(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Cell[CellGroupData[{ Cell["My Own Motivation", "Subsection"], Cell[TextData[{ "I am a \"working mathematician\":\nProof: AMS ", StyleBox["Subject Classification ", FontSize->14], StyleBox["13P10 \"Gr\[ODoubleDot]bner Bases\".\n", FontSize->14, CharacterEncoding->"WindowsANSI"] }], "Text"], Cell["\<\ I want to get support from formal reasoning system when I am \"working\".\ \>", "Text"], Cell["\<\ \"Working\" in mathematics is not \"isolated theorem proving\" but \"theory \ exploration\".\ \>", "Text"], Cell["Thus, I started the Theorema Project in 1996.", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], 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CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "emphasizes ", StyleBox["\"special\" provers (reasoners)", FontColor->RGBColor[1, 0, 1]], ": use nontrivial mathematics as \"black box\" for proving in special \ theories" }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "uses ", StyleBox["Mathematica ", FontColor->RGBColor[1, 0, 1]], "(as meta-programming language, as front-end, for accessing mathematical \ algorithms)" }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "\nAcknowledgement to ", StyleBox["Theorema", FontSlant->"Italic"], " Group members:\nT. Jebelean, T. Kutsia, (K. Nakagawa), F. Piroi, M. \ Rosenkranz, W. Windsteiger.\nPhD students: A. Craciun, L. Kovacs, N. Popov, \ C. Rosenkranz." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), 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First and Intermediate Principles in (Formal) Proving: A First Distinction\ \>", "Subsection"], Cell["\<\ Let's fix some logic, e.g. some inference rule system for (first order) \ predicate logic.\ \>", "Text"], Cell["Example of a proposition (to be proved):", "Text"], Cell[BoxData[GridBox[{ { RowBox[{\(\[ForAll] \+\(f, g : \[DoubleStruckCapitalN] \[Rule] \[DoubleStruckCapitalR]\)\ \), RowBox[{\(\[ForAll] \+\(a, b \[Element] \[DoubleStruckCapitalR]\)\), RowBox[{"(", RowBox[{GridBox[{ {\(limit[f, a]\)}, {\(limit[g, b]\)} }, ColumnAlignments->{Left}], "\[Implies]", \(limit[f + g, a + b]\)}], ")"}]}]}], \(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\((prop)\)\)} }, ColumnAlignments->{Left}]], "Input"], Cell[TextData[{ "\"", StyleBox["Proving ", FontColor->RGBColor[1, 0, 1]], StyleBox["from", FontSize->14, FontColor->RGBColor[1, 0, 1]], StyleBox[" first priciples", FontColor->RGBColor[1, 0, 1]], "\": Prove (prop) by the inference rules of predicate logic " }], "Text"], Cell["from the axioms of set theory", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell["and the definitions of the ingredient notions (quite many!)", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "\"", StyleBox["Proving ", FontColor->RGBColor[1, 0, 1]], StyleBox["from", FontSize->14, FontColor->RGBColor[1, 0, 1]], StyleBox[" intermediate priciples", FontColor->RGBColor[1, 0, 1]], "\": Prove (prop) by the inference rules of predicate logic " }], "Text"], Cell["(from the axioms of set theory)", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell["(the definitions of the ingredient notions)", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell["\<\ and intermediate knowledge on intermediate notions (already proved or \ assumed), e.g. knowledge on the operations on real numbers and the operations \ on real sequences.\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ 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(", StyleBox["by", FontSize->14, FontColor->RGBColor[0, 0, 1]], " \"intermediate principles\")." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell["Please note the parsing!", "SmallText"], Cell["\<\ There are all possible variants conceivable in between the two extremes. \ \>", "Text"], Cell[TextData[{ "In my personal view, the ", StyleBox["proved", FontColor->RGBColor[1, 0, 0]], StyleBox[" proof generators", FontColor->RGBColor[1, 0, 1]], " approach is more promising for making our systems attractive for the \ \"working mathematician\"." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { 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For the ", StyleBox["fixed inference rule system", FontColor->RGBColor[1, 0, 1]], ", write a ", StyleBox["proof checker", FontColor->RGBColor[1, 0, 1]], " that checks whether each step of a given proof is correct. " }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "B. Individual ", StyleBox["proofs", FontColor->RGBColor[1, 0, 1]], " for (interesting) propositions (higher up in the mathematical knowledge \ hierarchy) can now be proposed by the user and checked." }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "C. Since this is practically unattractive (unfeasible), various ", StyleBox["proof generators", FontColor->RGBColor[1, 0, 1]], " (of low and high sophistication) can be written that produce \"proofs\". \ However, these proofs are only considered as \"", StyleBox["proof proposals", FontColor->RGBColor[1, 0, 1]], "\". They are only accepted if ", StyleBox["checked by the proof checker", FontColor->RGBColor[1, 0, 1]], "." }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Ad A: The correctness of the checker can be seen \"by inspection\" because \ it is \"small\" (de Brujn's principle). (Alternatively, we could just say: \ The implementation ", StyleBox["is ", FontSlant->"Italic"], "the inference system.)" }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", "LastSlide"]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar"], Cell[CellGroupData[{ Cell["The Proved (Proof Generators) Approach", "Subsection"], Cell[TextData[{ "A. The basis is also a ", StyleBox["proof checker", FontColor->RGBColor[1, 0, 1]], " (which is nothing else than saying that the basis is an - implemented - \ inference system ", StyleBox["P", FontColor->RGBColor[0, 0, 1]], ")." }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "B. Of course, one could propose", StyleBox[" individual proofs", FontColor->RGBColor[1, 0, 1]], " to the checker. " }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "C. However, typically, for certain fixed \"theories\" (knowledge bases), \ ", StyleBox["special proof generators", FontColor->RGBColor[1, 0, 1]], " are written that use \"black box inference rules\" that are only valid in \ the specific theories." }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "D. The correctness of a ", StyleBox["proof generator must be proved", FontColor->RGBColor[1, 0, 1]], "." }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "E. The ", StyleBox["proofs", FontColor->RGBColor[1, 0, 1]], " produced by proved proof generators ", StyleBox["need not be checked", FontColor->RGBColor[1, 0, 1]], " any more by the initial proof checker!" }], "Text", CellDingbat->None, CellMargins->{{67.3125, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", "LastSlide"]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar"], Cell[CellGroupData[{ Cell["Remarks about the Proved Proof Generators Approach", "Subsection"], Cell[TextData[{ "Ad D: ", StyleBox["Correctness of special prover Q for theory T:", FontColor->RGBColor[1, 0, 1]], " One has to prove that, for all knowledge bases K and formulae F," }], "Text"], Cell[BoxData[ RowBox[{\(if\ \ \ \ K\), \( \[RightTee] \_Q\), RowBox[{\(F\ \ \ \ then\ \ \ \ \ K \[Union] T\), " ", SubscriptBox["\[RightTee]", StyleBox["P", FontColor->RGBColor[0, 0, 1]]], " ", \(\(F\)\(.\)\)}]}]], "Input"], Cell[TextData[{ "The (formal) proof of the correctness of Q must be done by using a ", StyleBox["prover whose correctness was already shown", FontColor->RGBColor[1, 0, 1]], " (i.e. at the beginning by using ", StyleBox["P", FontColor->RGBColor[0, 0, 1]], " alone)." }], "Text"], Cell[TextData[{ "Note that, typically, ", StyleBox["special provers are not", FontColor->RGBColor[1, 0, 1]], " just \"black box versions of ", StyleBox["repeated applications", FontColor->RGBColor[1, 0, 1]], " of the inference rules in ", StyleBox["P", FontColor->RGBColor[0, 0, 1]], "\". (Example: AC simplification by variable count.)" }], "Text"], Cell[TextData[{ "Proving the correctness of provers in a given logic and than applying them \ needs \"", StyleBox["reflection", FontColor->RGBColor[1, 0, 1]], "\"." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], 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have been doing", FontColor->RGBColor[1, 0, 1]], " all the time (although sometimes with low formal quality and, mostly, \ hiddenly)." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ In fact, as soon as one's attention is open for the proved proof generator \ approach, one detects its application in nearly every section of mathematical \ papers and textbooks.\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "It seems that the past success of building up mathematical theories \ without algorithm support mainly relies on the power of ", StyleBox["extending the proving systems while proving", FontColor->RGBColor[1, 0, 1]], "." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "In order to ", StyleBox["make the approach practically attractive", FontColor->RGBColor[1, 0, 1]], ":" }], "Text"], Cell["the systems must be \"opened\" in the sense that", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ the users may not only use the provers (proof generators) provided by the \ system \ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ but also must get the possibility to program their own proof generators and \ prove their correctness.\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]], Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ 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\[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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The formula in the scope of the universal quantifier is transformed \ into an equivalent formula that is a conjunction of disjunctions of \ equalities and negated equalities. The universal quantifier can then be \ distributed over the individual parts of the conjunction. 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This \ question can be answered by checking whether or not the (reduced) Groebner \ basis of\ \>", "ProofText", CellMargins->{{62.3125, 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->{{62.3125, 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->{{62.3125, 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->{{62.3125, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["The Groebner basis:", "ProofText", CellMargins->{{62.3125, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[Cell[BoxData[ \({1}\)]]], "ProofText", CellMargins->{{62.3125, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell[TextData[{ "Hence, ", StyleBox["(", "Label"], ButtonBox["Formula 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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->{{62.3125, 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->{{62.3125, Inherited}, {Inherited, Inherited}}, CellBracketOptions->{"Color"->RGBColor[0.705882, 0.054902, 0.611765]}], Cell["\<\ is exactly {1}. 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\"form intermediate principles\" and the \"by \ intermediate principles\" approach):\ \>", "Text"], Cell["from coefficient domains and power product domains", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["to the polynomial ring (functor) with reduction relation,", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ definition of Groebner bases (with or without reference to ideal theory - in \ which case set theory is needed),\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ proof of the main theorem of algorithmic Groebner bases theory (the theorem \ on S-polynomials),\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["algorithmic construction of Groebner bases,", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["computation of Groebner bases,", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ application of Groebner bases (for this, the various application areas must \ be formalized),\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ application as a special prover for proving theorems in geometry.\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{44.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ I proposed this benchmark 1996 at the 1st Calculemus Meeting in Rome.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ 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generator using Groebner bases as an \ \"intermediate inference rule\". \n- Polynomial rings and \"Groebner rings\" \ as functors, computation in logic.\n- Implementation of inductive proofs \ generator that comes close to the inductive proofs needed in Groebner bases \ theory. \n- Automated invention of main theorem of Groebner bases theory \ (modulo an inductive proof generator).\n- Groebner bases algorithm cannot yet \ be \"lifted\" from the status of being knowledge to the status of being an \ inference rule. (\"Reflection\" necessary.) " }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{63.9375, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "By ", StyleBox["L. Thery in Coq and by J.L. Ruiz-Reina et al. in ACL2", FontColor->RGBColor[1, 0, 1]], ": \n- Checked proof of main theorem of algorithmicGroebner bases theory \ (BB theorem on S-polys). \n- However, proofs need thousands of lines of user \ provided proof code. \n- Groebner bases algorithm cannot yet be \"lifted\" \ from the status of being knowledge to the status of being an inference rule." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{63.9375, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "In ", StyleBox["Mizar (by C. Schwarzweller and P. Rudnicki) and Nuprl (by P.B. \ Jackson)", FontColor->RGBColor[1, 0, 1]], ": Preparation towards proof checked algorithmic Groebner bases theory." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{63.9375, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["H. Persson ", FontColor->RGBColor[1, 0, 1]], "tried a Groebner bases algorithm synthesis in intuitionistic type theory \ but it is based on a fundamental misunderstanding of GB theory." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{63.9375, Inherited}, {Inherited, Inherited}}], Cell["But still a long (?) way to go.", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", "LastSlide"]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar"], Cell[CellGroupData[{ Cell["Self-Elimination", "Subsection"], Cell[TextData[{ "By recent work on algorithmic algorithm synthesis, I am now able to \ automatically ", StyleBox["invent", FontColor->RGBColor[1, 0, 1]], " my own Groebner bases algorithm, in particular the central ", StyleBox["notion", FontColor->RGBColor[1, 0, 1]], " of S-polynomial." }], "Text"], Cell["\<\ This new synthesis method is based on the idea of \"analysis of failing \ correctness proofs\".\ \>", "Text"], Cell["Details 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.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ 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knowledge on the auxiliary notion:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["is\[Dash]Gr\[ODoubleDot]bner\[Dash]basis", FontColor->RGBColor[0, 0, 1]], "[", "G", "]"}], "\[DoubleLeftRightArrow]", RowBox[{ RowBox[{"is\[Dash]confluent", "[", StyleBox[\( \[Rule] \_G\), FontColor->RGBColor[1, 0, 0]], "]"}], "."}]}]], "Input"], Cell[" etc.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], 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\)\), CounterBox["SlideShowNavigationBar", "LastSlide"]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar"], Cell[CellGroupData[{ Cell["Summary", "Subsection"], Cell[CellGroupData[{ Cell[" ", "Subsubsubsection"], Cell[TextData[{ "I think that our reasoning systems must get to a significantly", StyleBox[" higher level of automation", FontColor->RGBColor[1, 0, 1]], " in order to become attractive for \"working mathematicians\"." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "The resulting proofs must be presented in the form of ", StyleBox["proofs in ordinary math texts", FontColor->RGBColor[1, 0, 1]], "." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Coherent ", StyleBox["theory exploration", FontColor->RGBColor[1, 0, 1]], " rather than isolated theorem proving must be in the foreground." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "\"From\" and \"by\" \"", StyleBox["Intermediate principles", FontColor->RGBColor[1, 0, 1]], "\" play an important role in the structuring and automation process." }], "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["", "Subsubsubsection"], Cell[TextData[StyleBox["Bourbakism of 21st century:", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["includes algorithmic mathematics # computer-free math", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}], Cell["\<\ allows the quick build-up of changing \"views\" of mathematical areas # \ one fixed view\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{37.3125, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", "LastSlide"]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar"], Cell[CellGroupData[{ Cell["References", "Subsection"], 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. 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