internlm/Lean-Github huggingface dataset:
Theorem: partition.partition
Proof state before:
case right
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
  { inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
    inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
    inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
  partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
    (Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ result.snd[result.fst.val] = arr[first] ∧ ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
Tactic:
. apply And.intro
  . simp [*]
  . simp [*]
    apply p₃
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
⊢ ↑mid.val < n
Tactic:
assumption
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ result.fst = mid
Tactic:
rw [←eq]
Proof state after:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (_root_.partition arr first last fl).fst = mid
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (_root_.partition arr first last fl).fst = mid
Tactic:
unfold partition
Proof state after:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (let result := partitionImpl arr first last last fl ⋯;
      let mid := result.fst;
      let arr_1 := result.snd;
      (mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).fst =
    mid
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this⟩
⊢ (let result := partitionImpl arr first last last fl ⋯;
      let mid := result.fst;
      let arr_1 := result.snd;
      (mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).fst =
    mid
Tactic:
simp [afterLoop]
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ result.snd = swapped
Tactic:
rw [←eq]
Proof state after:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (_root_.partition arr first last fl).snd = swapped
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (_root_.partition arr first last fl).snd = swapped
Tactic:
unfold partition
Proof state after:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (let result := partitionImpl arr first last last fl ⋯;
      let mid := result.fst;
      let arr_1 := result.snd;
      (mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).snd =
    swapped
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝⟩
this : result.fst = mid
⊢ (let result := partitionImpl arr first last last fl ⋯;
      let mid := result.fst;
      let arr_1 := result.snd;
      (mid, (dbgTraceIfShared "swap2" arr_1).swap first ⟨↑mid.val, ⋯⟩)).snd =
    swapped
Tactic:
simp [afterLoop, dbgTraceIfShared]
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
⊢ arr'[first] = arr[first]
Tactic:
apply partitionImpl.first_eq
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ partitionImpl.LoopInvariant arr first last last last
Tactic:
apply partitionImpl.LoopInvariant.intro
Proof state after:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first]

case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first]

case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Tactic:
. intro k lk kl
  exact (Nat.lt_irrefl k (Nat.lt_trans kl lk)).elim
Proof state after:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Tactic:
. intro k lk kl
  exact (Nat.lt_irrefl k (Nat.lt_of_le_of_lt kl lk)).elim
Proof state after:
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Tactic:
. intro ll
  exact (Nat.lt_irrefl last ll).elim
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k < last → arr[k] < arr[first]
Tactic:
intro k lk kl
Proof state after:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k < last
⊢ arr[k] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k < last
⊢ arr[k] < arr[first]
Tactic:
exact (Nat.lt_irrefl k (Nat.lt_trans kl lk)).elim
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ ∀ (k : Fin n), last < k → k ≤ last → ¬arr[k] < arr[first]
Tactic:
intro k lk kl
Proof state after:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k ≤ last
⊢ ¬arr[k] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
k : Fin n
lk : last < k
kl : k ≤ last
⊢ ¬arr[k] < arr[first]
Tactic:
exact (Nat.lt_irrefl k (Nat.lt_of_le_of_lt kl lk)).elim
Proof state after:
no goals
Theorem: partition.partition
Proof state before:
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
⊢ last < last → arr[last] < arr[first]
Tactic:
intro ll
Proof state after:
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
ll : last < last
⊢ arr[last] < arr[first]
Theorem: partition.partition
Proof state before:
case inv₃
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
ll : last < last
⊢ arr[last] < arr[first]
Tactic:
exact (Nat.lt_irrefl last ll).elim
Proof state after:
no goals