Proof step 1 from proof always-noincrease-stable (conclusion) (back to tree)

□ (A'' ≪ A ∨ A'' = A) ⊦ ◇ □ A'' = A

User applied rule apply lemma

And obtained the following new goals

Proof step 2:
□ ⟨a := A0⟩ (◇ A0 ≪ a) ⊦
Proof step 3:
□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A

Rule argument

spec: <toplevel>, name:not-always-decrease-test, seq:□ ⟨a := A0⟩ (◇ A0 ≪ a) ⊦ , subst:[A0] → [A]

Additional information

Used terms: A

Proof step 2 from proof always-noincrease-stable (previous step) (back to tree)

□ ⟨a := A0⟩ (◇ A0 ≪ a) ⊦

Lemma not-always-decrease-test

Proof step 3 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A

User applied rule cut formula

And obtained the following new goals

Proof step 4:
□ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A
Proof step 17:
□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A

Rule argument

fma: □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

Additional information

Used formulas: □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

Proof step 4 from proof always-noincrease-stable (previous step) (back to tree)

□ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A

User applied rule execute always

And obtained the following new goals

Proof step 5:
□ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ (A'' ≪ A ∨ A'' = A), A'' ≪ A ∨ A'' = A ⊦ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A, □ A'' = A

Rule argument

pos: right 1

Additional information

Proof step 5 from proof always-noincrease-stable (previous step) (back to tree)

□ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ (A'' ≪ A ∨ A'' = A), A'' ≪ A ∨ A'' = A ⊦ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A, □ A'' = A

User applied rule assign left

And obtained the following new goals

Proof step 6:
□ (A ≪ a ∨ A = a), □ ⟨a0 := A⟩ (□ (A ≪ a0 ∨ A = a0)), □ (A'' ≪ A ∨ A'' = A), a = A, A'' ≪ A ∨ A'' = A ⊦ ⟨a0 := A⟩ (◇ A ≪ a0), ◇ □ A'' = A, □ A'' = A

Rule argument

pos: left 2

Additional information

Proof step 6 from proof always-noincrease-stable (previous step) (back to tree)

□ (A ≪ a ∨ A = a), □ ⟨a0 := A⟩ (□ (A ≪ a0 ∨ A = a0)), □ (A'' ≪ A ∨ A'' = A), a = A, A'' ≪ A ∨ A'' = A ⊦ ⟨a0 := A⟩ (◇ A ≪ a0), ◇ □ A'' = A, □ A'' = A

User applied rule assign right

And obtained the following new goals

Proof step 7:
□ (A ≪ a ∨ A = a), □ ⟨a1 := A⟩ (□ (A ≪ a1 ∨ A = a1)), □ (A'' ≪ A ∨ A'' = A), a = A, a0 = A, A'' ≪ A ∨ A'' = A ⊦ ◇ A ≪ a0, ◇ □ A'' = A, □ A'' = A

Rule argument

pos: right 1

Additional information

Proof step 7 from proof always-noincrease-stable (previous step) (back to tree)

□ (A ≪ a ∨ A = a), □ ⟨a1 := A⟩ (□ (A ≪ a1 ∨ A = a1)), □ (A'' ≪ A ∨ A'' = A), a = A, a0 = A, A'' ≪ A ∨ A'' = A ⊦ ◇ A ≪ a0, ◇ □ A'' = A, □ A'' = A

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
□ ⟨a1 := A⟩ (□ (A ≪ a1 ∨ A = a1)), □ (A'' ≪ A ∨ A'' = A), □ A = a, a = A, A'' ≪ A ∨ A'' = A ⊦ ◇ □ A'' = A, □ A'' = A

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ≪ a; (s)

Proof step 8 from proof always-noincrease-stable (previous step) (back to tree)

□ ⟨a1 := A⟩ (□ (A ≪ a1 ∨ A = a1)), □ (A'' ≪ A ∨ A'' = A), □ A = a, a = A, A'' ≪ A ∨ A'' = A ⊦ ◇ □ A'' = A, □ A'' = A

User applied rule weakening

And obtained the following new goals

Proof step 9:
□ A = a ⊦ □ A'' = A

Rule argument

poslist: left 1, left 2, left 4, left 5, right 1

Additional information

Proof step 9 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a ⊦ □ A'' = A

User applied rule induction

And obtained the following new goals

Proof step 10:
□ A = a, N = N'' +1 until A'' ≠ A, n = N, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

indvar: □ A'' = A wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: □ A'' = A

Proof step 10 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a, N = N'' +1 until A'' ≠ A, n = N, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

User applied rule step

And obtained the following new goals

Proof step 11:
□ A = a, A ≠ a ∨ n = N +1 ∧ N = N'' +1 until A'' ≠ A, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

args:[positions: , ]

Additional information

Proof step 11 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a, A ≠ a ∨ n = N +1 ∧ N = N'' +1 until A'' ≠ A, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
□ A = a, A ≠ a ∨ n = N + 1 ∧ N = N'' + 1 until A'' ≠ A, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 12 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a, A ≠ a ∨ n = N + 1 ∧ N = N'' + 1 until A'' ≠ A, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

User applied rule case distinction

And obtained the following new goals

Proof step 13:
□ A = a, A ≠ a, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦
Proof step 15:
N = N'' + 1 until A'' ≠ A, □ A = a, n = N + 1, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

pos: left 2

Additional information

Proof step 13 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a, A ≠ a, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
□ A = a, A ≠ a, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

Additional information

Proof step 14 from proof always-noincrease-stable (previous step) (back to tree)

□ A = a, A ≠ a, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

User applied rule always left

And closed the goal!

Rule argument

pos: left 1

Additional information

Proof step 15 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until A'' ≠ A, □ A = a, n = N + 1, boolvar → a = a0 ∧ n = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
N = N'' + 1 until A'' ≠ A, □ A = a, n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

Rule argument

Additional information

Proof step 16 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until A'' ≠ A, □ A = a, n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ A = a → ¬ N = N'' +1 until A'' ≠ A ⊦

User applied rule apply induction

And closed the goal!

Rule argument

subst: [N, A, a] → [N, A, a]

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: N, A, a

Proof step 17 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a)), □ ⟨a := A⟩ (◇ A ≪ a), ◇ □ A'' = A

User applied rule weakening

And obtained the following new goals

Proof step 18:
□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

Rule argument

poslist: right 2, right 3

Additional information

Proof step 18 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A) ⊦ □ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

User applied rule execute always

And obtained the following new goals

Proof step 19:
□ (A'' ≪ A ∨ A'' = A), A'' ≪ A ∨ A'' = A ⊦ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

Rule argument

pos: right 1

Additional information

Proof step 19 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A), A'' ≪ A ∨ A'' = A ⊦ ⟨a := A⟩ (□ (A ≪ a ∨ A = a))

User applied rule assign right

And obtained the following new goals

Proof step 20:
□ (A'' ≪ A ∨ A'' = A), a = A, A'' ≪ A ∨ A'' = A ⊦ □ (A ≪ a ∨ A = a)

Rule argument

pos: right 1

Additional information

Proof step 20 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A), a = A, A'' ≪ A ∨ A'' = A ⊦ □ (A ≪ a ∨ A = a)

User applied rule weakening

And obtained the following new goals

Proof step 21:
□ (A'' ≪ A ∨ A'' = A), a = A ⊦ □ (A ≪ a ∨ A = a)

Rule argument

poslist: left 3

Additional information

Proof step 21 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A), a = A ⊦ □ (A ≪ a ∨ A = a)

User applied rule induction

And obtained the following new goals

Proof step 22:
□ (A'' ≪ A ∨ A'' = A), N = N'' +1 until (¬ A ≪ a ∧ A ≠ a), a = A, n = N, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

Rule argument

indvar: □ (A ≪ a ∨ A = a) wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: □ (A ≪ a ∨ A = a)

Proof step 22 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A), N = N'' +1 until (¬ A ≪ a ∧ A ≠ a), a = A, n = N, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

User applied rule step

And obtained the following new goals

Proof step 23:
□ (A'' ≪ A ∨ A'' = A), ¬ a ≪ a ∧ a ≠ a ∨ n = N +1 ∧ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a), boolvar → a = a0 ∧ n = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

Rule argument

args:[positions: , ]

Additional information

Proof step 23 from proof always-noincrease-stable (previous step) (back to tree)

□ (A'' ≪ A ∨ A'' = A), ¬ a ≪ a ∧ a ≠ a ∨ n = N +1 ∧ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a), boolvar → a = a0 ∧ n = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 24:
N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a), □ (A'' ≪ A ∨ A'' = A), n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ ¬ a ≪ a; (s)

Proof step 24 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a), □ (A'' ≪ A ∨ A'' = A), n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦

User applied rule apply induction

And obtained the following new goals

Proof step 25:
N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a), □ (A'' ≪ A ∨ A'' = A), n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ ⟨a := A⟩ (N = N'' +1 until (¬ A ≪ a ∧ A ≠ a))

Rule argument

subst: [N, A, a] → [N, A, A]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: N, A, A

Proof step 25 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a), □ (A'' ≪ A ∨ A'' = A), n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ ⟨a := A⟩ (N = N'' +1 until (¬ A ≪ a ∧ A ≠ a))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a), □ (A'' ≪ A ∨ A'' = A), n = N + 1, boolvar → a = a0 ∧ N + 1 = n0, A ≪ a ∨ A = a, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ ⟨a := A⟩ (N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a))

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 26 from proof always-noincrease-stable (previous step) (back to tree)

User applied rule assign right

And obtained the following new goals

Proof step 27:
N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), a = A, n = N + 1, boolvar → a1 = a0 ∧ N + 1 = n0, A ≪ a1 ∨ A = a1, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

Rule argument

pos: right 1

Additional information

Proof step 27 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), a = A, n = N + 1, boolvar → a1 = a0 ∧ N + 1 = n0, A ≪ a1 ∨ A = a1, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), a = A, n = N + 1, A ≪ a1, boolvar → a1 = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

Rule argument

Additional information

Proof step 28 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), a = A, n = N + 1, A ≪ a1, boolvar → a1 = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

User applied rule subst equation

And obtained the following new goals

Proof step 29:
N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), n = N + 1, a ≪ a1, boolvar → a1 = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

Rule argument

pos: left 3, flags: rotate, discard

Additional information

Proof step 29 from proof always-noincrease-stable (previous step) (back to tree)

N = N'' + 1 until (¬ A ≪ a1 ∧ A ≠ a1), □ (A'' ≪ A ∨ A'' = A), n = N + 1, a ≪ a1, boolvar → a1 = a0 ∧ N + 1 = n0, \A ∀ N, A, a. N < n → □ (A'' ≪ A ∨ A'' = A) ∧ a = A → ¬ N = N'' +1 until (¬ A ≪ a ∧ A ≠ a) ⊦ N = N'' + 1 until (¬ A ≪ a ∧ A ≠ a)

User applied rule decompose

And closed the goal!

Rule argument

poslist: left 1, right 1

Additional information

Used simplifier rules: trans: ⊦ a ≪ a0 ∧ a0 ≪ a1 → a ≪ a1; (forward) from specification wellfounded

Proof Tree for always-noincrease-stable (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.