Evaluate winding numbers through Cauchy indices

In complex analysis, the winding number measures the number of times a path (counterclockwise) winds around a point, while the Cauchy index can approximate how the path winds. This entry provides a formalisation of the Cauchy index, which is then shown to be related to the winding number. In addition, this entry also offers a tactic that enables users to evaluate the winding number by calculating Cauchy indices. The connection between the winding number and the Cauchy index can be found in the literature [1] [2, Chapter 11]. 1 Some useful lemmas in topology theory Missing-Topology imports HOL−Analysis.Analysis begin 1.1 Misc lemma open-times-image: fixes S :: ′a::real-normed-field set assumes open S c 6=0 shows open ((( ∗ ) c) ‘ S ) 〈proof 〉 lemma image-linear-greaterThan: fixes x :: ′a::linordered-field assumes c 6=0 shows ((λx . c∗x+b) ‘ {x<..}) = (if c>0 then {c∗x+b <..} else {..< c∗x+b}) 〈proof 〉 lemma image-linear-lessThan: fixes x :: ′a::linordered-field assumes c 6=0 shows ((λx . c∗x+b) ‘ {..<x}) = (if c>0 then {..<c∗x+b} else {c∗x+b<..}) 〈proof 〉 lemma continuous-on-neq-split : fixes f :: ′a::linear-continuum-topology ⇒ ′b::linorder-topology