[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 25 from proof read-lookup-copy (previous step)

# ax0 = # kx0 + # kx + 1, new(ax0, fs), key ∈ ri, key1 ∈ ri, kx.toSet ⊆ dom ri, kx0.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ kx1, fs0, key0, ri0. # ax0 = # kx1 ∧ new(ax0, fs0) ∧ key0 ∈ ri0 ∧ kx1.toSet ⊆ dom ri0 ∧ (∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]])) → fs0[ax0, fs0[ri0[kx1]]][ri0[kx1, ax0][key0]] = fs0[ri0[key0]] ∨ dups(kx1) ⊦ fs[ri[key1]] = fs[ri[key]], dups(ax0), dups(kx0 + kx), dups(kx0), dups(kx), key ∈ kx0, key ∈ kx, key1 ∈ kx0, key1 ∈ kx, ax0[# kx0] ∈ ax0, ax0[# kx0] ∈ fs

User applied rule apply lemma

And obtained the following new goals

Proof step 26:
n ≤ # ax ⊦ ∃ ax0, ax1. ax = ax0 + ax1 ∧ # ax0 = n
Proof step 27:
# ax0 = # kx0 + # kx + 1, new(ax0, fs), key1 ∈ ri, key ∈ ri, kx0.toSet ⊆ dom ri, kx.toSet ⊆ dom ri, ∀ kx1, fs0, key0, ri0. # ax0 = # kx1 ∧ new(ax0, fs0) ∧ key0 ∈ ri0 ∧ kx1.toSet ⊆ dom ri0 ∧ (∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]])) → fs0[ax0, fs0[ri0[kx1]]][ri0[kx1, ax0][key0]] = fs0[ri0[key0]] ∨ dups(kx1), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∃ ax2, ax1. ax0 = ax2 + ax1 ∧ # ax2 = # kx0 ⊦ fs[ri[key1]] = fs[ri[key]], dups(kx), dups(kx0), dups(kx0 + kx), dups(ax0), ax0[# kx0] ∈ fs, ax0[# kx0] ∈ ax0, key1 ∈ kx, key1 ∈ kx0, key ∈ kx, key ∈ kx0

Rule argument

spec: list, inst: address-list-set name: split-first seq: n ≤ # ax ⊦ ∃ ax0, ax1. ax = ax0 + ax1 ∧ # ax0 = n subst: [n, ax] → [# kx0, ax0]

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ # kx = 0 ↔ kx = []; (s)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: # kx0, ax0