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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 781410, 19400]*) (*NotebookOutlinePosition[ 831770, 20991]*) (* CellTagsIndexPosition[ 828588, 20906]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Computer-Supported Mathematical Theory Exploration\ \>", "Subtitle", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["Bruno Buchberger\nRISC (Research Institute for \ Symbolic Computation)\nJohannes Kepler University\nand\nRICAM (Radon \ Institute for Computational and Applied Mathematics)\nAustrian Academy of \ Science \nLinz, Aust", FontSlant->"Plain"]], "Subsubtitle", TextAlignment->Center, TextJustification->0, CellTags->"Built\[Dash]in (PauleSchorn)"], Cell[TextData[StyleBox["Invited Talk at\nMathematica Gulf Conference, January \ 2004\nSultan Qaboos University\nDepartment of Mathematics and Statistics", FontSlant->"Plain"]], "Subsubtitle", TextAlignment->Center, TextJustification->0, CellTags->"Built\[Dash]in (PauleSchorn)"], Cell[TextData[{ StyleBox["Copyright", FontWeight->"Bold"], " Bruno Buchberger 2004." }], "Text"], Cell[TextData[{ StyleBox["Copyright Note", FontWeight->"Bold"], ": This file can be printed, stored and distributed under the following \ conditions:\n\n- the file is kept unchanged\n- this copyright note is \ included\n- a message is sent to Bruno.Buchberger@JKU.at.\n\nIf you use the \ material, please, cite this talk appropriately." }], "Text"], Cell[CellGroupData[{ Cell["A Paradigm Shift in Mathematical Software Systems", "Section"], Cell[CellGroupData[{ Cell["Goal of This Talk", "Subsection"], Cell["\<\ This workshop: demonstrating the power of current math software systems (like \ Mathematica).\ \>", "Text"], Cell["\<\ My talk yesterday: some of the algorithmics (Gr\[ODoubleDot]bner bases) \ behind current math systems.\ \>", "Text"], Cell["My talk today: possible future ingredients of math systems.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Current Math Software Systems = Solving and Simplifying", "Subsection"], Cell["Solving:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{x\ y\^2 - 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1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) - 75\/2\ \((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\ \[Sqrt]\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) - 55\/2\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) \ + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) + 75\/2\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\ \ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) \ + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) - 25\/2\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\ \/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \ \@21342)\)\^\(1/3\))\))\))\)\^\(3/2\))\), x \[Rule] \(-\(4\/5\)\) - 1\/2\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\) + 1\/2\ \[Sqrt]\((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \ \@21342)\)\^\(1/3\))\))\))\)}, {y \[Rule] 1\/51\ \((\(-150\) - 251\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\) - 55\/2\ \((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\) + 25\/2\ \((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\^\(3/2\) - 251\ \[Sqrt]\((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) - 55\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\ \ \[Sqrt]\((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) + 75\/2\ \((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\ \[Sqrt]\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) - 55\/2\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) \ + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) + 75\/2\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\)\ \ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) \ + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\))\))\) + 25\/2\ \((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\ \) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\ \/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \ \@21342)\)\^\(1/3\))\))\))\)\^\(3/2\))\), x \[Rule] \(-\(4\/5\)\) - 1\/2\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\))\) - 1\/2\ \[Sqrt]\((674\/75 - 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) - 1\/30\ \((165727 + 408\ \@21342)\)\^\(1/3\) + 1934/\((125\ \[Sqrt]\((337\/75 + 1\/30\ \((165727 - 408\ \@21342)\)\^\(1/3\) + 1\/30\ \((165727 + 408\ \ \@21342)\)\^\(1/3\))\))\))\)}}\)], "Output"] }, Open ]], Cell["How does this work?", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = {x\ y\^2 - 3 x\ y + 2, \(x\^2\) y - 5\ x\^2 - 2 x\ y\ + 3}\)], "Input"], Cell[BoxData[ \({2 - 3\ x\ y + x\ y\^2, 3 - 5\ x\^2 - 2\ x\ y + x\^2\ y}\)], "Output"] }, Open ]], Cell["Simplifying:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(GroebnerBasis[F]\)], "Input"], Cell[BoxData[ \({\(-20\) + 4\ y + 15\ y\^2 - 14\ y\^3 + 3\ y\^4, 4 + 20\ x - 10\ y + y\^2 + 3\ y\^3}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Why Does This Work? 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Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[{ StyleBox["(", "Label"], ButtonBox["Formula (Test): B1.1.not-exists", ButtonFunction:>( Theorema`Provers`Common`BasicProofCells`Private`ExtractTaggedCell[ "Formula (Test): B1.1.not-exists"]&), ButtonEvaluator->Automatic, ButtonStyle->"Hyperlink"], StyleBox[")", "Label"], " is transformed into the equivalent formula" }], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[{ Cell["", "Indent"], StyleBox["(Formula (Test): B1.1.not-exists-slack)", "Label", LineBreakWithin->False], Cell["", "Blank"], Cell[BoxData[ \(\[NotExists] \+\(x, y, \[Xi]\)\((\((\((Poly[1] = 0)\) \[And] \((Poly[2] = 0)\))\) \[And] \((\(-1\) + \[Xi]\ Poly[3] = 0)\))\)\)], "Conclusion", CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[".", "ProofText"] }], "Conclusion", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}, CellTags->"Formula (Test): B1.1.not-exists-slack"], Cell["\<\ Hence, we see that the proof problem is transformed into the question on \ whether or not a system of polynomial equations has a solution or not. This \ question can be answered by checking whether or not the (reduced) Groebner \ basis of\ \>", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({Poly[1], Poly[2], \(-1\) + \[Xi]\ Poly[3]}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["\<\ is exactly {1}. Hence, we compute the Groebner basis for the following polynomial list:\ \>", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({\(-1\) + x\^2\ \[Xi] + \((\(-1\))\) x y \[Xi] + x\^2\ y\ \[Xi] + \((\(-2\))\) \(y\^2\) \[Xi] + \((\(-2\))\) x \( y\^2\) \[Xi], \(-3\)\ x + x\^2\ y, x + y + x\ y}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["The Groebner basis:", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({1}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[{ "Hence, ", StyleBox["(", "Label"], ButtonBox["Formula (Test): 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This \ question can be answered by checking whether or not the (reduced) Groebner \ basis of\ \>", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({Poly[1], Poly[2], \(-1\) + \[Xi]1\ Poly[3], \(-1\) + \[Xi]2\ Poly[ 4]}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["\<\ is exactly {1}. Hence, we compute the Groebner basis for the following polynomial list:\ \>", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({\(-1\) + 3\ x\ \[Xi]1 + x\^2\ y\ \[Xi]1, \(-1\) + \((\(-2\))\) \(x\^2\) \[Xi]2 + \((\(-7\))\) x y \[Xi]2 + x\^2\ y\ \[Xi]2 + x\^3\ y\ \[Xi]2 + \((\(-2\))\) \(y\^2\) \[Xi]2 + \((\(-2\))\) x \( y\^2\) \[Xi]2 + 2\ x\^2\ y\^2\ \[Xi]2, \(-3\)\ x + x\^2\ y, x + y + x\ y}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["The Groebner basis:", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({1}\)]]], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[{ "Hence, ", StyleBox["(", "Label"], ButtonBox["Formula (Test): B1.2", ButtonFunction:>( Theorema`Provers`Common`BasicProofCells`Private`ExtractTaggedCell[ "Formula (Test): B1.2"]&), ButtonEvaluator->Automatic, ButtonStyle->"Hyperlink"], StyleBox[")", "Label"], " is ", "proved." }], "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["\<\ Since all of the individual subtheorems are proved, the original formula is \ proved.\ \>", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["\[EmptySquare]", "ProofText", CellMargins->{{49.5, Inherited}, {Inherited, 0}}, CellBracketOptions->{"Color"->RGBColor[0.643137, 0.756863, 1]}, TextAlignment->Right], Cell["\<\ Thousands of known (and unknown) geometric theorems and conjectures can be \ formulated as statements of this form and, hence, can be proved / disproved \ by the Gr\[ODoubleDot]bner bases method.\ \>", "Text"], Cell["\<\ The two proofs below are examples of proofs generated completely \ automatically by this method.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Example Pappus' Theorem", FontSize->18]], "Subsection", CellDingbat->None], Cell["Textbook formulation:", "Text", CellDingbat->"\[FilledSmallCircle]"], Cell[TextData[{ "Let A,\[VeryThinSpace]B, \[VeryThinSpace]C and A1,\[VeryThinSpace]B1, \ \[VeryThinSpace]C1 be on two lines and ", StyleBox["P\[NonBreakingSpace]=\[NonBreakingSpace]A\[VeryThinSpace]B1\ \[NonBreakingSpace]\[Intersection]\[NonBreakingSpace]A1\[VeryThinSpace]B", "DisplayFormula"], ", Q\[NonBreakingSpace]=\[NonBreakingSpace]A\[VeryThinSpace]C1\ \[NonBreakingSpace]\[Intersection]\[NonBreakingSpace]A1\[VeryThinSpace]C, ", StyleBox["S\[NonBreakingSpace]=\[NonBreakingSpace]B\[VeryThinSpace]C1\ \[NonBreakingSpace]\[Intersection]\[NonBreakingSpace]B1\[VeryThinSpace]C", "DisplayFormula"], ". 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Buchberger 2001) for Automated \ Algorithm Invention\ \>", "Subsection"], Cell[TextData[{ "The", StyleBox[" two main ideas:", FontColor->RGBColor[0, 0, 1]] }], "Text", CellMargins->{{12, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Use ", StyleBox["algorithm schemes ", FontColor->RGBColor[0, 0, 1]], "for guessing possible algorithms." }], "Text", CellMargins->{{59, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "For each algorithm scheme A, (automatically) ", StyleBox["attempt a proof ", FontColor->RGBColor[0, 0, 1]], "that A satisfies P. From the failing proof, derive (automatically) \ specifications for the subalgorithms." }], "Text", CellMargins->{{59, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Find appropriate subalgorithms in an algorithm library or repeat failing \ correctness proof method for inventing the ", StyleBox["subalgorithms", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellMargins->{{12, Inherited}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell["Inventing the Algorithm by Lazy Thinking: First Round", "Subsection"], Cell["\<\ The method first selects an algorithm scheme from a library of algorithm \ schemes, e.g. the algorithm scheme \"divide and conquer\":\ \>", "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(\[ForAll] \+\(is\[Dash]tuple[X]\)\), FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["(", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox[\(sorted[X]\), FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["{", FontColor->RGBColor[0, 0, 1]], StyleBox[GridBox[{ {\(special[X]\), "\[DoubleLeftArrow]", \(is\[Dash]trivial\[Dash]tuple[ X]\)}, {\(merged[sorted[left\[Dash]split[X]], sorted[right\[Dash]split[X]]]\), "\[DoubleLeftArrow]", "otherwise"} }, ColumnAlignments->{Left}], FontColor->RGBColor[0, 0, 1]], StyleBox["}", ShowContents->False, FontColor->RGBColor[0, 0, 1]]}]}], StyleBox[")", FontColor->RGBColor[0, 0, 1]]}]}]], "Input"], Cell[TextData[{ "The following proof attempt is automatically generated in ", StyleBox["Theorema", FontSlant->"Italic"], ":" }], "Text"], Cell[TextData[StyleBox["Proof Attempt Begin ", FontSlant->"Italic"]], "Text"], Cell["\<\ For proving the correctness theorem, we use well-founded induction w.r.t. \ \[Succeeds] on X:\ \>", "Text", Background->GrayLevel[0.900008]], Cell["We assume", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(is\[Dash]tuple[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]\)], "Input",\ Background->GrayLevel[0.900008]], Cell["and the induction hypothesis", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\[ForAll] \+\(is\[Dash]tuple[Y]\)\)\+\(\[LeftAngleBracket]xo\&_\ \[RightAngleBracket] \[Succeeds] Y\)is\[Dash]sorted\[Dash]version[Y, sorted[Y]]\)], "Input", Background->GrayLevel[0.900008]], Cell["and we show", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(is\[Dash]sorted\[Dash]version[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket], sorted[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\)\(.\)\)\)], \ "Input", Background->GrayLevel[0.900008]], Cell["\<\ We use the algorithm scheme for 'sorted' and distinguish two cases:\ \>", "Text", Background->GrayLevel[0.900008]], Cell["CASE", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(is\[Dash]trivial\[Dash]tuple[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]]\)\(:\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell["In this case, we have to show", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(is\[Dash]sorted\[Dash]version[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket], special[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\)], "Input", Background->GrayLevel[0.900008]], Cell["\<\ i.e., by the definition of 'is\[Dash]sorted\[Dash]version', we have to show\ \>", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((G1)\)\ \ \ \ \ \ is\[Dash]tuple[ special[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\)\(,\)\)\)], \ "Input", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((G2)\)\ \ \ \ \ \ special[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]] \[TildeTilde] \[LeftAngleBracket]xo\&_\ \[RightAngleBracket]\)\(,\)\)\)], "Input", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\((G2)\)\ \ \ \ \ \ \(\(is\[Dash]sorted[ special[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\)\(.\)\)\)], \ "Input", Background->GrayLevel[0.900008]], Cell["(G1) is true because of the type requirement for 'special'. ", "Text", Background->GrayLevel[0.900008]], Cell["For (G2), by the fact that", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\[ForAll] \+\(is\[Dash]trivial\[Dash]tuple[X], \ \ is\[Dash]tuple[Y]\)\((X \[TildeTilde] Y \[DoubleLeftRightArrow] \((X = Y)\))\)\)\(,\)\)\)], "Input", Background->GrayLevel[0.900008]], Cell["it suffices to prove that", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(special[\[LeftAngleBracket]xo\&_\[RightAngleBracket]] = \(\(\ \[LeftAngleBracket]xo\&_\[RightAngleBracket]\)\(.\)\)\)], "Input", FontColor->RGBColor[1, 0, 0], Background->GrayLevel[0.900008]], Cell["Here we are stuck.", "Text", Background->GrayLevel[0.900008]], Cell["Proof Attempt End", "Text", FontSlant->"Italic", Background->GrayLevel[0.900008]], Cell["Requirement Generation", "Subsubsubsection"], Cell[TextData[{ "Now we analyze the failing proof situation and conjecture the requirement \ (can be done completely automatically in ", StyleBox["Theorema", 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\(\(is\[Dash]sorted[ merged[sorted[ left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]], sorted[right\[Dash]split[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]]]]]]\)\(.\)\)\)], "Input", FontColor->RGBColor[1, 0, 0], Background->GrayLevel[0.900008]], Cell["\<\ >From the case assumption, by the type requirements on 'left\[Dash]split' and \ 'right\[Dash]split', the property that 'left\[Dash]split' and \ 'right\[Dash]split' produce shorter tuples, and the induction hypothesis we \ obtain\ \>", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(is\[Dash]sorted\[Dash]version[ left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]], sorted[left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]]\ \)\(,\)\)\)], "Input", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(is\[Dash]sorted\[Dash]version[ right\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]], sorted[right\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]\ ]]\)\(.\)\)\)], "Input", Background->GrayLevel[0.900008]], Cell["\<\ >From this, by the definition of 'is\[Dash]sorted\[Dash]version', we obtain\ \>", "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((AL1)\)\ \ \ \ \ is\[Dash]tuple[ sorted[left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\ ]\)\(,\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((AL2)\)\ \ \ \ \ left\[Dash]split[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]] \[TildeTilde] sorted[left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\)\ \(,\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((AL3)\)\ \ \ \ \ is\[Dash]sorted[ sorted[left\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\ ]\)\(,\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((AR1)\)\ \ \ \ \ is\[Dash]tuple[ sorted[right\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]\ ]]\)\(,\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(\(\((AR2)\)\ \ \ \ \ right\[Dash]split[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]] \[TildeTilde] sorted[right\[Dash]split[\[LeftAngleBracket]xo\&_\[RightAngleBracket]]]\ \)\(,\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell[BoxData[ \(\((AR3)\)\ \ \ \ \ \(\(is\[Dash]sorted[ sorted[right\[Dash]split[\[LeftAngleBracket]xo\&_\ \[RightAngleBracket]]]]\)\(.\)\)\)], "Input", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[0.900008]], Cell["\<\ (H1) follows from (AL1) and (AR1) by the type requirement on 'merged'.\ \>", "Text", Background->GrayLevel[0.900008]], Cell["Now we are stuck.", "Text"], Cell[TextData[StyleBox["Proof Attempt End", FontSlant->"Italic"]], "Text"], Cell["Requirement Generation", "Subsubsubsection"], Cell[TextData[{ "It is not so near at hand but, after some thinking, relatively easy to \ conjecture (and our current ", StyleBox["Theorema", FontSlant->"Italic"], " conjecture generating algorithm can do this automatically) that the \ following requirement on the functions 'left-split', 'right-split' und \ 'merged'" }], "Text"], Cell[BoxData[ RowBox[{\(\(\[ForAll] \+\(is\[Dash]tuple[X, Y, Z]\)\)\+\(\[Not] is\[Dash]trivial\[Dash]tuple[X]\)\), RowBox[{"(", RowBox[{ RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(left\[Dash]split[X] \[TildeTilde] Y\)}, {\(right\[Dash]split[X] \[TildeTilde] Z\)}, {\(is\[Dash]sorted[Y]\)}, {\(is\[Dash]sorted[Z]\)} }, ColumnAlignments->{Left}], StyleBox["}", ShowContents->False]}], "\[Implies]", RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(merged[Y, Z] \[TildeTilde] X\)}, {\(is\[Dash]sorted[merged[Y, Z]]\)} }, ColumnAlignments->{Left}], StyleBox["}", ShowContents->False]}]}], ")"}]}]], "Input", FontColor->RGBColor[1, 0, 0]], Cell["\<\ will make it possible to get over the failing proof situation.\ \>", "Text"], Cell["\<\ We add this requirement to the knowledge base and proceed to the next \ invention round.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Inventing the Algorithm by Lazy Thinking: Last Round", "Subsection"], Cell["\<\ We now do exactly the same proof attempt once more (or we just jump to the \ proof situation where the previous proof attempt got stuck.)\ \>", "Text"], Cell["\<\ This time, the inductive proof will succeed using the added requirement on \ 'left-split', 'right-split' und 'merged' for proving (H2) and (H3).\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["What Do We Have Now?", "Subsection"], Cell["\<\ If we now collect the requirements on the functions'special', 'left-split', \ 'right-split', and 'merged', we see that we invented and proved the following \ theorem:\ \>", "Text"], Cell["If", "Text"], Cell[BoxData[ RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ { RowBox[{ RowBox[{ StyleBox[\(\[ForAll] \+\(is\[Dash]tuple[X]\)\), FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["(", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox[\(sorted[X]\), FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["{", FontColor->RGBColor[0, 0, 1]], StyleBox[GridBox[{ {\(special[X]\), "\[DoubleLeftArrow]", \ \(is\[Dash]trivial\[Dash]tuple[X]\)}, {\(merged[sorted[left\[Dash]split[X]], sorted[right\[Dash]split[X]]]\), "\[DoubleLeftArrow]", "otherwise"} }, ColumnAlignments->{Left}], FontColor->RGBColor[0, 0, 1]], StyleBox["}", ShowContents->False, FontColor->RGBColor[0, 0, 1]]}]}], StyleBox[")", FontColor->RGBColor[0, 0, 1]]}]}], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[0, 0, 1]]}]}, { RowBox[{\(\(\[ForAll] \+\(is\[Dash]tuple[ X]\)\)\+\(\[Not] is\[Dash]trivial\[Dash]tuple[X]\)\), RowBox[{"\[And]", RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(left\[Dash]split[X] \[Precedes] 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StyleBox["}", ShowContents->False]}]], "Input"], Cell["then", "Text"], Cell[BoxData[ RowBox[{\(\[ForAll] \+\(is\[Dash]tuple[X]\)\), RowBox[{\(is\[Dash]sorted\[Dash]version[X, \ sorted[X]]\), Cell[ ""]}]}]], "Input"], Cell["\<\ In other words, we found completely automatically that the divide-and-conquer \ algorithm is a correct sorting algorithm, if we take (or invent) any \ subalgorithms 'merged', 'left\[Dash]split', 'right\[Dash]split' that satisfy \ the specifications formulated in the above theorem!\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Recent Breakthrough", "Subsection"], Cell["\<\ Recently, I managed to obtain by the above automatic algorithm synthesis \ method my own Gr\[ODoubleDot]bner bases algorithm, see\ \>", "Text"], Cell["\<\ B. Buchberger. Towards the Automated Synthesis of a Gr\[ODoubleDot]bner Bases Algorithm. RACSAM (Review of the Royal Spanish Academy of Science), to appear, 10 pages.\ \ \>", "Text"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusion", "Section"], Cell[TextData[{ "I believe that, ", StyleBox["going into the direction of systems like ", FontColor->RGBColor[0, 0, 1]], StyleBox["Theorema", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], ", the following will soon be possible:" }], "Text"], Cell[TextData[{ "Computer-support of ", StyleBox["all aspects of doing mathematics", FontColor->RGBColor[0, 0, 1]], " will reach higher and higher levels including inventing, exploring, \ proving." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{56, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "In particular, ", StyleBox["nonalgorithmic and algorithmic mathematics", FontColor->RGBColor[0, 0, 1]], " will be supportable in one common logical and software-technological \ frame." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{56, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Mathematical software systems, in addition to providing algorithm \ libraries, will have to provide ", StyleBox["huge mathematical knowledge libraries", FontColor->RGBColor[0, 0, 1]], ". Building up and using such libraries, essentially, is a task of formal \ logic. " }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{56, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "The ", StyleBox["education of mathematicians ", FontColor->RGBColor[0, 0, 1]], "will have to shift towards including in-depth training on the formal / \ logical aspects of mathematics." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{56, Inherited}, {Inherited, Inherited}}] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[CellGroupData[{ Cell["On Gr\[ODoubleDot]bner Bases", "Subsubsection"], Cell["\<\ [Buchberger 1970] B. Buchberger. Ein algorithmisches Kriterium f\[UDoubleDot]r die L\ \[ODoubleDot]sbarkeit eines algebraischen Gleichungssystems (An Algorithmical \ Criterion for the Solvability of Algebraic Systems of Equations). Aequationes \ mathematicae 4/3, 1970, pp. 374-383. (English translation in: [Buchberger, \ Winkler 1998], pp. 535 -545.) Published version of the PhD Thesis of B. \ Buchberger, University of Innsbruck, Austria, 1965.\ \>", "Text"], Cell["\<\ [Buchberger 1998] B. Buchberger. Introduction to Gr\[ODoubleDot]bner Bases. In: [Buchberger, \ Winkler 1998], pp.3-31.\ \>", "Text"], Cell["\<\ [Buchberger, Winkler, 1998] B. Buchberger, F. Winkler (eds.). Gr\[ODoubleDot]bner Bases and Applications, \ Proceedings of the International Conference \"33 Years of Gr\[ODoubleDot]bner \ Bases\", 1998, RISC, Austria, London Mathematical Society Lecture Note \ Series, Vol. 251, Cambridge University Press, 1998.\ \>", "Text"], Cell["\<\ [Becker, Weispfenning 1993] T. Becker, V. Weispfenning. Gr\[ODoubleDot]bner Bases: A Computational \ Approach to Commutative Algebra, Springer, New York, 1993.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["On Mathematical Knowledge Management", "Subsubsection"], Cell["\<\ B. Buchberger, G. Gonnet, M. Hazewinkel (eds.) Mathematical Knowledge Management. Special Issue of Annals of Mathematics and Artificial Intelligence, Vol. 38, \ No. 1-3, May 2003, Kluwer Academic Publisher, 232 pages.\ \>", "Text"], Cell["\<\ A.Asperti, B. Buchberger,J.H.Davenport (eds.) Mathematical Knowledge Management. Proceedings of the Second International Conference on Mathematical Knowledge \ Management (MKM 2003), Bertinoro, Italy, Feb.16-18, 2003, Lecture Notes in \ Computer Science, Vol.2594, Springer, Berlin-Heidelberg-NewYork, 2003, 223 \ pages.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["On Theorema", "Subsubsection"], Cell["\<\ [Buchberger et al. 2000] B. Buchberger, C. Dupre, T. Jebelean, F. Kriftner, K. Nakagawa, D. Vasaru, W. \ Windsteiger. The Theorema Project: A Progress Report. In: M. Kerber and M. \ Kohlhase (eds.), Symbolic Computation and Automated Reasoning (Proceedings of \ CALCULEMUS 2000, Symposium on the Integration of Symbolic Computation and \ Mechanized Reasoning, August 6-7, 2000, St. Andrews, Scotland), A.K. Peters, \ Natick, Massachusetts, ISBN 1-56881-145-4, pp. 98-113.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["On Theory Exploration and Algorithm Synthesis", "Subsubsection"], Cell[TextData[{ "[Buchberger 2000] \nB. Buchberger. Theory Exploration with ", StyleBox["Theorema", FontSlant->"Italic"], ". \nAnalele Universitatii Din Timisoara, Ser. Matematica-Informatica, Vol. \ XXXVIII, Fasc.2, 2000, (Proceedings of SYNASC 2000, 2nd International \ Workshop on Symbolic and Numeric Algorithms in Scientific Computing, Oct. \ 4-6, 2000, Timisoara, Rumania, T. Jebelean, V. Negru, A. Popovici eds.), ISSN \ 1124-970X, pp. 9-32. " }], "Text"], Cell["\<\ [Buchberger 2003] B. Buchberger. Algorithm Invention and Verification by Lazy Thinking. In: D. Petcu, V. Negru, D. Zaharie, T. Jebelean (eds), Proceedings of SYNASC \ 2003 (Symbolic and Numeric Algorithms for Scientific Computing, Timisoara, \ Romania, October 1\[Dash]4, 2003), Mirton Publishing, ISBN 973\[Dash]661\ \[Dash]104\[Dash]3, pp. 2\[Dash]26.\ \>", "Text"], Cell["\<\ [Buchberger 2004] B. Buchberger. The Four Parallel Threads of Formal Theory Exploration. \ Technical Report of the SFB (Special Research Area) Scientific Computing, \ Johannes Kepler University, Linz, in preparation.\ \>", "Text"], Cell["\<\ [Buchberger, Craciun 2003] B. Buchberger, A. Craciun. Algorithm Synthesis by Lazy Thinking: Examples and \ Implementation in Theorema. in: Fairouz Kamareddine (ed.), Proc. of the \ Mathematical Knowledge Management Workshop, Edinburgh, Nov. 25, 2003, \ Electronic Notes on Theoretical Computer Science, volume dedicated to the \ MKM 03 Symposium, Elsevier, ISBN 044451290X, to appear.\ \>", "Text"], Cell["\<\ [Buchberger 2004] B. Buchberger. 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*) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)