Proof step 1 from proof precise-Sep (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬ precise(adP ⊗_ st), ∀ a. a ∈ st → precise(adP(a, st[a])) ⊦

Rule argument

Additional information

Proof step 2 from proof precise-Sep (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 3:

  ¬ precise(adP ⊗_ st), ∀ a. a ∈ st → precise(adP(a, st[a])), ∀ st0, adP. # st0 < # st → ¬ precise(adP ⊗_ st0) ∧ (∀ a. a ∈ st0 → precise(adP(a, st0[a]))) → false ⊦

Rule argument

Additional information

Used simplifier rules: <; (wellfounded) Used terms: # st

Proof step 3 from proof precise-Sep (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 4:

  st ≠ ∅, ¬ precise(adP ⊗_ st), ∀ a. a ∈ st → precise(adP(a, st[a])), ∀ st0, adP. # st0 < # st → ¬ precise(adP ⊗_ st0) ∧ (∀ a. a ∈ st0 → precise(adP(a, st0[a]))) → false ⊦

Rule argument

Additional information

Used simplifier rules: Sep-emp: ⊦ adP ⊗_ ∅ = emp; ⊦ precise(emp); (s) Used formulas: st = ∅

Proof step 4 from proof precise-Sep (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

  ⊦ st = ∅ ↔ ¬(∃ a, d, st0. st = st0[a, d] ∧ ¬ a ∈ st0)

• Proof step 6:

  ¬ precise(adP ⊗_ st), ∀ a. a ∈ st → precise(adP(a, st[a])), ∃ a, d, st0. st = st0[a, d] ∧ ¬ a ∈ st0, ∀ st0, adP. # st0 < # st → ¬ precise(adP ⊗_ st0) ∧ (∀ a. a ∈ st0 → precise(adP(a, st0[a]))) → false ⊦

Rule argument

Additional information

Used terms: st

Proof step 5 from proof precise-Sep (previous step) (back to tree)

A spec-lemma

Proof step 6 from proof precise-Sep (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦

Rule argument

Additional information

Used simplifier rules: Sep-ins: ⊦ ¬ a ∈ st → adP ⊗_ st[a, d] = adP(a, d) * adP ⊗_ st;

Proof step 7 from proof precise-Sep (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 8:

  ⊦ precise(hP0) ∧ precise(hP1) → precise(hP0 * hP1)

• Proof step 9:

  ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦ precise(adP(a, d)) ∧ precise(adP ⊗_ st0)

Rule argument

Additional information

Used terms: adP ⊗_ st0, adP(a, d)

Proof step 8 from proof precise-Sep (previous step) (back to tree)

A spec-lemma

Proof step 9 from proof precise-Sep (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦ precise(adP(a, d)) ∧ precise(adP ⊗_ st0)

Rule argument

Additional information

Proof step 10 from proof precise-Sep (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 11:

  ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ a ∈ st0, a ∈ st0[a, d] → precise(adP(a, st0[a, d][a])), ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦ precise(adP(a, d)) ∧ precise(adP ⊗_ st0)

Rule argument

Additional information

Used terms: a

Proof step 11 from proof precise-Sep (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  precise(adP(a, d)), ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ precise(adP ⊗_ st0), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[a, d]; (s)

Proof step 12 from proof precise-Sep (previous step) (back to tree)

User applied rule apply induction

And obtained the following new goals

• Proof step 13:

  precise(adP(a, d)), ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ precise(adP ⊗_ st0), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦ ¬ precise(adP ⊗_ st0) ∧ (∀ a. a ∈ st0 → precise(adP(a, st0[a])))

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ st → # st[a, d] = # st + 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: st0, adP

Proof step 13 from proof precise-Sep (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  precise(adP(a, d)), a0 ∈ st0, ¬ precise(adP(a, d) * adP ⊗_ st0), ¬ precise(adP ⊗_ st0), ¬ precise(adP(a0, st0[a0])), ¬ a ∈ st0, ∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → ¬ precise(adP ⊗_ st1) ∧ (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → false ⊦

Rule argument

Additional information

Proof step 14 from proof precise-Sep (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ b ∈ st ∧ ¬ a ∈ st → st[a, d][b] = st[b]; (s) ⊦ a ∈ st[b, d] ↔ a = b ∨ a ∈ st; (ws) Used terms: a0