[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 12 from proof precise-Sep (previous step)

precise(aP(a)), ∀ a0. a0 ∈ s0 → precise(aP(a0)), ∀ s1, aP. # s1 < #(s0 ++ a) → (∀ a. a ∈ s1 → precise(aP(a))) → precise(aP ⊗_ s1) ⊦ precise(aP(a) * aP ⊗_ s0), a ∈ s0

User applied rule apply induction

And obtained the following new goals

Proof step 13:
precise(aP(a)), ∀ a0. a0 ∈ s0 → precise(aP(a0)), ∀ s1, aP. # s1 < #(s0 ++ a) → (∀ a. a ∈ s1 → precise(aP(a))) → precise(aP ⊗_ s1) ⊦ precise(aP(a) * aP ⊗_ s0), a ∈ s0, ∀ a. a ∈ s0 → precise(aP(a))

Rule argument

subst: [s1, aP] → [s0, aP]

Additional information

Used simplifier rules: ⊦ precise(P0) ∧ precise(P1) → precise(P0 * P1); (s)%sep-ass: ⊦ (P0 * P1) * P2 = P0 * P1 * P2; from specification heap-sep%sep-com: ⊦ P0 * P1 = P1 * P0; from specification heap-sep% ⊦ ¬ a ∈ s → #(s ++ a) = # s + 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: s0, aP