How to systematically normalize inequalities to < (lt) and <= (le) in Coq?
In proving facts about inequalities (for real numbers), there is <, <=, >, and >=. It's kind of tedious to have to write down and use theorems/lemmas for both these forms. Currently, I am converting these to just < and <= manually by first assert and then prove a trivial subgoal. I was wondering if it is possible to automatically normalize all ineqaulities to just < and <= in the hypotheses and the goal?
gt and ge are functions that call lt and le respectively on swapped arguments. To get rid of them, just unfold them. unfold gt, ge. You may want to unfold lt as well: it's defined in terms of le. Since the definition of gt uses lt, unfold gt first. unfold gt, ge, lt. You can tell Coq to try this when attempting to prove a goal with auto. Hint Unfold gt ge lt.
Split conjunction goal into subgoals
Idiomatically expressing “The Following Are Equivalent” in Coq
What are the possible ways to define parallel composition in Coq apart from using list?
Defining a finite automata Coq
Proving even + even = even with mutual induction using tactics
Rewriting hypothesis with a more concrete expression
Coq rewriting using lambda arguments
How to rewrite a goal using function definition?
coqtop -lv (version 8.6) no longer displaying proof subgoals?
Coq beginner - Prove a basic lemma
How to do induction differently?
Is is possible to implement a Coq tactic that inspects a HintDb? If so, how?
Need help on proving proposition in Coq
Folding back after unfolding
Could coq derive this solution automatically? Can coq manipulate algebraic expressions when performing the search for a proof?
A special case of Lob's theorem using Coq