internlm/Lean-Github huggingface dataset:
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k' : Nat
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
k_eq_succ : k = ⟨k' + 1, k'_succ_in_bounds⟩
k'_in_bounds : k' < acc.snd.fst.length
k'_ne_j : ⟨k', k'_in_bounds⟩ ≠ j
⊢ ¬(acc.snd.fst.get ⟨k', ⋯⟩).fst.val = ↑i
Tactic:
exact h3 ⟨k', k'_in_bounds⟩ k'_ne_j
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
have k_val_ne_zero : k.1 ≠ 0 := by
  intro k_eq_zero
  simp only [List.length_cons, ← k_eq_zero, ne_eq, not_true] at k_ne_zero
Proof state after:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
rcases Nat.exists_eq_succ_of_ne_zero k_val_ne_zero with ⟨k', k_eq_k'_succ⟩
Proof state after:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k'.succ
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k'.succ
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
rw [Nat.succ_eq_add_one] at k_eq_k'_succ
Proof state after:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
have k'_succ_in_bounds : k' + 1 < (l :: acc.2.1).length := by rw [← k_eq_k'_succ]; exact k.2
Proof state after:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ ∃ k' k'_succ_in_bounds, k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
apply Exists.intro k' ∘ Exists.intro k'_succ_in_bounds
Proof state after:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ k = ⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ k = ⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
apply Fin.eq_of_val_eq
Proof state after:
case intro.a
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ ↑k = ↑⟨k' + 1, k'_succ_in_bounds⟩
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
case intro.a
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_val_ne_zero : ↑k ≠ 0
k' : Nat
k_eq_k'_succ : ↑k = k' + 1
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
⊢ ↑k = ↑⟨k' + 1, k'_succ_in_bounds⟩
Tactic:
exact k_eq_k'_succ
Proof state after:
no goals
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
⊢ ↑k ≠ 0
Tactic:
intro k_eq_zero
Proof state after:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_eq_zero : ↑k = 0
⊢ False
Theorem: LRAT.DefaultFormula.confirmRupHint_preserves_invariant_helper
Proof state before:
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × List (Literal (PosFin n)) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.derivedLits_invariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
  Eq.mpr (id (congrArg (fun _a => _a = n) Array.modify_preserves_size)) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin acc.snd.fst.length
j_eq_i : (acc.snd.fst.get j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (acc.snd.fst.get j).snd f.assignments[↑i]
h2 : ¬hasAssignment (acc.snd.fst.get j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin acc.snd.fst.length), k ≠ j → (acc.snd.fst.get k).fst.val ≠ ↑i
l' : Literal (PosFin n) := acc.snd.fst.get j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_ne_i : ¬l.fst.val = ↑i
k : Fin (l :: acc.snd.fst).length
k_ne_j_succ : k ≠ ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬↑k = 0
k_eq_zero : ↑k = 0
⊢ False
Tactic:
simp only [List.length_cons, ← k_eq_zero, ne_eq, not_true] at k_ne_zero
Proof state after:
no goals