--- a/NEWS Wed Oct 08 10:15:04 2014 +0200
+++ b/NEWS Wed Oct 08 11:09:17 2014 +0200
@@ -126,6 +126,12 @@
generated code in target languages in HOL/Library/Code_Test. See
HOL/Codegenerator_Test/Code_Test* for examples.
+* Library/Sum_of_Squares: simplified and improved "sos" method. Always
+use local CSDP executable, which is much faster than the NEOS server.
+The "sos_cert" functionality is invoked as "sos" with additional
+argument. Minor INCOMPATIBILITY.
+
+
*** ML ***
* Tactical PARALLEL_ALLGOALS is the most common way to refer to
--- a/src/HOL/Library/Sum_of_Squares.thy Wed Oct 08 10:15:04 2014 +0200
+++ b/src/HOL/Library/Sum_of_Squares.thy Wed Oct 08 11:09:17 2014 +0200
@@ -3,8 +3,8 @@
Author: Philipp Meyer, TU Muenchen
*)
-header {* A decision method for universal multivariate real arithmetic with addition,
- multiplication and ordering using semidefinite programming *}
+header {* A decision procedure for universal multivariate real arithmetic
+ with addition, multiplication and ordering using semidefinite programming *}
theory Sum_of_Squares
imports Complex_Main
@@ -15,27 +15,4 @@
ML_file "Sum_of_Squares/positivstellensatz_tools.ML"
ML_file "Sum_of_Squares/sos_wrapper.ML"
-text {*
- Proof method sos invocations:
- \begin{itemize}
-
- \item remote solver: @{text "(sos remote_csdp)"}
-
- \item local solver: @{text "(sos csdp)"}
-
- The latter requires a local executable from
- @{url "https://projects.coin-or.org/Csdp"} and the Isabelle settings variable
- variable @{text ISABELLE_CSDP} pointing to it.
-
- \end{itemize}
-
- By default, method sos calls @{text remote_csdp}. This can take of
- the order of a minute for one sos call, because sos calls CSDP
- repeatedly. If you install CSDP locally, sos calls typically takes
- only a few seconds.
-
- The sos method generates a certificate which can be used to repeat
- the proof without calling an external prover.
-*}
-
end
--- a/src/HOL/Library/Sum_of_Squares/etc/settings Wed Oct 08 10:15:04 2014 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,8 +0,0 @@
-# -*- shell-script -*- :mode=shellscript:
-
-ISABELLE_SUM_OF_SQUARES="$COMPONENT"
-ISABELLE_NEOS_SERVER="http://neos-server.org:3332"
-
-# local SDP Solver, cf. https://projects.coin-or.org/Csdp
-#ISABELLE_CSDP="/usr/local/bin/csdp"
-
--- a/src/HOL/Library/Sum_of_Squares/neos_csdp_client Wed Oct 08 10:15:04 2014 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,85 +0,0 @@
-#!/usr/bin/env python
-import sys
-import signal
-import xmlrpclib
-import time
-import re
-import os
-
-# Neos server config
-neos = xmlrpclib.Server(os.getenv("ISABELLE_NEOS_SERVER"))
-
-jobNumber = 0
-password = ""
-inputfile = None
-outputfile = None
-# interrupt handler
-def cleanup(signal, frame):
- sys.stdout.write("Caught interrupt, cleaning up\n")
- if jobNumber != 0:
- neos.killJob(jobNumber, password)
- if inputfile != None:
- inputfile.close()
- if outputfile != None:
- outputfile.close()
- sys.exit(21)
-
-signal.signal(signal.SIGHUP, cleanup)
-signal.signal(signal.SIGINT, cleanup)
-signal.signal(signal.SIGQUIT, cleanup)
-signal.signal(signal.SIGTERM, cleanup)
-
-if len(sys.argv) <> 3:
- sys.stderr.write("Usage: neos_csdp_client <input_filename> <output_filename>\n")
- sys.exit(19)
-
-xml_pre = "<document>\n<category>sdp</category>\n<solver>csdp</solver>\n<inputMethod>SPARSE_SDPA</inputMethod>\n<dat><![CDATA["
-xml_post = "]]></dat>\n</document>\n"
-xml = xml_pre
-inputfile = open(sys.argv[1],"r")
-buffer = 1
-while buffer:
- buffer = inputfile.read()
- xml += buffer
-inputfile.close()
-xml += xml_post
-
-(jobNumber,password) = neos.submitJob(xml)
-
-if jobNumber == 0:
- sys.stdout.write("error submitting job: %s" % password)
- sys.exit(20)
-else:
- sys.stdout.write("jobNumber = %d\tpassword = %s\n" % (jobNumber,password))
-
-offset=0
-messages = ""
-status="Waiting"
-while status == "Running" or status=="Waiting":
- time.sleep(1)
- (msg,offset) = neos.getIntermediateResults(jobNumber,password,offset)
- messages += msg.data
- sys.stdout.write(msg.data)
- status = neos.getJobStatus(jobNumber, password)
-
-msg = neos.getFinalResults(jobNumber, password).data
-sys.stdout.write("---------- Begin CSDP Output -------------\n");
-sys.stdout.write(msg)
-
-# extract solution
-result = msg.split("Solution:")
-if len(result) > 1:
- solution = result[1].strip()
- if solution != "":
- outputfile = open(sys.argv[2],"w")
- outputfile.write(solution)
- outputfile.close()
-
-# extract return code
-p = re.compile(r"^Error: Command exited with non-zero status (\d+)$", re.MULTILINE)
-m = p.search(messages)
-if m:
- sys.exit(int(m.group(1)))
-else:
- sys.exit(0)
-
--- a/src/HOL/Library/Sum_of_Squares/sos_wrapper.ML Wed Oct 08 10:15:04 2014 +0200
+++ b/src/HOL/Library/Sum_of_Squares/sos_wrapper.ML Wed Oct 08 11:09:17 2014 +0200
@@ -6,9 +6,7 @@
signature SOS_WRAPPER =
sig
- datatype prover_result = Success | Failure | Error
- val dest_dir: string Config.T
- val prover_name: string Config.T
+ val sos_tac: Proof.context -> string option -> int -> tactic
end
structure SOS_Wrapper: SOS_WRAPPER =
@@ -21,33 +19,33 @@
| str_of_result Error = "Error"
-(*** calling provers ***)
+fun filename name =
+ File.tmp_path (Path.basic (name ^ serial_string ()))
-val dest_dir = Attrib.setup_config_string @{binding sos_dest_dir} (K "")
+fun find_failure rv =
+ case rv of
+ 0 => (Success, "SDP solved")
+ | 1 => (Failure, "SDP is primal infeasible")
+ | 2 => (Failure, "SDP is dual infeasible")
+ | 3 => (Success, "SDP solved with reduced accuracy")
+ | 4 => (Failure, "Maximum iterations reached")
+ | 5 => (Failure, "Stuck at edge of primal feasibility")
+ | 6 => (Failure, "Stuck at edge of dual infeasibility")
+ | 7 => (Failure, "Lack of progress")
+ | 8 => (Failure, "X, Z, or O was singular")
+ | 9 => (Failure, "Detected NaN or Inf values")
+ | _ => (Error, "return code is " ^ string_of_int rv)
-fun filename dir name =
+val exe = Path.explode "$ISABELLE_CSDP"
+
+fun run_solver ctxt input =
let
- val probfile = Path.basic (name ^ serial_string ())
- val dir_path = Path.explode dir
- in
- if dir = "" then
- File.tmp_path probfile
- else if File.exists dir_path then
- Path.append dir_path probfile
- else error ("No such directory: " ^ dir)
- end
-
-fun run_solver ctxt name exe find_failure input =
- let
- val _ = warning ("Calling solver: " ^ name)
-
(* create input file *)
- val dir = Config.get ctxt dest_dir
- val input_file = filename dir "sos_in"
+ val input_file = filename "sos_in"
val _ = File.write input_file input
(* call solver *)
- val output_file = filename dir "sos_out"
+ val output_file = filename "sos_out"
val (output, rv) =
Isabelle_System.bash_output
(if File.exists exe then
@@ -59,10 +57,8 @@
val result = if File.exists output_file then File.read output_file else ""
(* remove temporary files *)
- val _ =
- if dir = "" then
- (File.rm input_file; if File.exists output_file then File.rm output_file else ())
- else ()
+ val _ = File.rm input_file
+ val _ = if File.exists output_file then File.rm output_file else ()
val _ =
if Config.get ctxt Sum_of_Squares.trace
@@ -78,78 +74,24 @@
end
-(*** various provers ***)
-
-(* local csdp client *)
-
-fun find_csdp_failure rv =
- case rv of
- 0 => (Success, "SDP solved")
- | 1 => (Failure, "SDP is primal infeasible")
- | 2 => (Failure, "SDP is dual infeasible")
- | 3 => (Success, "SDP solved with reduced accuracy")
- | 4 => (Failure, "Maximum iterations reached")
- | 5 => (Failure, "Stuck at edge of primal feasibility")
- | 6 => (Failure, "Stuck at edge of dual infeasibility")
- | 7 => (Failure, "Lack of progress")
- | 8 => (Failure, "X, Z, or O was singular")
- | 9 => (Failure, "Detected NaN or Inf values")
- | _ => (Error, "return code is " ^ string_of_int rv)
-
-val csdp = (Path.explode "$ISABELLE_CSDP", find_csdp_failure)
-
-
-(* remote neos server *)
-
-fun find_neos_failure rv =
- case rv of
- 20 => (Error, "error submitting job")
- | 21 => (Error, "interrupt")
- | _ => find_csdp_failure rv
-
-val neos_csdp = (Path.explode "$ISABELLE_SUM_OF_SQUARES/neos_csdp_client", find_neos_failure)
-
-
-(* named provers *)
-
-fun prover "remote_csdp" = neos_csdp
- | prover "csdp" = csdp
- | prover name = error ("Unknown prover: " ^ name)
-
-val prover_name = Attrib.setup_config_string @{binding sos_prover_name} (K "remote_csdp")
-
-fun call_solver ctxt opt_name =
- let
- val name = the_default (Config.get ctxt prover_name) opt_name
- val (exe, find_failure) = prover name
- in run_solver ctxt name exe find_failure end
-
-
-(* certificate output *)
-
-fun output_line cert =
- "To repeat this proof with a certificate use this command:\n" ^
- Active.sendback_markup [] ("apply (sos_cert \"" ^ cert ^ "\")")
-
-val print_cert = warning o output_line o Positivstellensatz_Tools.print_cert
-
-
(* method setup *)
-fun sos_solver print method = SIMPLE_METHOD' o Sum_of_Squares.sos_tac print method
+fun print_cert cert =
+ warning
+ ("To repeat this proof with a certificate use this proof method:\n" ^
+ Active.sendback_markup [] ("(sos \"" ^ Positivstellensatz_Tools.print_cert cert ^ "\")"))
+
+fun sos_tac ctxt NONE =
+ Sum_of_Squares.sos_tac print_cert
+ (Sum_of_Squares.Prover (run_solver ctxt)) ctxt
+ | sos_tac ctxt (SOME cert) =
+ Sum_of_Squares.sos_tac ignore
+ (Sum_of_Squares.Certificate (Positivstellensatz_Tools.read_cert ctxt cert)) ctxt
val _ = Theory.setup
(Method.setup @{binding sos}
- (Scan.lift (Scan.option Parse.xname)
- >> (fn opt_name => fn ctxt =>
- sos_solver print_cert
- (Sum_of_Squares.Prover (call_solver ctxt opt_name)) ctxt))
- "prove universal problems over the reals using sums of squares" #>
- Method.setup @{binding sos_cert}
- (Scan.lift Parse.string
- >> (fn cert => fn ctxt =>
- sos_solver ignore
- (Sum_of_Squares.Certificate (Positivstellensatz_Tools.read_cert ctxt cert)) ctxt))
- "prove universal problems over the reals using sums of squares with certificates")
+ (Scan.lift (Scan.option Parse.string)
+ >> (fn arg => fn ctxt => SIMPLE_METHOD' (sos_tac ctxt arg)))
+ "prove universal problems over the reals using sums of squares")
end
--- a/src/HOL/ROOT Wed Oct 08 10:15:04 2014 +0200
+++ b/src/HOL/ROOT Wed Oct 08 11:09:17 2014 +0200
@@ -600,11 +600,8 @@
ML
SAT_Examples
Nominal2_Dummy
+ SOS
SOS_Cert
- theories [condition = ISABELLE_CSDP]
- SOS
- theories [condition = ISABELLE_FULL_TEST]
- SOS_Remote
theories [skip_proofs = false]
Meson_Test
theories [condition = SVC_HOME]
--- a/src/HOL/ex/SOS.thy Wed Oct 08 10:15:04 2014 +0200
+++ b/src/HOL/ex/SOS.thy Wed Oct 08 11:09:17 2014 +0200
@@ -10,121 +10,121 @@
begin
lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
- by (sos csdp)
+ by sos
lemma "a1 >= 0 & a2 >= 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) --> a1 * a2 - b1 * b2 >= (0::real)"
- by (sos csdp)
+ by sos
lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
- by (sos csdp)
+ by sos
lemma "(0::real) <= x & x <= 1 & 0 <= y & y <= 1 --> x^2 + y^2 < 1 |(x - 1)^2 + y^2 < 1 | x^2 + (y - 1)^2 < 1 | (x - 1)^2 + (y - 1)^2 < 1"
- by (sos csdp)
+ by sos
lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z"
- by (sos csdp)
+ by sos
lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
- by (sos csdp)
+ by sos
lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
- by (sos csdp)
+ by sos
lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
- by (sos csdp)
+ by sos
lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
- by (sos csdp)
+ by sos
lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
- by (sos csdp)
+ by sos
text \<open>One component of denominator in dodecahedral example.\<close>
lemma "2 <= x & x <= 125841 / 50000 & 2 <= y & y <= 125841 / 50000 & 2 <= z & z <= 125841 / 50000 --> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= (0::real)"
- by (sos csdp)
+ by sos
text \<open>Over a larger but simpler interval.\<close>
lemma "(2::real) <= x & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 0 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
- by (sos csdp)
+ by sos
text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close>
lemma "2 <= (x::real) & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
- by (sos csdp)
+ by sos
text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close>
lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
- by (sos csdp)
+ by sos
lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
- by (sos csdp)
+ by sos
lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
- by (sos csdp)
+ by sos
lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
- by (sos csdp)
+ by sos
lemma "(0::real) < x --> 0 < 1 + x + x^2"
- by (sos csdp)
+ by sos
lemma "(0::real) <= x --> 0 < 1 + x + x^2"
- by (sos csdp)
+ by sos
lemma "(0::real) < 1 + x^2"
- by (sos csdp)
+ by sos
lemma "(0::real) <= 1 + 2 * x + x^2"
- by (sos csdp)
+ by sos
lemma "(0::real) < 1 + abs x"
- by (sos csdp)
+ by sos
lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
- by (sos csdp)
+ by sos
lemma "abs ((1::real) + x^2) = (1::real) + x^2"
- by (sos csdp)
+ by sos
lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
- by (sos csdp)
+ by sos
lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
- by (sos csdp)
+ by sos
lemma "(1::real) < x --> x^2 < y --> 1 < y"
- by (sos csdp)
+ by sos
lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos csdp)
+ by sos
lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos csdp)
+ by sos
lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
- by (sos csdp)
+ by sos
lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x"
- by (sos csdp)
+ by sos
lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)"
- by (sos csdp)
+ by sos
(* lemma "((x::real) - y - 2 * x^4 = 0) & 0 <= x & x <= 2 & 0 <= y & y <= 3 --> y^2 - 7 * y - 12 * x + 17 >= 0" by sos *) (* Too hard?*)
lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
- by (sos csdp)
+ by sos
lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
- by (sos csdp)
+ by sos
lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
- by (sos csdp)
+ by sos
lemma "4*r^2 = p^2 - 4*q & r >= (0::real) & x^2 + p*x + q = 0 --> 2*(x::real) = - p + 2*r | 2*x = -p - 2*r"
- by (sos csdp)
+ by sos
end
--- a/src/HOL/ex/SOS_Cert.thy Wed Oct 08 10:15:04 2014 +0200
+++ b/src/HOL/ex/SOS_Cert.thy Wed Oct 08 11:09:17 2014 +0200
@@ -10,121 +10,121 @@
begin
lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
- by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
+ by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
lemma "a1 >= 0 & a2 >= 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) --> a1 * a2 - b1 * b2 >= (0::real)"
- by (sos_cert "(((A<0 * R<1) + (([~1/2*a1*b2 + ~1/2*a2*b1] * A=0) + (([~1/2*a1*a2 + 1/2*b1*b2] * A=1) + (((A<0 * R<1) * ((R<1/2 * [b2]^2) + (R<1/2 * [b1]^2))) + ((A<=0 * (A<=1 * R<1)) * ((R<1/2 * [b2]^2) + ((R<1/2 * [b1]^2) + ((R<1/2 * [a2]^2) + (R<1/2 * [a1]^2))))))))))")
+ by (sos "(((A<0 * R<1) + (([~1/2*a1*b2 + ~1/2*a2*b1] * A=0) + (([~1/2*a1*a2 + 1/2*b1*b2] * A=1) + (((A<0 * R<1) * ((R<1/2 * [b2]^2) + (R<1/2 * [b1]^2))) + ((A<=0 * (A<=1 * R<1)) * ((R<1/2 * [b2]^2) + ((R<1/2 * [b1]^2) + ((R<1/2 * [a2]^2) + (R<1/2 * [a1]^2))))))))))")
lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
- by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
+ by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
lemma "(0::real) <= x & x <= 1 & 0 <= y & y <= 1 --> x^2 + y^2 < 1 |(x - 1)^2 + y^2 < 1 | x^2 + (y - 1)^2 < 1 | (x - 1)^2 + (y - 1)^2 < 1"
- by (sos_cert "((R<1 + (((A<=3 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=7 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=5 * R<1)) * (R<1 * [1]^2)))))))")
+ by (sos "((R<1 + (((A<=3 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=7 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=5 * R<1)) * (R<1 * [1]^2)))))))")
lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z"
- by (sos_cert "(((A<0 * R<1) + (((A<0 * R<1) * (R<1/2 * [1]^2)) + (((A<=2 * R<1) * (R<1/2 * [~1*x + y]^2)) + (((A<=1 * R<1) * (R<1/2 * [~1*x + z]^2)) + (((A<=1 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + (((A<=0 * R<1) * (R<1/2 * [~1*y + z]^2)) + (((A<=0 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + ((A<=0 * (A<=1 * (A<=3 * R<1))) * (R<1/2 * [1]^2))))))))))")
+ by (sos "(((A<0 * R<1) + (((A<0 * R<1) * (R<1/2 * [1]^2)) + (((A<=2 * R<1) * (R<1/2 * [~1*x + y]^2)) + (((A<=1 * R<1) * (R<1/2 * [~1*x + z]^2)) + (((A<=1 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + (((A<=0 * R<1) * (R<1/2 * [~1*y + z]^2)) + (((A<=0 * (A<=2 * (A<=3 * R<1))) * (R<1/2 * [1]^2)) + ((A<=0 * (A<=1 * (A<=3 * R<1))) * (R<1/2 * [1]^2))))))))))")
lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
- by (sos_cert "(((A<0 * R<1) + (([~3] * A=0) + (R<1 * ((R<2 * [~1/2*x + ~1/2*y + z]^2) + (R<3/2 * [~1*x + y]^2))))))")
+ by (sos "(((A<0 * R<1) + (([~3] * A=0) + (R<1 * ((R<2 * [~1/2*x + ~1/2*y + z]^2) + (R<3/2 * [~1*x + y]^2))))))")
lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
- by (sos_cert "(((A<0 * R<1) + (([~4] * A=0) + (R<1 * ((R<3 * [~1/3*w + ~1/3*x + ~1/3*y + z]^2) + ((R<8/3 * [~1/2*w + ~1/2*x + y]^2) + (R<2 * [~1*w + x]^2)))))))")
+ by (sos "(((A<0 * R<1) + (([~4] * A=0) + (R<1 * ((R<3 * [~1/3*w + ~1/3*x + ~1/3*y + z]^2) + ((R<8/3 * [~1/2*w + ~1/2*x + y]^2) + (R<2 * [~1*w + x]^2)))))))")
lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
- by (sos_cert "(((A<0 * R<1) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))")
+ by (sos "(((A<0 * R<1) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))")
lemma "(x::real) > 1 & y > 1 --> x * y > x + y - 1"
- by (sos_cert "((((A<0 * A<1) * R<1) + ((A<=0 * R<1) * (R<1 * [1]^2))))")
+ by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * R<1) * (R<1 * [1]^2))))")
lemma "abs(x) <= 1 --> abs(64 * x^7 - 112 * x^5 + 56 * x^3 - 7 * x) <= (1::real)"
- by (sos_cert "((((A<0 * R<1) + ((A<=1 * R<1) * (R<1 * [~8*x^3 + ~4*x^2 + 4*x + 1]^2)))) & ((((A<0 * A<1) * R<1) + ((A<=1 * (A<0 * R<1)) * (R<1 * [8*x^3 + ~4*x^2 + ~4*x + 1]^2)))))")
+ by (sos "((((A<0 * R<1) + ((A<=1 * R<1) * (R<1 * [~8*x^3 + ~4*x^2 + 4*x + 1]^2)))) & ((((A<0 * A<1) * R<1) + ((A<=1 * (A<0 * R<1)) * (R<1 * [8*x^3 + ~4*x^2 + ~4*x + 1]^2)))))")
text \<open>One component of denominator in dodecahedral example.\<close>
lemma "2 <= x & x <= 125841 / 50000 & 2 <= y & y <= 125841 / 50000 & 2 <= z & z <= 125841 / 50000 --> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= (0::real)"
- by (sos_cert "(((A<0 * R<1) + ((R<1 * ((R<5749028157/5000000000 * [~25000/222477*x + ~25000/222477*y + ~25000/222477*z + 1]^2) + ((R<864067/1779816 * [419113/864067*x + 419113/864067*y + z]^2) + ((R<320795/864067 * [419113/1283180*x + y]^2) + (R<1702293/5132720 * [x]^2))))) + (((A<=4 * (A<=5 * R<1)) * (R<3/2 * [1]^2)) + (((A<=3 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<3/2 * [1]^2)) + (((A<=1 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=1 * (A<=3 * R<1)) * (R<1/2 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<3/2 * [1]^2)))))))))))))")
+ by (sos "(((A<0 * R<1) + ((R<1 * ((R<5749028157/5000000000 * [~25000/222477*x + ~25000/222477*y + ~25000/222477*z + 1]^2) + ((R<864067/1779816 * [419113/864067*x + 419113/864067*y + z]^2) + ((R<320795/864067 * [419113/1283180*x + y]^2) + (R<1702293/5132720 * [x]^2))))) + (((A<=4 * (A<=5 * R<1)) * (R<3/2 * [1]^2)) + (((A<=3 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<3/2 * [1]^2)) + (((A<=1 * (A<=5 * R<1)) * (R<1/2 * [1]^2)) + (((A<=1 * (A<=3 * R<1)) * (R<1/2 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<3/2 * [1]^2)))))))))))))")
text \<open>Over a larger but simpler interval.\<close>
lemma "(2::real) <= x & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 0 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
- by (sos_cert "((R<1 + ((R<1 * ((R<1 * [~1/6*x + ~1/6*y + ~1/6*z + 1]^2) + ((R<1/18 * [~1/2*x + ~1/2*y + z]^2) + (R<1/24 * [~1*x + y]^2)))) + (((A<0 * R<1) * (R<1/12 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1/6 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1/6 * [1]^2)))))))))))")
+ by (sos "((R<1 + ((R<1 * ((R<1 * [~1/6*x + ~1/6*y + ~1/6*z + 1]^2) + ((R<1/18 * [~1/2*x + ~1/2*y + z]^2) + (R<1/24 * [~1*x + y]^2)))) + (((A<0 * R<1) * (R<1/12 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<1/6 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<1/6 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1/6 * [1]^2)))))))))))")
text \<open>We can do 12. I think 12 is a sharp bound; see PP's certificate.\<close>
lemma "2 <= (x::real) & x <= 4 & 2 <= y & y <= 4 & 2 <= z & z <= 4 --> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)"
- by (sos_cert "(((A<0 * R<1) + (((A<=4 * R<1) * (R<2/3 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1 * [1]^2)) + (((A<=3 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * R<1) * (R<2/3 * [1]^2)) + (((A<=2 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * R<1) * (R<2/3 * [1]^2)) + (((A<=0 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=0 * (A<=3 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<8/3 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))))))))))))))))")
+ by (sos "(((A<0 * R<1) + (((A<=4 * R<1) * (R<2/3 * [1]^2)) + (((A<=4 * (A<=5 * R<1)) * (R<1 * [1]^2)) + (((A<=3 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * R<1) * (R<2/3 * [1]^2)) + (((A<=2 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=2 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=2 * (A<=3 * R<1)) * (R<1 * [1]^2)) + (((A<=1 * (A<=4 * R<1)) * (R<1/3 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * R<1) * (R<2/3 * [1]^2)) + (((A<=0 * (A<=5 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=4 * R<1)) * (R<8/3 * [1]^2)) + (((A<=0 * (A<=3 * R<1)) * (R<1/3 * [1]^2)) + (((A<=0 * (A<=2 * R<1)) * (R<8/3 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2))))))))))))))))))")
text \<open>Inequality from sci.math (see "Leon-Sotelo, por favor").\<close>
lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x + y <= x^2 + y^2"
- by (sos_cert "(((A<0 * R<1) + (([1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")
+ by (sos "(((A<0 * R<1) + (([1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")
lemma "0 <= (x::real) & 0 <= y & (x * y = 1) --> x * y * (x + y) <= x^2 + y^2"
- by (sos_cert "(((A<0 * R<1) + (([~1*x + ~1*y + 1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")
+ by (sos "(((A<0 * R<1) + (([~1*x + ~1*y + 1] * A=0) + (R<1 * ((R<1 * [~1/2*x + ~1/2*y + 1]^2) + (R<3/4 * [~1*x + y]^2))))))")
lemma "0 <= (x::real) & 0 <= y --> x * y * (x + y)^2 <= (x^2 + y^2)^2"
- by (sos_cert "(((A<0 * R<1) + (R<1 * ((R<1 * [~1/2*x^2 + y^2 + ~1/2*x*y]^2) + (R<3/4 * [~1*x^2 + x*y]^2)))))")
+ by (sos "(((A<0 * R<1) + (R<1 * ((R<1 * [~1/2*x^2 + y^2 + ~1/2*x*y]^2) + (R<3/4 * [~1*x^2 + x*y]^2)))))")
lemma "(0::real) <= a & 0 <= b & 0 <= c & c * (2 * a + b)^3/ 27 <= x \<longrightarrow> c * a^2 * b <= x"
- by (sos_cert "(((A<0 * R<1) + (((A<=3 * R<1) * (R<1 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/27 * [~1*a + b]^2)) + ((A<=0 * (A<=2 * R<1)) * (R<8/27 * [~1*a + b]^2))))))")
+ by (sos "(((A<0 * R<1) + (((A<=3 * R<1) * (R<1 * [1]^2)) + (((A<=1 * (A<=2 * R<1)) * (R<1/27 * [~1*a + b]^2)) + ((A<=0 * (A<=2 * R<1)) * (R<8/27 * [~1*a + b]^2))))))")
lemma "(0::real) < x --> 0 < 1 + x + x^2"
- by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
+ by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
lemma "(0::real) <= x --> 0 < 1 + x + x^2"
- by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
+ by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
lemma "(0::real) < 1 + x^2"
- by (sos_cert "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
+ by (sos "((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
lemma "(0::real) <= 1 + 2 * x + x^2"
- by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [x + 1]^2))))")
+ by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [x + 1]^2))))")
lemma "(0::real) < 1 + abs x"
- by (sos_cert "((R<1 + (((A<=1 * R<1) * (R<1/2 * [1]^2)) + ((A<=0 * R<1) * (R<1/2 * [1]^2)))))")
+ by (sos "((R<1 + (((A<=1 * R<1) * (R<1/2 * [1]^2)) + ((A<=0 * R<1) * (R<1/2 * [1]^2)))))")
lemma "(0::real) < 1 + (1 + x)^2 * (abs x)"
- by (sos_cert "(((R<1 + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [x + 1]^2))))) & ((R<1 + (((A<0 * R<1) * (R<1 * [x + 1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
+ by (sos "(((R<1 + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [x + 1]^2))))) & ((R<1 + (((A<0 * R<1) * (R<1 * [x + 1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))")
lemma "abs ((1::real) + x^2) = (1::real) + x^2"
- by (sos_cert "(() & (((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<1 * R<1) * (R<1/2 * [1]^2))))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2)))))))")
+ by (sos "(() & (((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<1 * R<1) * (R<1/2 * [1]^2))))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2)))))))")
lemma "(3::real) * x + 7 * a < 4 \<and> 3 < 2 * x \<longrightarrow> a < 0"
- by (sos_cert "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
+ by (sos "((R<1 + (((A<1 * R<1) * (R<2 * [1]^2)) + (((A<0 * R<1) * (R<3 * [1]^2)) + ((A<=0 * R<1) * (R<14 * [1]^2))))))")
lemma "(0::real) < x --> 1 < y --> y * x <= z --> x < z"
- by (sos_cert "((((A<0 * A<1) * R<1) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
+ by (sos "((((A<0 * A<1) * R<1) + (((A<=1 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2)))))")
lemma "(1::real) < x --> x^2 < y --> 1 < y"
- by (sos_cert "((((A<0 * A<1) * R<1) + ((R<1 * ((R<1/10 * [~2*x + y + 1]^2) + (R<1/10 * [~1*x + y]^2))) + (((A<1 * R<1) * (R<1/2 * [1]^2)) + (((A<0 * R<1) * (R<1 * [x]^2)) + (((A<=0 * R<1) * ((R<1/10 * [x + 1]^2) + (R<1/10 * [x]^2))) + (((A<=0 * (A<1 * R<1)) * (R<1/5 * [1]^2)) + ((A<=0 * (A<0 * R<1)) * (R<1/5 * [1]^2)))))))))")
+ by (sos "((((A<0 * A<1) * R<1) + ((R<1 * ((R<1/10 * [~2*x + y + 1]^2) + (R<1/10 * [~1*x + y]^2))) + (((A<1 * R<1) * (R<1/2 * [1]^2)) + (((A<0 * R<1) * (R<1 * [x]^2)) + (((A<=0 * R<1) * ((R<1/10 * [x + 1]^2) + (R<1/10 * [x]^2))) + (((A<=0 * (A<1 * R<1)) * (R<1/5 * [1]^2)) + ((A<=0 * (A<0 * R<1)) * (R<1/5 * [1]^2)))))))))")
lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
+ by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
lemma "(b::real)^2 < 4 * a * c --> ~(a * x^2 + b * x + c = 0)"
- by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
+ by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
lemma "((a::real) * x^2 + b * x + c = 0) --> b^2 >= 4 * a * c"
- by (sos_cert "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
+ by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [2*a*x + b]^2))))")
lemma "(0::real) <= b & 0 <= c & 0 <= x & 0 <= y & (x^2 = c) & (y^2 = a^2 * c + b) --> a * c <= y * x"
- by (sos_cert "(((A<0 * (A<0 * R<1)) + (((A<=2 * (A<=3 * (A<0 * R<1))) * (R<2 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2)))))")
+ by (sos "(((A<0 * (A<0 * R<1)) + (((A<=2 * (A<=3 * (A<0 * R<1))) * (R<2 * [1]^2)) + ((A<=0 * (A<=1 * R<1)) * (R<1 * [1]^2)))))")
lemma "abs(x - z) <= e & abs(y - z) <= e & 0 <= u & 0 <= v & (u + v = 1) --> abs((u * x + v * y) - z) <= (e::real)"
- by (sos_cert "((((A<0 * R<1) + (((A<=3 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=1 * (A<=5 * R<1)) * (R<1 * [1]^2))))) & ((((A<0 * A<1) * R<1) + (((A<=3 * (A<=5 * (A<0 * R<1))) * (R<1 * [1]^2)) + ((A<=1 * (A<=4 * (A<0 * R<1))) * (R<1 * [1]^2))))))")
+ by (sos "((((A<0 * R<1) + (((A<=3 * (A<=6 * R<1)) * (R<1 * [1]^2)) + ((A<=1 * (A<=5 * R<1)) * (R<1 * [1]^2))))) & ((((A<0 * A<1) * R<1) + (((A<=3 * (A<=5 * (A<0 * R<1))) * (R<1 * [1]^2)) + ((A<=1 * (A<=4 * (A<0 * R<1))) * (R<1 * [1]^2))))))")
(* lemma "((x::real) - y - 2 * x^4 = 0) & 0 <= x & x <= 2 & 0 <= y & y <= 3 --> y^2 - 7 * y - 12 * x + 17 >= 0" by sos *) (* Too hard?*)
lemma "(0::real) <= x --> (1 + x + x^2)/(1 + x^2) <= 1 + x"
- by (sos_cert "(((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2))))))")
+ by (sos "(((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + ((A<0 * R<1) * (R<1 * [1]^2))))))")
lemma "(0::real) <= x --> 1 - x <= 1 / (1 + x + x^2)"
- by (sos_cert "(((R<1 + (([~4/3] * A=0) + ((R<1 * ((R<1/3 * [3/2*x + 1]^2) + (R<7/12 * [x]^2))) + ((A<=0 * R<1) * (R<1/3 * [1]^2)))))) & (((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))))")
+ by (sos "(((R<1 + (([~4/3] * A=0) + ((R<1 * ((R<1/3 * [3/2*x + 1]^2) + (R<7/12 * [x]^2))) + ((A<=0 * R<1) * (R<1/3 * [1]^2)))))) & (((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2)))) & ((R<1 + ((R<1 * (R<1 * [x]^2)) + (((A<0 * R<1) * (R<1 * [1]^2)) + ((A<=0 * R<1) * (R<1 * [1]^2))))))))")
lemma "(x::real) <= 1 / 2 --> - x - 2 * x^2 <= - x / (1 - x)"
- by (sos_cert "((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2))))")
+ by (sos "((((A<0 * A<1) * R<1) + ((A<=0 * (A<0 * R<1)) * (R<1 * [x]^2))))")
lemma "4*r^2 = p^2 - 4*q & r >= (0::real) & x^2 + p*x + q = 0 --> 2*(x::real) = - p + 2*r | 2*x = -p - 2*r"
- by (sos_cert "((((((A<0 * A<1) * R<1) + ([~4] * A=0))) & ((((A<0 * A<1) * R<1) + ([4] * A=0)))) & (((((A<0 * A<1) * R<1) + ([4] * A=0))) & ((((A<0 * A<1) * R<1) + ([~4] * A=0)))))")
+ by (sos "((((((A<0 * A<1) * R<1) + ([~4] * A=0))) & ((((A<0 * A<1) * R<1) + ([4] * A=0)))) & (((((A<0 * A<1) * R<1) + ([4] * A=0))) & ((((A<0 * A<1) * R<1) + ([~4] * A=0)))))")
end
--- a/src/HOL/ex/SOS_Remote.thy Wed Oct 08 10:15:04 2014 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,37 +0,0 @@
-(* Title: HOL/ex/SOS_Remote.thy
- Author: Amine Chaieb, University of Cambridge
- Author: Philipp Meyer, TU Muenchen
-
-Examples for Sum_of_Squares: remote CSDP server.
-*)
-
-theory SOS_Remote
-imports "~~/src/HOL/Library/Sum_of_Squares"
-begin
-
-lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x \<Longrightarrow> a < 0"
- by (sos remote_csdp)
-
-lemma "a1 >= 0 & a2 >= 0 \<and> (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) \<and> (a1 * b1 + a2 * b2 = 0) --> a1 * a2 - b1 * b2 >= (0::real)"
- by (sos remote_csdp)
-
-lemma "(3::real) * x + 7 * a < 4 & 3 < 2 * x --> a < 0"
- by (sos remote_csdp)
-
-lemma "(0::real) <= x & x <= 1 & 0 <= y & y <= 1 --> x^2 + y^2 < 1 |(x - 1)^2 + y^2 < 1 | x^2 + (y - 1)^2 < 1 | (x - 1)^2 + (y - 1)^2 < 1"
- by (sos remote_csdp)
-
-lemma "(0::real) <= x & 0 <= y & 0 <= z & x + y + z <= 3 --> x * y + x * z + y * z >= 3 * x * y * z"
- by (sos remote_csdp)
-
-lemma "((x::real)^2 + y^2 + z^2 = 1) --> (x + y + z)^2 <= 3"
- by (sos remote_csdp)
-
-lemma "(w^2 + x^2 + y^2 + z^2 = 1) --> (w + x + y + z)^2 <= (4::real)"
- by (sos remote_csdp)
-
-lemma "(x::real) >= 1 & y >= 1 --> x * y >= x + y - 1"
- by (sos remote_csdp)
-
-end
-