By Yves Bertot
Coq is an interactive evidence assistant for the advance of mathematical theories and officially qualified software program. it truly is in accordance with a thought known as the calculus of inductive buildings, a variation of style theory.
This ebook presents a practical creation to the improvement of proofs and licensed courses utilizing Coq. With its huge choice of examples and routines it's a useful software for researchers, scholars, and engineers drawn to formal equipment and the advance of zero-fault software.
Read or Download Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions PDF
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Additional resources for Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions
This chapter shows how to perform simple proofs in a powerful logic. Inductive Constructions The Calculus of Inductive Constructions is introduced in Chap. 6, where we describe how to define data structures such as natural numbers, lists, and trees. New tools are associated with these types: tactics for proofs by induction, simplification rules, and so on. Chapter 7 is not related to inductive constructions but is included there because the tactics it describes make the exposition in chapter 8 easier to present.
Check plus. plus: nat-+nat-+nat Check Zplus. Zplus : Z-+ Z-+ Z Check negb. negb : bool-+ bool Check orb. orb : bool-+ bool-+ bool The following dialogue shows what happens when using an identifier that was not previously declared or defined: Check zero. Error: The reference "zero" was not found in the current environment Function Application The main control structure of our language is the application of functions to arguments. Let us consider an environment E and a context r and two expressions el and e2 with respective types A-+B and A in EUr; then the application of el to e2 is the term written "el e2" and this term has type B in the environment and context being considered.
This function was defined with the help of auxiliary locally defined values that were replaced with local bindings when exiting the section. We show how a term using this function is evaluated with and without (-conversion: Eval cbv beta delta [h] in (h 56 78). = let s := 56+78 in let d := 56-78 in s*s + d*d :Z Eval cbv beta zeta delta [h] in (h 56 78). = (56+78}*(56+78}+(56-78}*(56-78) :Z L-reduction (pronounced iota-reduction) is related to inductive objects and is presented in greater details in another part of the book (Sect.