Download Automated Deduction in Geometry: 8th International Workshop, by Susanne Apel, Jürgen Richter-Gebert (auth.), Pascal Schreck, PDF

By Susanne Apel, Jürgen Richter-Gebert (auth.), Pascal Schreck, Julien Narboux, Jürgen Richter-Gebert (eds.)

This booklet constitutes the completely refereed post-workshop complaints of the eighth overseas Workshop on computerized Deduction in Geometry, ADG 2010, held in Munich, Germany in July 2010.
The thirteen revised complete papers provided have been rigorously chosen in the course of rounds of reviewing and development from the lectures given on the workshop. issues addressed via the papers are occurrence geometry utilizing a few form of combinatoric argument; computing device algebra; software program implementation; in addition to good judgment and evidence assistants.

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Extra info for Automated Deduction in Geometry: 8th International Workshop, ADG 2010, Munich, Germany, July 22-24, 2010, Revised Selected Papers

Example text

So ({a, k, l} , {c, k, l}) and ({c, u, v)} , ({a, u, v}) are the graph edges corresponding [c,u,v] to the edge (a, c) in both triangles. We want to express [a,k,l] as a product [c,k,l] [a,u,v] of biquadratic fractions. 2 we can successively exchange the points spanning the cutting line via biquadratic fractions. Together with the fact that B is consistent we can conclude, that in each intermediate step, the triples occurring lie in B. We exemplify this in the case where f is extended to z and {l, k, z} and {u, v, z} and {a, c, z} are H-collinear (left picture in Figure 4—the other case can be done similarly).

Fleuriot of the properties hold independently of HRω . For instance, aside from proving that =ω is an equivalence relation and that ≤ω is reflexive, anti-symmetric, and transitive, we also mechanize the following (expected) theorems directly over Z∗ :2 – – – – ¬ x <ω x x ≤ω y ↔ x < ω y ∨ x =ω y x <ω y ↔ x < ω y ∨ x = ω y x =ω x ; y =ω y ; x ≤ω y =⇒ x ≤ω y which, though “easy to see” [16], required some effort to prove formally. We also show that the algebraic operations are well-behaved over HRω by deriving all the expected closure rules: – – – – x ∈ HRω =⇒ −ω x ∈ HRω x ∈ HRω ; y ∈ HRω =⇒ x + y ∈ HRω x ∈ HRω ⇒ inverseω x ∈ HRω x ∈ HRω ; y ∈ HRω =⇒ x ×ω y ∈ HRω While the first two rules are trivially proved, the last two require somewhat more work as they involve case-splits on the variables involved.

K = w) and fulfill the requirements in the second possibility of a Ceva-Menelaus proof. The deduction is again indicated by the scheme {a,c,w} {w,b,k} {a,w,c} {a, b, c} −→ {a, w, b} −→ {a, w, k} −→ {a, c, k}. The only thing left to show is that the number of Menelaus triangles is even. The total number of triangles is even, since otherwise, there could not exist a matching of edges. So it is sufficient to show that the number of Ceva triangels is even. So we can reformulate our claim: Form the Ceva or Menelaus expression (as indicated in (2) or (3) and with this order of indices inside the brackets ) and c,d] multiply them.

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