internlm/Lean-Github huggingface dataset:
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ (assignments.modify (↑i) (removeAssignment true))[↑i] = addAssignment false f.assignments[↑i]
Tactic:
have i_in_bounds : i.1 < assignments.size := hsize ▸ i.2
Proof state after:
case right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
i_in_bounds : ↑i < assignments.size
⊢ (assignments.modify (↑i) (removeAssignment true))[↑i] = addAssignment false f.assignments[↑i]
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
i_in_bounds : ↑i < assignments.size
⊢ (assignments.modify (↑i) (removeAssignment true))[↑i] = addAssignment false f.assignments[↑i]
Tactic:
rw [Array.get_modify_at_idx i_in_bounds, ih3, ih4]
Proof state after:
case right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
i_in_bounds : ↑i < assignments.size
⊢ removeAssignment true both = addAssignment false unassigned
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
i_in_bounds : ↑i < assignments.size
⊢ removeAssignment true both = addAssignment false unassigned
Tactic:
decide
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ¬hasAssignment false f.assignments[↑i] = true ∧ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Tactic:
constructor
Proof state after:
case right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ¬hasAssignment false f.assignments[↑i] = true

case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ¬hasAssignment false f.assignments[↑i] = true

case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Tactic:
. simp only [hasAssignment, hasNegAssignment, ih4, ite_false, not_false_eq_true]
Proof state after:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Tactic:
. intro k k_ge_idx_add_one k_ne_j2
  intro h1
  by_cases units[k.1].2
  . next h2 =>
    have k_ne_j1 : k ≠ j1 := by
      rw [← idx_eq_j1]
      intro k_eq_idx
      rw [k_eq_idx] at k_ge_idx_add_one
      exact Nat.lt_irrefl idx.1 $ Nat.lt_of_succ_le k_ge_idx_add_one
    have h3 := units_nodup k j1 k_ne_j1
    simp only [Fin.getElem_fin, ih1, ← h1, ← h2, ne_eq] at h3
    exact h3 rfl
  . next h2 =>
    have h3 := units_nodup k j2 k_ne_j2
    simp only [Bool.not_eq_true] at h2
    simp only [Fin.getElem_fin, ih2, ← h1, ← h2, ne_eq] at h3
    exact h3 rfl
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ¬hasAssignment false f.assignments[↑i] = true
Tactic:
simp only [hasAssignment, hasNegAssignment, ih4, ite_false, not_false_eq_true]
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
Tactic:
intro k k_ge_idx_add_one k_ne_j2
Proof state after:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
⊢ units[k].fst.val ≠ ↑i
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
⊢ units[k].fst.val ≠ ↑i
Tactic:
intro h1
Proof state after:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
⊢ False
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case right.right.right.right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
⊢ False
Tactic:
by_cases units[k.1].2
Proof state after:
case pos
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : units[↑k].snd = true
⊢ False

case neg
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : ¬units[↑k].snd = true
⊢ False
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case pos
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : units[↑k].snd = true
⊢ False

case neg
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : ¬units[↑k].snd = true
⊢ False
Tactic:
. next h2 =>
  have k_ne_j1 : k ≠ j1 := by
    rw [← idx_eq_j1]
    intro k_eq_idx
    rw [k_eq_idx] at k_ge_idx_add_one
    exact Nat.lt_irrefl idx.1 $ Nat.lt_of_succ_le k_ge_idx_add_one
  have h3 := units_nodup k j1 k_ne_j1
  simp only [Fin.getElem_fin, ih1, ← h1, ← h2, ne_eq] at h3
  exact h3 rfl
Proof state after:
case neg
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : ¬units[↑k].snd = true
⊢ False
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case neg
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : ¬units[↑k].snd = true
⊢ False
Tactic:
. next h2 =>
  have h3 := units_nodup k j2 k_ne_j2
  simp only [Bool.not_eq_true] at h2
  simp only [Fin.getElem_fin, ih2, ← h1, ← h2, ne_eq] at h3
  exact h3 rfl
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.clear_insert_inductive_case
Proof state before:
case pos
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
k : Fin units.size
k_ge_idx_add_one : ↑k ≥ ↑idx + 1
k_ne_j2 : k ≠ j2
h1 : units[k].fst.val = ↑i
h✝ : units[↑k].snd = true
⊢ False
Tactic:
next h2 =>
have k_ne_j1 : k ≠ j1 := by
  rw [← idx_eq_j1]
  intro k_eq_idx
  rw [k_eq_idx] at k_ge_idx_add_one
  exact Nat.lt_irrefl idx.1 $ Nat.lt_of_succ_le k_ge_idx_add_one
have h3 := units_nodup k j1 k_ne_j1
simp only [Fin.getElem_fin, ih1, ← h1, ← h2, ne_eq] at h3
exact h3 rfl
Proof state after:
no goals