Commit 12daccee by Guillaume Melquiond

### Add a realization of real.square.

```Remark about the theory: if one considers that the square root is always
nonnegative (by definition in Coq), then Lemma Sqrt_positive has an
extraneous hypothesis.```
parent 5459fb63
 (* This file is generated by Why3's Coq driver *) (* Beware! Only edit allowed sections below *) Require Import ZArith. Require Import Rbase. Require Import R_sqrt. (*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/theories".*) (*Add Rec LoadPath "/home/guillaume/bin/why3/share/why3/modules".*) Require real.Real. Definition sqr(x:R): R := (x * x)%R. Definition sqrt: R -> R. (* YOU MAY EDIT THE PROOF BELOW *) exact sqrt. Defined. (* DO NOT EDIT BELOW *) (* YOU MAY EDIT THE CONTEXT BELOW *) (* DO NOT EDIT BELOW *) Lemma Sqrt_positive : forall (x:R), (0%R <= x)%R -> (0%R <= (sqrt x))%R. (* YOU MAY EDIT THE PROOF BELOW *) intros x _. apply sqrt_pos. Qed. (* DO NOT EDIT BELOW *) (* YOU MAY EDIT THE CONTEXT BELOW *) (* DO NOT EDIT BELOW *) Lemma Sqrt_square : forall (x:R), (0%R <= x)%R -> ((sqr (sqrt x)) = x). (* YOU MAY EDIT THE PROOF BELOW *) exact sqrt_sqrt. Qed. (* DO NOT EDIT BELOW *) (* YOU MAY EDIT THE CONTEXT BELOW *) (* DO NOT EDIT BELOW *) Lemma Square_sqrt : forall (x:R), (0%R <= x)%R -> ((sqrt (x * x)%R) = x). (* YOU MAY EDIT THE PROOF BELOW *) exact sqrt_square. Qed. (* DO NOT EDIT BELOW *)
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