: 0
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[grind.split] w = 1, generation: 0
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[grind.split] x = 1, generation: 0
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-/
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#guard_msgs in
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example (f g : Int → Int) (x y z w : Int)
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: 0 ≤ x → x ≤ 1 → 0 ≤ w →
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g 0 = z → g 1 = z → g 2 = z →
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f 0 = y → f 1 = y →
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g w ≠ z → f x = y := by
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set_option trace.grind.split true in
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grind
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/--
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trace: [grind.split] x = 0, generation: 0
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[grind.split] x = 1, generation: 0
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-/
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#guard_msgs in
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example (f g : Int → Int) (x y z w : Int)
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: 0 ≤ x → x ≤ 1 → 0 ≤ w →
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g 0 = z → g 1 = z → g 2 = z →
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f 0 = y → f 1 = y →
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g w ≠ z → f x = y := by
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fail_if_success grind [#23ad] -- not possible to solve using this set of anchors.
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set_option trace.grind.split true in
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grind only [#23ad, #beb4] -- Only these two splits were performed.
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/--
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trace: [grind.split] x = 0, generation: 0
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[grind.split] x = 1, generation: 0
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-/
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#guard_msgs in
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example (f g : Int → Int) (x y z w : Int)
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: 0 ≤ x → x ≤ 1 → 0 ≤ w →
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g 0 = z → g 1 = z → g 2 = z →
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f 0 = y → f 1 = y →
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g w ≠ z → f x = y := by
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set_option trace.grind.split true in
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grind => finish only [#23ad, #beb4] -- Only these two splits were performed.
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/--
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trace: [grind.ematch.instance] h: f (f a) = f a
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[grind.ematch.instance] h: f (f (f a)) = f (f a)
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[grind.ematch.instance] h: f (f (f (f a))) = f (f (f a))
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[grind.ematch.instance] h_1: g (g (g b)) = g (g b)
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[grind.ematch.instance] h_1: g (g b) = g b
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-/
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#guard_msgs in
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example (f g : Int → Int)
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(_ : ∀ x, f (f x) = f x)
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(_ : ∀ x, g (g x) = g x)
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(a b : Int)
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(_ : g (g b) = b)
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: f (f (f a)) = f a := by
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set_option trace.grind.ematch.instance true in
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grind
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/--
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trace: [grind.ematch.instance] h✝³: f (f a) = f a
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[grind.ematch.instance] h✝³: f (f (f a)) = f (f a)
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[grind.ematch.instance] h✝³: f (f (f (f a))) = f (f (f a))
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-/
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#guard_msgs in
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example (f g : Int → Int)
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(_ : ∀ x, f (f x) = f x)
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(_ : ∀ x, g (g x) = g x)
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(a b : Int)
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(_ : g (g b) = b)
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: f (f (f a)) = f a := by
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set_option trace.grind.ematch.instance true in
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grind only [#99cb]
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/--
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trace: [grind.ematch.instance] h✝³: f (f a) = f a
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[grind.ematch.instance] h✝³: f (f (f a)) = f (f a)
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[grind.ematch.instance] h✝³: f (f (f (f a))) = f (f (f a))
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-/
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#guard_msgs in
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example (f g : Int → Int)
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(_ : ∀ x, f (f x) = f x)
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(_ : ∀ x, g (g x) = g x)
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(a b : Int)
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(_ : g (g b) = b)
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: f (f (f a)) = f a := by
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set_option trace.grind.ematch.instance true in
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grind => finish only [#99cb]
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