[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 rem-preserves-monls (previous step)

nl0[m0] = rem(⌜nl0⌝ , np)[n], ⌜n⌝ + ⌜1⌝ = # rem(⌜nl0⌝ , np), m0 < # nl0, m ≤ nl0[m0], mon<(⌜nl0⌝ + ⌜m '⌝ ), ∀ n0. n0 < n → rem(⌜nl0⌝ , np)[n0] < rem(⌜nl0⌝ , np)[n0 + 1], ∀ nl1, np. # nl1 < #(nl0 + m ') → mon<(⌜nl1⌝ ) → mon<(rem(⌜nl1⌝ , np)) ⊦ nl0 = [], np(m0), np(# nl0)

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: inst: name: monls-ls seq: ⊦ mon<(nσ) ∧ m < n ∧ ⌜n⌝ < # nσ → nσ[m] < nσ[n] subst: [nσ, m, n] → [⌜nl0⌝ + ⌜m '⌝ , m0, # nl0] flags: current, []

Additional information

Used simplifier rules: size-append: ⊦ #(nσ + nσ0) = # nσ + # nσ0; from specification stream[natstream]%size-fin: ⊦ # ⌜nl⌝ = ⌜# nl⌝ ; from specification stream[natstream]%len-02: ⊦ # m ' = 0 +1; from specification list[natstream]%append-assoc: ⊦ (nσ + nσ0) + nσ1 = nσ + nσ0 + nσ1; from specification stream[natstream]%less-succ-leq: ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; from specification nat-inf%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%ref-1: ⊦ ¬ n < n; from specification nat-basic1%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%get-append2-fin: ⊦ (⌜nl⌝ + ⌜nl0⌝ )[m] = (nl + nl0)[m]; from specification stream[natstream]%get-butlast: ⊦ n < # nl → (nl + m ')[n] = nl[n]; from specification list-del[natstream]%a: ⊦ (nl + nl0) + nl1 = nl + nl0 + nl1; from specification list[natstream]%get-last-03: ⊦ (nl + m ')[# nl] = m; from specification list-del[natstream]%le-ls-simp: ⊦ m ≤ n → (n < m ↔ false); from specification nat Used terms: ⌜nl0⌝ + ⌜m '⌝ , m0, # nl0