mirror of
https://github.com/AdaCore/why3.git
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680 lines
38 KiB
XML
680 lines
38 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
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"https://www.why3.org/why3session.dtd">
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<why3session shape_version="6">
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<prover id="1" name="Alt-Ergo" version="2.6.0" timelimit="5" steplimit="0" memlimit="2000"/>
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<file format="whyml" proved="true">
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<path name=".."/><path name="multiplication.mlw"/>
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<theory name="MultiplicationSingle" proved="true">
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<goal name="multiplication_errors_basic'vc" expl="VC for multiplication_errors_basic" proved="true">
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<transf name="split_vc" proved="true" >
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<goal name="multiplication_errors_basic'vc.0" expl="postcondition" proved="true">
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<transf name="assert" proved="true" arg1="let t = to_real c in
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let t1 = eta *. 0.0 in
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let t2 = 1.0 +. eps in
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let t3 = to_real a *. to_real b in
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let t4 = t3 *. t in
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abs (to_real result -. t4)
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<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. t2)) *. abs t4)
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+. ((((((0.0 +. (0.0 *. eps)) *. abs t3) +. ((eta +. t1) *. abs t))
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+. t1)
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*. t2)
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+. eta))">
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<goal name="multiplication_errors_basic'vc.0.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real result -. ((to_real a *. to_real b) *. to_real c))
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<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
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*. abs ((to_real a *. to_real b) *. to_real c))
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+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
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+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
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+. (eta *. 0.0))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors_basic'vc.0.0.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real result -. ((to_real a *. to_real b) *. to_real c))
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<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
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*. (abs (to_real a *. to_real b) *. abs (to_real c)))
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+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
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+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
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+. (eta *. 0.0))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors_basic'vc.0.0.0.0" expl="asserted formula" proved="true">
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<transf name="apply" proved="true" arg1="umul_single_error_propagation" arg2="with" arg3="a **. b,c">
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<goal name="multiplication_errors_basic'vc.0.0.0.0.0" expl="apply premises" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real (a **. b) -. (to_real a *. to_real b))
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<=. ((eps *. abs (to_real a *. to_real b)) +. eta)">
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<goal name="multiplication_errors_basic'vc.0.0.0.0.0.0" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.105258" steps="1020"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.0.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.042842" steps="141"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.049198" steps="89"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.2" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.069144" steps="101"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.3" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.040703" steps="92"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.4" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.040260" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.5" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.043195" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.6" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.037765" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.7" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.045188" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.0.8" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.028241" steps="83"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.0.1" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.083270" steps="559"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.0.1" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.121663" steps="792"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors_basic'vc.0.1" expl="postcondition" proved="true">
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<proof prover="1"><result status="valid" time="0.070553" steps="414"/></proof>
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</goal>
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</transf>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc" expl="VC for multiplication_errors" proved="true">
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<transf name="split_vc" proved="true" >
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<goal name="multiplication_errors'vc.0" expl="postcondition" proved="true">
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<transf name="assert" proved="true" arg1="let t1 = to_real d in
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let t2 = to_real c in
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let t8 = 1.0 +. eps in
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let t9 = t8 *. eta in
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let t10 = eps +. (eps *. t8) in
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let t11 = t1 *. (to_real e *. to_real f) in
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let t12 = (to_real a *. to_real b) *. t2 in
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let t13 = (t9 *. abs t1) +. eta in
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let t14 = (t9 *. abs t2) +. eta in
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let t15 = t12 *. t11 in
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abs (to_real result -. t15)
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<=. (((eps +. (((t10 +. t10) +. (t10 *. t10)) *. t8)) *. abs t15)
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+. ((((((t13 +. (t13 *. t10)) *. abs t12)
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+. ((t14 +. (t14 *. t10)) *. abs t11))
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+. (t14 *. t13))
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*. t8)
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+. eta))">
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<goal name="multiplication_errors'vc.0.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs
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(to_real result
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-. (((to_real a *. to_real b) *. to_real c)
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*. (to_real d *. (to_real e *. to_real f))))
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<=. (((eps
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+. ((((eps +. (eps *. (1.0 +. eps))) +. (eps +. (eps *. (1.0 +. eps))))
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+. ((eps +. (eps *. (1.0 +. eps)))
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. (1.0 +. eps)))
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*. abs
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(((to_real a *. to_real b) *. to_real c)
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*. (to_real d *. (to_real e *. to_real f))))
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+. ((((((((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
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+. (((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. abs ((to_real a *. to_real b) *. to_real c))
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+. ((((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. abs (to_real d *. (to_real e *. to_real f))))
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+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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*. ((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs
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(to_real result
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-. (((to_real a *. to_real b) *. to_real c)
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*. (to_real d *. (to_real e *. to_real f))))
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<=. (((eps
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+. ((((eps +. (eps *. (1.0 +. eps))) +. (eps +. (eps *. (1.0 +. eps))))
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+. ((eps +. (eps *. (1.0 +. eps)))
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. (1.0 +. eps)))
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*. (abs ((to_real a *. to_real b) *. to_real c)
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*. abs (to_real d *. (to_real e *. to_real f))))
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+. ((((((((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
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+. (((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. abs ((to_real a *. to_real b) *. to_real c))
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+. ((((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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*. (eps +. (eps *. (1.0 +. eps)))))
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*. abs (to_real d *. (to_real e *. to_real f))))
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+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
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*. ((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0.0" expl="asserted formula" proved="true">
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<transf name="apply" proved="true" arg1="umul_single_error_propagation" arg2="with" arg3="(a **. b) **. c,d **. (e **. f)">
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<goal name="multiplication_errors'vc.0.0.0.0.0" expl="apply premises" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real ((a **. b) **. c) -. ((to_real a *. to_real b) *. to_real c))
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<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
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*. abs ((to_real a *. to_real b) *. to_real c))
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+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
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+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
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+. (eta *. 0.0))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0.0.0.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real ((a **. b) **. c) -. ((to_real a *. to_real b) *. to_real c))
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<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
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*. (abs (to_real a *. to_real b) *. abs (to_real c)))
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+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
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+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
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+. (eta *. 0.0))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0" expl="asserted formula" proved="true">
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<transf name="apply" proved="true" arg1="umul_single_error_propagation" arg2="with" arg3="a **. b,c">
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0" expl="apply premises" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real (a **. b) -. (to_real a *. to_real b))
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<=. ((eps *. abs (to_real a *. to_real b)) +. eta)">
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0.0" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.107169" steps="1020"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.043141" steps="141"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.051137" steps="89"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.2" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.054969" steps="101"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.3" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.059122" steps="92"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.4" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.047284" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.5" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.056552" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.6" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.045509" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.7" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.063405" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.8" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.042096" steps="83"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.0.1" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.134119" steps="559"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.0.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.100312" steps="239"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1" expl="apply premises" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real (d **. (e **. f)) -. (to_real d *. (to_real e *. to_real f)))
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<=. (((eps +. (((0.0 +. eps) +. (0.0 *. eps)) *. (1.0 +. eps)))
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*. abs (to_real d *. (to_real e *. to_real f)))
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+. ((((((eta +. (eta *. 0.0)) *. abs (to_real d))
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+. ((0.0 +. (0.0 *. eps)) *. abs (to_real e *. to_real f)))
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+. (0.0 *. eta))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0.0.1.0" expl="asserted formula" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real (d **. (e **. f)) -. (to_real d *. (to_real e *. to_real f)))
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<=. (((eps +. (((0.0 +. eps) +. (0.0 *. eps)) *. (1.0 +. eps)))
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*. (abs (to_real d) *. abs (to_real e *. to_real f)))
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+. ((((((eta +. (eta *. 0.0)) *. abs (to_real d))
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+. ((0.0 +. (0.0 *. eps)) *. abs (to_real e *. to_real f)))
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+. (0.0 *. eta))
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*. (1.0 +. eps))
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+. eta))">
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0" expl="asserted formula" proved="true">
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<transf name="apply" proved="true" arg1="umul_single_error_propagation" arg2="with" arg3="d,e **. f">
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.0" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.041165" steps="89"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1" expl="apply premises" proved="true">
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<transf name="assert" proved="true" arg1="abs (to_real (e **. f) -. (to_real e *. to_real f))
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<=. ((eps *. abs (to_real e *. to_real f)) +. eta)">
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1.0" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.109763" steps="1020"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1.1" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.061415" steps="141"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.2" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.053984" steps="92"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.3" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.072308" steps="101"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.4" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.040001" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.5" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.037152" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.6" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.065069" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.7" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.067950" steps="84"/></proof>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.8" expl="apply premises" proved="true">
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<proof prover="1"><result status="valid" time="0.071104" steps="83"/></proof>
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</goal>
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</transf>
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</goal>
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<goal name="multiplication_errors'vc.0.0.0.0.1.0.1" expl="asserted formula" proved="true">
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<proof prover="1"><result status="valid" time="0.100635" steps="569"/></proof>
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</goal>
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|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.072260" steps="239"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.2" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.050077" steps="102"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.3" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.047419" steps="102"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.4" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.045738" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.5" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.042283" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.6" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.040666" steps="91"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.7" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.037386" steps="91"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.8" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.036885" steps="83"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="2.621282" steps="5379"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="2.471189" steps="4574"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.1" expl="postcondition" proved="true">
|
|
<proof prover="1"><result status="valid" time="1.890650" steps="3830"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
</theory>
|
|
<theory name="MultiplicationDouble" proved="true">
|
|
<goal name="multiplication_errors_basic'vc" expl="VC for multiplication_errors_basic" proved="true">
|
|
<transf name="split_vc" proved="true" >
|
|
<goal name="multiplication_errors_basic'vc.0" expl="postcondition" proved="true">
|
|
<transf name="assert" proved="true" arg1="let t = to_real c in
|
|
let t1 = eta *. 0.0 in
|
|
let t2 = 1.0 +. eps in
|
|
let t3 = to_real a *. to_real b in
|
|
let t4 = t3 *. t in
|
|
abs (to_real result -. t4)
|
|
<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. t2)) *. abs t4)
|
|
+. ((((((0.0 +. (0.0 *. eps)) *. abs t3) +. ((eta +. t1) *. abs t))
|
|
+. t1)
|
|
*. t2)
|
|
+. eta))">
|
|
<goal name="multiplication_errors_basic'vc.0.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real result -. ((to_real a *. to_real b) *. to_real c))
|
|
<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
|
|
*. abs ((to_real a *. to_real b) *. to_real c))
|
|
+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
|
|
+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
|
|
+. (eta *. 0.0))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors_basic'vc.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real result -. ((to_real a *. to_real b) *. to_real c))
|
|
<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
|
|
*. (abs (to_real a *. to_real b) *. abs (to_real c)))
|
|
+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
|
|
+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
|
|
+. (eta *. 0.0))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="apply" proved="true" arg1="umul_double_error_propagation" arg2="with" arg3="a **. b,c">
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.0" expl="apply premises" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real (a **. b) -. (to_real a *. to_real b))
|
|
<=. ((eps *. abs (to_real a *. to_real b)) +. eta)">
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.0.0" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.110392" steps="1020"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.043614" steps="141"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.064864" steps="89"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.2" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.055980" steps="101"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.3" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.062339" steps="92"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.4" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.044936" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.5" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.042066" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.6" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.044831" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.7" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.030915" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.0.8" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.042523" steps="83"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.078971" steps="559"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.086812" steps="792"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors_basic'vc.0.1" expl="postcondition" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.075224" steps="414"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc" expl="VC for multiplication_errors" proved="true">
|
|
<transf name="split_vc" proved="true" >
|
|
<goal name="multiplication_errors'vc.0" expl="postcondition" proved="true">
|
|
<transf name="assert" proved="true" arg1="let t1 = to_real d in
|
|
let t2 = to_real c in
|
|
let t8 = 1.0 +. eps in
|
|
let t9 = t8 *. eta in
|
|
let t10 = eps +. (eps *. t8) in
|
|
let t11 = t1 *. (to_real e *. to_real f) in
|
|
let t12 = (to_real a *. to_real b) *. t2 in
|
|
let t13 = (t9 *. abs t1) +. eta in
|
|
let t14 = (t9 *. abs t2) +. eta in
|
|
let t15 = t12 *. t11 in
|
|
abs (to_real result -. t15)
|
|
<=. (((eps +. (((t10 +. t10) +. (t10 *. t10)) *. t8)) *. abs t15)
|
|
+. ((((((t13 +. (t13 *. t10)) *. abs t12)
|
|
+. ((t14 +. (t14 *. t10)) *. abs t11))
|
|
+. (t14 *. t13))
|
|
*. t8)
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs
|
|
(to_real result
|
|
-. (((to_real a *. to_real b) *. to_real c)
|
|
*. (to_real d *. (to_real e *. to_real f))))
|
|
<=. (((eps
|
|
+. ((((eps +. (eps *. (1.0 +. eps))) +. (eps +. (eps *. (1.0 +. eps))))
|
|
+. ((eps +. (eps *. (1.0 +. eps)))
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. (1.0 +. eps)))
|
|
*. abs
|
|
(((to_real a *. to_real b) *. to_real c)
|
|
*. (to_real d *. (to_real e *. to_real f))))
|
|
+. ((((((((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. abs ((to_real a *. to_real b) *. to_real c))
|
|
+. ((((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. abs (to_real d *. (to_real e *. to_real f))))
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
*. ((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs
|
|
(to_real result
|
|
-. (((to_real a *. to_real b) *. to_real c)
|
|
*. (to_real d *. (to_real e *. to_real f))))
|
|
<=. (((eps
|
|
+. ((((eps +. (eps *. (1.0 +. eps))) +. (eps +. (eps *. (1.0 +. eps))))
|
|
+. ((eps +. (eps *. (1.0 +. eps)))
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. (1.0 +. eps)))
|
|
*. (abs ((to_real a *. to_real b) *. to_real c)
|
|
*. abs (to_real d *. (to_real e *. to_real f))))
|
|
+. ((((((((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. abs ((to_real a *. to_real b) *. to_real c))
|
|
+. ((((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
*. (eps +. (eps *. (1.0 +. eps)))))
|
|
*. abs (to_real d *. (to_real e *. to_real f))))
|
|
+. (((((1.0 +. eps) *. eta) *. abs (to_real c)) +. eta)
|
|
*. ((((1.0 +. eps) *. eta) *. abs (to_real d)) +. eta)))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="apply" proved="true" arg1="umul_double_error_propagation" arg2="with" arg3="(a **. b) **. c,d **. (e **. f)">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0" expl="apply premises" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real ((a **. b) **. c) -. ((to_real a *. to_real b) *. to_real c))
|
|
<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
|
|
*. abs ((to_real a *. to_real b) *. to_real c))
|
|
+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
|
|
+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
|
|
+. (eta *. 0.0))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real ((a **. b) **. c) -. ((to_real a *. to_real b) *. to_real c))
|
|
<=. (((eps +. (((eps +. 0.0) +. (eps *. 0.0)) *. (1.0 +. eps)))
|
|
*. (abs (to_real a *. to_real b) *. abs (to_real c)))
|
|
+. ((((((0.0 +. (0.0 *. eps)) *. abs (to_real a *. to_real b))
|
|
+. ((eta +. (eta *. 0.0)) *. abs (to_real c)))
|
|
+. (eta *. 0.0))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0" expl="asserted formula" proved="true">
|
|
<transf name="apply" proved="true" arg1="umul_double_error_propagation" arg2="with" arg3="a **. b,c">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0" expl="apply premises" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real (a **. b) -. (to_real a *. to_real b))
|
|
<=. ((eps *. abs (to_real a *. to_real b)) +. eta)">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0.0" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.111890" steps="1020"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.065520" steps="141"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.068939" steps="89"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.2" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.053268" steps="101"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.3" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.067821" steps="92"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.4" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.069933" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.5" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.044568" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.6" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.050097" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.7" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.051977" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.0.8" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.044966" steps="83"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.100255" steps="559"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.073698" steps="239"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1" expl="apply premises" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real (d **. (e **. f)) -. (to_real d *. (to_real e *. to_real f)))
|
|
<=. (((eps +. (((0.0 +. eps) +. (0.0 *. eps)) *. (1.0 +. eps)))
|
|
*. abs (to_real d *. (to_real e *. to_real f)))
|
|
+. ((((((eta +. (eta *. 0.0)) *. abs (to_real d))
|
|
+. ((0.0 +. (0.0 *. eps)) *. abs (to_real e *. to_real f)))
|
|
+. (0.0 *. eta))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0" expl="asserted formula" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real (d **. (e **. f)) -. (to_real d *. (to_real e *. to_real f)))
|
|
<=. (((eps +. (((0.0 +. eps) +. (0.0 *. eps)) *. (1.0 +. eps)))
|
|
*. (abs (to_real d) *. abs (to_real e *. to_real f)))
|
|
+. ((((((eta +. (eta *. 0.0)) *. abs (to_real d))
|
|
+. ((0.0 +. (0.0 *. eps)) *. abs (to_real e *. to_real f)))
|
|
+. (0.0 *. eta))
|
|
*. (1.0 +. eps))
|
|
+. eta))">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0" expl="asserted formula" proved="true">
|
|
<transf name="apply" proved="true" arg1="umul_double_error_propagation" arg2="with" arg3="d,e **. f">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.0" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.066678" steps="89"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1" expl="apply premises" proved="true">
|
|
<transf name="assert" proved="true" arg1="abs (to_real (e **. f) -. (to_real e *. to_real f))
|
|
<=. ((eps *. abs (to_real e *. to_real f)) +. eta)">
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1.0" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.149677" steps="1020"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.1.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.059920" steps="141"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.2" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.045417" steps="92"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.3" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.056132" steps="101"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.4" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.037990" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.5" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.042664" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.6" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.053716" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.7" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.040140" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.0.8" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.041696" steps="83"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.109968" steps="569"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.1.1" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.070900" steps="239"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.2" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.069181" steps="102"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.3" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.064175" steps="102"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.4" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.050132" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.5" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.058196" steps="84"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.6" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.071636" steps="91"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.7" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.047329" steps="91"/></proof>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.0.8" expl="apply premises" proved="true">
|
|
<proof prover="1"><result status="valid" time="0.054859" steps="83"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="3.022375" steps="5329"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.0.1" expl="asserted formula" proved="true">
|
|
<proof prover="1"><result status="valid" time="3.378756" steps="5409"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
<goal name="multiplication_errors'vc.0.1" expl="postcondition" proved="true">
|
|
<proof prover="1"><result status="valid" time="2.093901" steps="3830"/></proof>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
</transf>
|
|
</goal>
|
|
</theory>
|
|
</file>
|
|
</why3session>
|