[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 13 from proof put-sw-in (previous step)

m0 < # x0, m ≤ # x0, ∀ m, n, a. m < # x0 ∧ n < # x0 → (a ∈ x0[m, x0[n]][n, x0[m]] ↔ a ∈ x0) ⊦ a ∈ (a0 ' + x0)[m, x0[m0]][m0 + 1, (a0 ' + x0)[m]] ↔ a = a0 ∨ a ∈ x0, x0 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.rest[n - 1]
Proof step 15:
a0 ' + x0 ≠ [] ∧ m ≠ 0, m0 < # x0, m ≤ # x0, ∀ m, n, a. m < # x0 ∧ n < # x0 → (a ∈ x0[m, x0[n]][n, x0[m]] ↔ a ∈ x0) ⊦ a ∈ (a0 ' + x0)[m, x0[m0]][m0 + 1, (a0 ' + x0).rest[m - 1]] ↔ a = a0 ∨ a ∈ x0, x0 = []

Rule argument

spec: list-del inst: name: get-rec seq: ⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.rest[n - 1] subst: [n, x] → [m, a0 ' + x0] flags: rewrite, current

Additional information

Used simplifier rules: put-zero: ⊦ (b ' + x)[0, a] = a + x;%get-zero-01: ⊦ (a ' + x)[0] = a;%put-succ: ⊦ (b ' + x)[n + 1, a] = b + x[n, a];%put-in: ⊦ a ≠ x[m] ∧ m < # x → (a ∈ x[m, b] ↔ a = b ∨ a ∈ x);%get-in: ⊦ m < # x → (¬ x[m] ∈ x ↔ false);%e-01: ⊦ a ' + x ≠ []; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%c: ⊦ a + x = a ' + x; from specification list%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%in-rec: ⊦ a ∈ b ' + x ↔ ¬ ¬(a = b ∨ a ∈ x); from specification list%lez: ⊦ 0 ≤ m; from specification nat Used terms: m, a0 ' + x0