Proof step 1 from proof rmdup (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬(rmdup(x) ⊆ y ↔ x ⊆ y) ⊦

Rule argument

Additional information

Proof step 2 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ⊦ (∀ a. a ∈ rmdup(x) → a ∈ y) ↔ (∀ a. a ∈ x → a ∈ y)

Rule argument

Additional information

Used simplifier rules: subset: ⊦ x ⊆ y ↔ (∀ a. a ∈ x → a ∈ y);

Proof step 3 from proof rmdup (previous step) (back to tree)

Heuristic structural induction applied rule structural induction

And obtained the following new goals

• Proof step 4:

  ⊦ (∀ a. a ∈ rmdup(x) → a ∈ []) ↔ (∀ a. a ∈ x → a ∈ [])

• Proof step 10:

  ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0) ⊦ (∀ a0. a0 ∈ rmdup(x) → a0 ∈ a + y0) ↔ (∀ a0. a0 ∈ x → a0 ∈ a + y0)

Rule argument

Additional information

Used simplifier rules: list('a) freely generated by [], +;;

Proof step 4 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  ⊦ (∀ a. ¬ a ∈ rmdup(x)) ↔ (∀ a. ¬ a ∈ x)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ []; (s)

Proof step 5 from proof rmdup (previous step) (back to tree)

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

• Proof step 6:

  ∀ a. ¬ a ∈ rmdup(x) ⊦ ∀ a. ¬ a ∈ x

• Proof step 8:

  ∀ a. ¬ a ∈ x ⊦ ∀ a. ¬ a ∈ rmdup(x)

Rule argument

Additional information

Proof step 6 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  a ∈ x, ∀ a. ¬ a ∈ rmdup(x) ⊦

Rule argument

Additional information

Proof step 7 from proof rmdup (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: rm-in-03: ⊦ ¬ a ∈ rmdup(x) → ¬ a ∈ x; (forward) from specification polylist-dup Used terms: a

Proof step 8 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  a ∈ rmdup(x), a ∈ x, ∀ a. ¬ a ∈ x ⊦

Rule argument

Additional information

Used simplifier rules: rm-in-01: ⊦ a ∈ rmdup(x) → a ∈ x; (forward) from specification polylist-dup

Proof step 9 from proof rmdup (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: a

Proof step 10 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0) ⊦ (∀ a0. a0 ∈ rmdup(x) → a0 = a ∨ a0 ∈ y0) ↔ (∀ a0. a0 ∈ x → a0 = a ∨ a0 ∈ y0)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a + x = a ' + x; (s)

Proof step 11 from proof rmdup (previous step) (back to tree)

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

• Proof step 12:

  ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0), ∀ a0. a0 ∈ rmdup(x) → a0 = a ∨ a0 ∈ y0 ⊦ ∀ a0. a0 ∈ x → a0 = a ∨ a0 ∈ y0

• Proof step 14:

  ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0), ∀ a0. a0 ∈ x → a0 = a ∨ a0 ∈ y0 ⊦ ∀ a0. a0 ∈ rmdup(x) → a0 = a ∨ a0 ∈ y0

Rule argument

Additional information

Proof step 12 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  a0 ≠ a, a0 ∈ x, ¬ a0 ∈ y0, ∀ a0. a0 ∈ rmdup(x) → a0 = a ∨ a0 ∈ y0, ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0) ⊦

Rule argument

Additional information

Proof step 13 from proof rmdup (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: rm-in-03: ⊦ ¬ a ∈ rmdup(x) → ¬ a ∈ x; (forward) from specification polylist-dup Used terms: a0

Proof step 14 from proof rmdup (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  a0 ≠ a, a0 ∈ rmdup(x), ¬ a0 ∈ y0, ∀ a0. a0 ∈ x → a0 = a ∨ a0 ∈ y0, ∀ x. (∀ a. a ∈ rmdup(x) → a ∈ y0) ↔ (∀ a. a ∈ x → a ∈ y0) ⊦

Rule argument

Additional information

Proof step 15 from proof rmdup (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: rm-in-01: ⊦ a ∈ rmdup(x) → a ∈ x; (forward) from specification polylist-dup Used terms: a0