internlm/Lean-Github huggingface dataset:
Theorem: interval_shift₂
Proof state before:
case inr
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
⊢ s ≤ w ∨ w ≤ s ∧ y < s
Tactic:
cases Nat.lt_or_ge y s with
| inl hys => exact Or.inr ⟨hws, hys⟩
| inr hsy => exact Or.inl (Nat.le_trans hsy hyw)
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inr.inl
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ s ≤ w ∨ w ≤ s ∧ y < s
Tactic:
exact Or.inr ⟨hws, hys⟩
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inr.inr
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hsy : y ≥ s
⊢ s ≤ w ∨ w ≤ s ∧ y < s
Tactic:
exact Or.inl (Nat.le_trans hsy hyw)
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inl
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (w - s) = w - s ∧ trunc n (s - w) = s - w
Tactic:
constructor
Proof state after:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (w - s) = w - s

case inl.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (w - s) = w - s

case inl.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (s - w) = s - w
Tactic:
. have hfr : trunc n (round (x - t)) = round (x - t) := trunc_round npos hfaithful₁ (x - t)
  have hrx : round (x - t) ≤ x := round_sub_le npos hfaithful₀ hfaithful₁ t hfx
  apply s₁ npos hsw hwx hfs hfx hfw
  rewrite [hs]
  exact s₂ npos hfaithful₀ hfaithful₁ hfx hfr hrx
Proof state after:
case inl.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inl.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (s - w) = s - w
Tactic:
. rw [tsub_eq_zero_of_le hsw, trunc_zero]
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (w - s) = w - s
Tactic:
have hfr : trunc n (round (x - t)) = round (x - t) := trunc_round npos hfaithful₁ (x - t)
Proof state after:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
⊢ trunc n (w - s) = w - s
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
⊢ trunc n (w - s) = w - s
Tactic:
have hrx : round (x - t) ≤ x := round_sub_le npos hfaithful₀ hfaithful₁ t hfx
Proof state after:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (w - s) = w - s
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (w - s) = w - s
Tactic:
apply s₁ npos hsw hwx hfs hfx hfw
Proof state after:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (x - s) = x - s
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (x - s) = x - s
Tactic:
rewrite [hs]
Proof state after:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (x - round (x - round (x - t))) = x - round (x - round (x - t))
Theorem: interval_shift₂
Proof state before:
case inl.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
hfr : trunc n (round (x - t)) = round (x - t)
hrx : round (x - t) ≤ x
⊢ trunc n (x - round (x - round (x - t))) = x - round (x - round (x - t))
Tactic:
exact s₂ npos hfaithful₀ hfaithful₁ hfx hfr hrx
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inl.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hsw : s ≤ w
⊢ trunc n (s - w) = s - w
Tactic:
rw [tsub_eq_zero_of_le hsw, trunc_zero]
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inr.intro
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (w - s) = w - s ∧ trunc n (s - w) = s - w
Tactic:
constructor
Proof state after:
case inr.intro.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (w - s) = w - s

case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inr.intro.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (w - s) = w - s

case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (s - w) = s - w
Tactic:
. rw [tsub_eq_zero_of_le hws, trunc_zero]
Proof state after:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (s - w) = s - w
Tactic:
. have hyy' : y < round (x - round (x - y)) := Nat.lt_of_lt_of_le hys hsy'
  have hwy' : w ≤ round (x - round (x - y)) := Nat.le_trans hws hsy'
  apply s₁ npos hws hsy' hfw (trunc_round npos hfaithful₁ _) hfs
  exact s₃ npos hfaithful₀ hfaithful₁ hfx hfy hfw hyx hyw hxyulp hyy' hwy'
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inr.intro.left
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (w - s) = w - s
Tactic:
rw [tsub_eq_zero_of_le hws, trunc_zero]
Proof state after:
no goals
Theorem: interval_shift₂
Proof state before:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
⊢ trunc n (s - w) = s - w
Tactic:
have hyy' : y < round (x - round (x - y)) := Nat.lt_of_lt_of_le hys hsy'
Proof state after:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
⊢ trunc n (s - w) = s - w
Tactic:
have hwy' : w ≤ round (x - round (x - y)) := Nat.le_trans hws hsy'
Proof state after:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
hwy' : w ≤ round (x - round (x - y))
⊢ trunc n (s - w) = s - w
Theorem: interval_shift₂
Proof state before:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
hwy' : w ≤ round (x - round (x - y))
⊢ trunc n (s - w) = s - w
Tactic:
apply s₁ npos hws hsy' hfw (trunc_round npos hfaithful₁ _) hfs
Proof state after:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
hwy' : w ≤ round (x - round (x - y))
⊢ trunc n (round (x - round (x - y)) - w) = round (x - round (x - y)) - w
Theorem: interval_shift₂
Proof state before:
case inr.intro.right
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfw : trunc n w = w
hxyulp : ulp n x ≤ 2 * y
ht : t = round (y - round (y - z))
hs₂ : t ≤ x → s = round (x - round (x - t))
hyw : y ≤ w
hwx : w ≤ x
hty : t ≤ y
hyx : y ≤ x
htx : t ≤ x
hs : s = round (x - round (x - t))
hfs : trunc n s = s
hft : trunc n t = t
hsy' : s ≤ round (x - round (x - y))
hws : w ≤ s
hys : y < s
hyy' : y < round (x - round (x - y))
hwy' : w ≤ round (x - round (x - y))
⊢ trunc n (round (x - round (x - y)) - w) = round (x - round (x - y)) - w
Tactic:
exact s₃ npos hfaithful₀ hfaithful₁ hfx hfy hfw hyx hyw hxyulp hyy' hwy'
Proof state after:
no goals