enrich path-basic with comment: This specification gives various functions on paths.; functions . .last : path → V; comment: the last node of a path; . .butlast : path → path; comment: specified for paths which are not nodes: the path without the last step; . + . : path × path → path; comment: replaces the last element of first path with full second path. Used to append paths, when the last element of first path is first of second; . to . : path × nat → path; comment: for n ≤ # pa the first n steps of path; . from . : path × nat → path; comment: for n ≤ # pa the path with the first n steps cut off. ; . [ . ] : path × nat → V; comment: for n ≤ # pa the pa[n] is the nth node of path ; . [ . ]i : path × nat → J; comment: for n < # pa pa[n]i is the nth input on path ; . [ . ]o : path × nat → P; comment: for n < # pa pa[n]o is the nth input on path ; Step : path × nat → path; comment: one step path of step n; predicates Path : path; MPath : path; FPath : path; BPath : path; variables j0 : J; p0 : P; v0, v' : V; axioms to-base : ⊦ pa to 0 = pa .first '; used for : ls, s; to-rec-base : ⊦ v ' to (n + 1) = v '; used for : ls, s; to-rec-rec : ⊦ mkpa(v, j, p, pa) to (n + 1) = mkpa(v, j, p, pa to n); used for : ls, s; from-base : ⊦ pa from 0 = pa; used for : ls, s; from-rec-base : ⊦ v ' from (n + 1) = v '; used for : ls, s; from-rec-rec : ⊦ mkpa(v, j, p, pa) from (n + 1) = pa from n; used for : ls, s; getv-base : ⊦ pa[0] = pa .first; used for : s, ls; getv-rec : ⊦ (mkpa(v, j, p, pa)[n + 1]) = pa[n]; used for : s, ls; geti-base : ⊦ (mkpa(v, j, p, pa)[0]i) = j; used for : s, ls; geti-rec : ⊦ (mkpa(v, j, p, pa)[n + 1]i) = pa[n]i; used for : s, ls; geto-base : ⊦ (mkpa(v, j, p, pa)[0]o) = p; used for : s, ls; geto-rec : ⊦ (mkpa(v, j, p, pa)[n + 1]o) = pa[n]o; used for : s, ls; butlast-base : ⊦ mkpa(v, j, p, v0 ') .butlast = v '; used for : s, ls; butlast-rec : ⊦ mkpa(v, j, p, mkpa(v0, j0, p0, pa)) .butlast = mkpa(v, j, p, mkpa(v0, j0, p0, pa) .butlast); used for : s, ls; append-base : ⊦ v ' + pa = pa; used for : s, ls; append-rec : ⊦ mkpa(v, j, p, pa) + pa0 = mkpa(v, j, p, pa + pa0); used for : s, ls; last-base : ⊦ v ' .last = v; used for : s, ls; last-rec : ⊦ mkpa(v, j, p, pa) .last = pa .last; used for : s, ls; Path-base : ⊦ Path(v '); used for : s, ls; Path-rec : ⊦ Path(mkpa(v, j, p, pa)) ↔ (∃ ρ. St(ρ)(v, j, p, pa .first) ∧ Path(pa)); used for : s, ls; FPath-def : ⊦ FPath(pa) ↔ Path(pa) ∧ init(pa .first); used for : s, ls; BPath-def : ⊦ BPath(pa) ↔ Path(pa) ∧ final(pa .last); used for : s, ls; MPath-def : ⊦ MPath(pa) ↔ Path(pa) ∧ init(pa .first) ∧ final(pa .last); used for : s, ls; Step-base : ⊦ Step(mkpa(v, j, p, pa), 0) = mkpa(v, j, p, pa .first '); used for : ls, s; Step-rec : ⊦ Step(mkpa(v, j, p, pa), n + 1) = Step(pa, n); used for : ls, s; end enrich