lagrange_inequality: removed a Coq proof

parent 1bc757e2
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require Reals.R_sqrt.
Require BuiltIn.
Require real.Real.
Require real.Square.
(* Why3 assumption *)
Definition dot (x1:R) (x2:R) (y1:R) (y2:R): R :=
((x1 * y1)%R + (x2 * y2)%R)%R.
(* Why3 assumption *)
Definition norm2 (x1:R) (x2:R): R :=
((Reals.RIneq.Rsqr x1) + (Reals.RIneq.Rsqr x2))%R.
Axiom norm2_pos : forall (x1:R) (x2:R), (0%R <= (norm2 x1 x2))%R.
Axiom Lagrange : forall (a1:R) (a2:R) (b1:R) (b2:R), (((norm2 a1
a2) * (norm2 b1 b2))%R = ((Reals.RIneq.Rsqr (dot a1 a2 b1
b2)) + (Reals.RIneq.Rsqr ((a1 * b2)%R - (a2 * b1)%R)%R))%R).
Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R),
((Reals.RIneq.Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1
y2))%R)%R.
(* Why3 assumption *)
Definition norm (x1:R) (x2:R): R := (Reals.R_sqrt.sqrt (norm2 x1 x2)).
Axiom norm_pos : forall (x1:R) (x2:R), (0%R <= (norm x1 x2))%R.
Require Import Why3.
Ltac ae := why3 "Alt-Ergo,0.99.1," timelimit 3; admit.
Import R_sqrt.
Open Scope R_scope.
(* Why3 goal *)
Theorem sqr_le_sqrt : forall (x:R) (y:R), ((Reals.RIneq.Rsqr x) <= y)%R ->
(x <= (Reals.R_sqrt.sqrt y))%R.
(* Why3 intros x y h1. *)
intros x y h1.
assert (0 <= Rsqr x) by ae.
assert (0 <= y) by ae.
assert (h : (x < 0 \/ x >= 0)%R).
ae.
destruct h.
ae.
replace x with (sqrt (Rsqr x)).
apply sqrt_le_1.
ae.
ae.
ae.
ae.
Admitted.
......@@ -31,7 +31,6 @@
<proof prover="11"><result status="valid" time="0.00"/></proof>
</goal>
<goal name="sqr_le_sqrt" proved="true">
<proof prover="0" edited="lagrange_inequality_CauchySchwarzInequality_sqr_le_sqrt_1.v"><result status="valid" time="0.97"/></proof>
<proof prover="6"><result status="valid" time="0.04"/></proof>
</goal>
<goal name="CauchySchwarz" proved="true">
......
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