src/HOL/ROOT
author immler
Tue Jul 10 09:38:35 2018 +0200 (11 months ago)
changeset 68607 67bb59e49834
parent 68569 c64319959bab
child 68617 75129a73aca3
permissions -rw-r--r--
make theorem, corollary, and proposition %important for HOL-Analysis manual
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chapter HOL
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session HOL (main) = Pure +
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  description {*
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    Classical Higher-order Logic.
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  *}
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  theories
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    Main (global)
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    Complex_Main (global)
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  document_files
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    "root.bib"
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    "root.tex"
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session "HOL-Proofs" (timing) = Pure +
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  description {*
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    HOL-Main with explicit proof terms.
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  *}
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  options [quick_and_dirty = false, record_proofs = 2, parallel_proofs = 0, parallel_limit = 500]
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  sessions
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    "HOL-Library"
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  theories
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    "HOL-Library.Realizers"
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session "HOL-Library" (main timing) in Library = HOL +
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  description {*
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    Classical Higher-order Logic -- batteries included.
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  *}
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  theories
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    Library
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    (*conflicting type class instantiations and dependent applications*)
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    Finite_Lattice
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    List_Lexorder
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    Prefix_Order
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    Product_Lexorder
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    Product_Order
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    Subseq_Order
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    (*conflicting syntax*)
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    Datatype_Records
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    (*data refinements and dependent applications*)
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    AList_Mapping
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    Code_Binary_Nat
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    Code_Prolog
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    Code_Real_Approx_By_Float
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    Code_Target_Numeral
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    DAList
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    DAList_Multiset
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    RBT_Mapping
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    RBT_Set
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    (*printing modifications*)
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    OptionalSugar
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    (*prototypic tools*)
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    Predicate_Compile_Quickcheck
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    (*legacy tools*)
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    Old_Datatype
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    Old_Recdef
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    Realizers
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    Refute
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  document_files "root.bib" "root.tex"
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session "HOL-Analysis" (main timing) in Analysis = HOL +
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  options [document_tags = "theorem%important,corollary%important,proposition%important,%unimportant",
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    document_variants = "document:manual=-proof,-ML,-unimportant"]
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  sessions
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    "HOL-Library"
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    "HOL-Computational_Algebra"
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  theories
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    Analysis
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  document_files
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    "root.tex"
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session "HOL-Analysis-ex" in "Analysis/ex" = "HOL-Analysis" +
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  theories
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    Approximations
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session "HOL-Computational_Algebra" (main timing) in "Computational_Algebra" = "HOL-Library" +
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  theories
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    Computational_Algebra
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    (*conflicting type class instantiations and dependent applications*)
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    Field_as_Ring
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session "HOL-Hahn_Banach" in Hahn_Banach = "HOL-Library" +
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  description {*
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    Author:     Gertrud Bauer, TU Munich
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    The Hahn-Banach theorem for real vector spaces.
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    This is the proof of the Hahn-Banach theorem for real vectorspaces,
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    following H. Heuser, Funktionalanalysis, p. 228 -232. The Hahn-Banach
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    theorem is one of the fundamental theorems of functional analysis. It is a
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    conclusion of Zorn's lemma.
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    Two different formaulations of the theorem are presented, one for general
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    real vectorspaces and its application to normed vectorspaces.
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    The theorem says, that every continous linearform, defined on arbitrary
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    subspaces (not only one-dimensional subspaces), can be extended to a
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    continous linearform on the whole vectorspace.
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  *}
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  sessions
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    "HOL-Analysis"
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  theories
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    Hahn_Banach
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  document_files "root.bib" "root.tex"
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session "HOL-Induct" in Induct = "HOL-Library" +
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  description {*
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    Examples of (Co)Inductive Definitions.
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    Comb proves the Church-Rosser theorem for combinators (see
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    http://www.cl.cam.ac.uk/ftp/papers/reports/TR396-lcp-generic-automatic-proof-tools.ps.gz).
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    Mutil is the famous Mutilated Chess Board problem (see
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    http://www.cl.cam.ac.uk/ftp/papers/reports/TR394-lcp-mutilated-chess-board.dvi.gz).
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    PropLog proves the completeness of a formalization of propositional logic
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    (see
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    http://www.cl.cam.ac.uk/Research/Reports/TR312-lcp-set-II.ps.gz).
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    Exp demonstrates the use of iterated inductive definitions to reason about
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    mutually recursive relations.
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  *}
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  theories [quick_and_dirty]
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    Common_Patterns
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  theories
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    Nested_Datatype
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    QuoDataType
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    QuoNestedDataType
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    Term
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    SList
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    ABexp
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    Infinitely_Branching_Tree
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    Ordinals
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    Sigma_Algebra
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    Comb
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    PropLog
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    Com
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  document_files "root.tex"
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session "HOL-IMP" (timing) in IMP = "HOL-Library" +
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  options [document_variants = document]
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  theories
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    BExp
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    ASM
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    Finite_Reachable
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    Denotational
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    Compiler2
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    Poly_Types
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    Sec_Typing
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    Sec_TypingT
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    Def_Init_Big
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    Def_Init_Small
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    Fold
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    Live
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    Live_True
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    Hoare_Examples
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    Hoare_Sound_Complete
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    VCG
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    Hoare_Total
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    VCG_Total_EX
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    VCG_Total_EX2
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    Collecting1
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    Collecting_Examples
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    Abs_Int_Tests
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    Abs_Int1_parity
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    Abs_Int1_const
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    Abs_Int3
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    Procs_Dyn_Vars_Dyn
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    Procs_Stat_Vars_Dyn
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    Procs_Stat_Vars_Stat
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    C_like
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    OO
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  document_files "root.bib" "root.tex"
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session "HOL-IMPP" in IMPP = HOL +
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  description {*
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    Author:     David von Oheimb
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    Copyright   1999 TUM
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    IMPP -- An imperative language with procedures.
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    This is an extension of IMP with local variables and mutually recursive
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    procedures. For documentation see "Hoare Logic for Mutual Recursion and
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    Local Variables" (http://isabelle.in.tum.de/Bali/papers/FSTTCS99.html).
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  *}
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  theories EvenOdd
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session "HOL-Data_Structures" (timing) in Data_Structures = HOL +
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  options [document_variants = document]
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  sessions
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    "HOL-Number_Theory"
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  theories [document = false]
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    Less_False
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  theories
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    Sorting
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    Balance
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    Tree_Map
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    AVL_Map
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    RBT_Map
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    Tree23_Map
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    Tree234_Map
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    Brother12_Map
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    AA_Map
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    Set2_Join_RBT
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    Trie
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    Leftist_Heap
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    Binomial_Heap
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  document_files "root.tex" "root.bib"
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session "HOL-Import" in Import = HOL +
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  theories HOL_Light_Maps
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  theories [condition = HOL_LIGHT_BUNDLE] HOL_Light_Import
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session "HOL-Number_Theory" (main timing) in Number_Theory = "HOL-Computational_Algebra" +
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  description {*
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    Fundamental Theorem of Arithmetic, Chinese Remainder Theorem, Fermat/Euler
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    Theorem, Wilson's Theorem, some lemmas for Quadratic Reciprocity.
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  *}
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  sessions
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    "HOL-Algebra"
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  theories
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    Number_Theory
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  document_files
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    "root.tex"
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session "HOL-Hoare" in Hoare = HOL +
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  description {*
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    Verification of imperative programs (verification conditions are generated
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    automatically from pre/post conditions and loop invariants).
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  *}
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  theories Hoare
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  document_files "root.bib" "root.tex"
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session "HOL-Hoare_Parallel" (timing) in Hoare_Parallel = HOL +
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  description {*
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    Verification of shared-variable imperative programs a la Owicki-Gries.
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    (verification conditions are generated automatically).
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  *}
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  theories Hoare_Parallel
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  document_files "root.bib" "root.tex"
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session "HOL-Codegenerator_Test" in Codegenerator_Test = "HOL-Number_Theory" +
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  sessions
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    "HOL-Data_Structures"
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    "HOL-ex"
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  theories
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    Generate
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    Generate_Binary_Nat
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    Generate_Target_Nat
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    Generate_Efficient_Datastructures
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    Code_Lazy_Test
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    Code_Test_PolyML
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    Code_Test_Scala
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  theories [condition = ISABELLE_GHC]
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    Code_Test_GHC
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  theories [condition = ISABELLE_MLTON]
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    Code_Test_MLton
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  theories [condition = ISABELLE_OCAMLC]
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    Code_Test_OCaml
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  theories [condition = ISABELLE_SMLNJ]
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    Code_Test_SMLNJ
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session "HOL-Metis_Examples" (timing) in Metis_Examples = "HOL-Library" +
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  description {*
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Jasmin Blanchette, TU Muenchen
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    Testing Metis and Sledgehammer.
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  *}
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  sessions
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    "HOL-Decision_Procs"
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  theories
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    Abstraction
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    Big_O
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    Binary_Tree
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    Clausification
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    Message
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    Proxies
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    Tarski
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    Trans_Closure
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    Sets
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session "HOL-Nitpick_Examples" in Nitpick_Examples = "HOL-Library" +
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  description {*
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2009
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  *}
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  theories [quick_and_dirty] Nitpick_Examples
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session "HOL-Algebra" (main timing) in Algebra = "HOL-Computational_Algebra" +
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  description {*
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    Author: Clemens Ballarin, started 24 September 1999, and many others
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    The Isabelle Algebraic Library.
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  *}
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  theories
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    (* Orders and Lattices *)
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    Galois_Connection    (* Knaster-Tarski theorem and Galois connections *)
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    (* Groups *)
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    FiniteProduct        (* Product operator for commutative groups *)
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    Sylow                (* Sylow's theorem *)
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    Bij                  (* Automorphism Groups *)
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    Multiplicative_Group
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    Zassenhaus            (* The Zassenhaus lemma *)
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    (* Rings *)
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    Divisibility         (* Rings *)
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    IntRing              (* Ideals and residue classes *)
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    UnivPoly             (* Polynomials *)
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    (* Main theory *)
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    Algebra             
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  document_files "root.bib" "root.tex"
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session "HOL-Auth" (timing) in Auth = "HOL-Library" +
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  description {*
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    A new approach to verifying authentication protocols.
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  *}
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  theories
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    Auth_Shared
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    Auth_Public
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    "Smartcard/Auth_Smartcard"
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    "Guard/Auth_Guard_Shared"
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    "Guard/Auth_Guard_Public"
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  document_files "root.tex"
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session "HOL-UNITY" (timing) in UNITY = "HOL-Auth" +
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  description {*
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1998  University of Cambridge
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    Verifying security protocols using Chandy and Misra's UNITY formalism.
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  *}
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  theories
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    (*Basic meta-theory*)
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    UNITY_Main
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    (*Simple examples: no composition*)
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    "Simple/Deadlock"
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    "Simple/Common"
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    "Simple/Network"
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    "Simple/Token"
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    "Simple/Channel"
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    "Simple/Lift"
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    "Simple/Mutex"
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    "Simple/Reach"
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    "Simple/Reachability"
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    (*Verifying security protocols using UNITY*)
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    "Simple/NSP_Bad"
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    (*Example of composition*)
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    "Comp/Handshake"
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    (*Universal properties examples*)
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    "Comp/Counter"
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    "Comp/Counterc"
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    "Comp/Priority"
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    "Comp/TimerArray"
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    "Comp/Progress"
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    "Comp/Alloc"
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    "Comp/AllocImpl"
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    "Comp/Client"
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    (*obsolete*)
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    ELT
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  document_files "root.tex"
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session "HOL-Unix" in Unix = "HOL-Library" +
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  options [print_mode = "no_brackets,no_type_brackets"]
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  theories Unix
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  document_files "root.bib" "root.tex"
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session "HOL-ZF" in ZF = "HOL-Library" +
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  theories
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    MainZF
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    Games
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  document_files "root.tex"
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session "HOL-Imperative_HOL" (timing) in Imperative_HOL = "HOL-Library" +
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  options [print_mode = "iff,no_brackets"]
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  theories Imperative_HOL_ex
wenzelm@56781
   382
  document_files "root.bib" "root.tex"
wenzelm@48481
   383
wenzelm@65550
   384
session "HOL-Decision_Procs" (timing) in Decision_Procs = "HOL-Algebra" +
wenzelm@51544
   385
  description {*
wenzelm@51544
   386
    Various decision procedures, typically involving reflection.
wenzelm@51544
   387
  *}
wenzelm@65543
   388
  theories
wenzelm@65543
   389
    Decision_Procs
wenzelm@48481
   390
lars@63000
   391
session "HOL-Proofs-ex" in "Proofs/ex" = "HOL-Proofs" +
wenzelm@65573
   392
  sessions
wenzelm@65573
   393
    "HOL-Isar_Examples"
wenzelm@52424
   394
  theories
wenzelm@52424
   395
    Hilbert_Classical
wenzelm@62363
   396
    Proof_Terms
wenzelm@52424
   397
    XML_Data
wenzelm@48481
   398
wenzelm@63888
   399
session "HOL-Proofs-Extraction" (timing) in "Proofs/Extraction" = "HOL-Proofs" +
wenzelm@51421
   400
  description {*
wenzelm@51421
   401
    Examples for program extraction in Higher-Order Logic.
wenzelm@51421
   402
  *}
wenzelm@62354
   403
  options [parallel_proofs = 0, quick_and_dirty = false]
wenzelm@65543
   404
  sessions
wenzelm@65543
   405
    "HOL-Computational_Algebra"
wenzelm@48481
   406
  theories
wenzelm@48481
   407
    Greatest_Common_Divisor
wenzelm@48481
   408
    Warshall
wenzelm@48481
   409
    Higman_Extraction
wenzelm@48481
   410
    Pigeonhole
wenzelm@48481
   411
    Euclid
wenzelm@56781
   412
  document_files "root.bib" "root.tex"
wenzelm@48481
   413
wenzelm@63888
   414
session "HOL-Proofs-Lambda" (timing) in "Proofs/Lambda" = "HOL-Proofs" +
wenzelm@51403
   415
  description {*
wenzelm@51403
   416
    Lambda Calculus in de Bruijn's Notation.
wenzelm@51403
   417
wenzelm@51403
   418
    This session defines lambda-calculus terms with de Bruijn indixes and
wenzelm@51403
   419
    proves confluence of beta, eta and beta+eta.
wenzelm@51403
   420
wenzelm@51403
   421
    The paper "More Church-Rosser Proofs (in Isabelle/HOL)" describes the whole
wenzelm@51403
   422
    theory (see http://www.in.tum.de/~nipkow/pubs/jar2001.html).
wenzelm@51403
   423
  *}
wenzelm@62354
   424
  options [print_mode = "no_brackets",
wenzelm@62242
   425
    parallel_proofs = 0, quick_and_dirty = false]
wenzelm@65543
   426
  sessions
wenzelm@65543
   427
    "HOL-Library"
wenzelm@48481
   428
  theories
wenzelm@48481
   429
    Eta
wenzelm@48481
   430
    StrongNorm
wenzelm@48481
   431
    Standardization
wenzelm@48481
   432
    WeakNorm
wenzelm@56781
   433
  document_files "root.bib" "root.tex"
wenzelm@48481
   434
wenzelm@48738
   435
session "HOL-Prolog" in Prolog = HOL +
wenzelm@48481
   436
  description {*
wenzelm@48481
   437
    Author:   David von Oheimb (based on a lecture on Lambda Prolog by Nadathur)
wenzelm@51403
   438
wenzelm@51403
   439
    A bare-bones implementation of Lambda-Prolog.
wenzelm@51403
   440
wenzelm@51403
   441
    This is a simple exploratory implementation of Lambda-Prolog in HOL,
wenzelm@51403
   442
    including some minimal examples (in Test.thy) and a more typical example of
wenzelm@51403
   443
    a little functional language and its type system.
wenzelm@48481
   444
  *}
wenzelm@48481
   445
  theories Test Type
wenzelm@48481
   446
wenzelm@65550
   447
session "HOL-MicroJava" (timing) in MicroJava = "HOL-Library" +
wenzelm@51403
   448
  description {*
wenzelm@51403
   449
    Formalization of a fragment of Java, together with a corresponding virtual
wenzelm@51403
   450
    machine and a specification of its bytecode verifier and a lightweight
wenzelm@51403
   451
    bytecode verifier, including proofs of type-safety.
wenzelm@51403
   452
  *}
wenzelm@65573
   453
  sessions
wenzelm@65573
   454
    "HOL-Eisbach"
wenzelm@59446
   455
  theories
wenzelm@59446
   456
    MicroJava
wenzelm@56781
   457
  document_files
wenzelm@56781
   458
    "introduction.tex"
wenzelm@56781
   459
    "root.bib"
wenzelm@56781
   460
    "root.tex"
wenzelm@48481
   461
wenzelm@48738
   462
session "HOL-NanoJava" in NanoJava = HOL +
wenzelm@51403
   463
  description {*
wenzelm@51403
   464
    Hoare Logic for a tiny fragment of Java.
wenzelm@51403
   465
  *}
wenzelm@48481
   466
  theories Example
wenzelm@56781
   467
  document_files "root.bib" "root.tex"
wenzelm@48481
   468
wenzelm@65573
   469
session "HOL-Bali" (timing) in Bali = "HOL-Library" +
wenzelm@48481
   470
  theories
wenzelm@48481
   471
    AxExample
wenzelm@48481
   472
    AxSound
wenzelm@48481
   473
    AxCompl
wenzelm@48481
   474
    Trans
wenzelm@60751
   475
    TypeSafe
wenzelm@56781
   476
  document_files "root.tex"
wenzelm@48481
   477
wenzelm@48738
   478
session "HOL-IOA" in IOA = HOL +
wenzelm@48481
   479
  description {*
wenzelm@51403
   480
    Author:     Tobias Nipkow and Konrad Slind and Olaf Müller
wenzelm@51403
   481
    Copyright   1994--1996  TU Muenchen
wenzelm@48481
   482
blanchet@55417
   483
    The meta-theory of I/O-Automata in HOL. This formalization has been
wenzelm@51403
   484
    significantly changed and extended, see HOLCF/IOA. There are also the
wenzelm@51403
   485
    proofs of two communication protocols which formerly have been here.
wenzelm@48481
   486
wenzelm@48481
   487
    @inproceedings{Nipkow-Slind-IOA,
wenzelm@48481
   488
    author={Tobias Nipkow and Konrad Slind},
wenzelm@48481
   489
    title={{I/O} Automata in {Isabelle/HOL}},
wenzelm@48481
   490
    booktitle={Proc.\ TYPES Workshop 1994},
wenzelm@48481
   491
    publisher=Springer,
wenzelm@48481
   492
    series=LNCS,
wenzelm@48481
   493
    note={To appear}}
wenzelm@48481
   494
    ftp://ftp.informatik.tu-muenchen.de/local/lehrstuhl/nipkow/ioa.ps.gz
wenzelm@48481
   495
wenzelm@48481
   496
    and
wenzelm@48481
   497
wenzelm@48481
   498
    @inproceedings{Mueller-Nipkow,
wenzelm@48481
   499
    author={Olaf M\"uller and Tobias Nipkow},
wenzelm@48481
   500
    title={Combining Model Checking and Deduction for {I/O}-Automata},
wenzelm@48481
   501
    booktitle={Proc.\ TACAS Workshop},
wenzelm@48481
   502
    organization={Aarhus University, BRICS report},
wenzelm@48481
   503
    year=1995}
wenzelm@48481
   504
    ftp://ftp.informatik.tu-muenchen.de/local/lehrstuhl/nipkow/tacas.dvi.gz
wenzelm@48481
   505
  *}
wenzelm@48481
   506
  theories Solve
wenzelm@48481
   507
wenzelm@48738
   508
session "HOL-Lattice" in Lattice = HOL +
wenzelm@48481
   509
  description {*
wenzelm@48481
   510
    Author:     Markus Wenzel, TU Muenchen
wenzelm@48481
   511
wenzelm@48481
   512
    Basic theory of lattices and orders.
wenzelm@48481
   513
  *}
wenzelm@48481
   514
  theories CompleteLattice
wenzelm@56781
   515
  document_files "root.tex"
wenzelm@48481
   516
wenzelm@66752
   517
session "HOL-ex" (timing) in ex = "HOL-Number_Theory" +
wenzelm@51421
   518
  description {*
wenzelm@51421
   519
    Miscellaneous examples for Higher-Order Logic.
wenzelm@51421
   520
  *}
wenzelm@65548
   521
  theories
wenzelm@57507
   522
    Adhoc_Overloading_Examples
wenzelm@48481
   523
    Antiquote
wenzelm@65549
   524
    Argo_Examples
wenzelm@65549
   525
    Arith_Examples
wenzelm@65549
   526
    Ballot
wenzelm@65549
   527
    BinEx
wenzelm@65549
   528
    Birthday_Paradox
haftmann@67909
   529
    Bit_Lists
wenzelm@65549
   530
    Bubblesort
wenzelm@65549
   531
    CTL
wenzelm@65549
   532
    Cartouche_Examples
wenzelm@65563
   533
    Case_Product
wenzelm@65549
   534
    Chinese
wenzelm@65549
   535
    Classical
wenzelm@65549
   536
    Code_Binary_Nat_examples
wenzelm@65549
   537
    Code_Timing
wenzelm@65549
   538
    Coercion_Examples
wenzelm@65549
   539
    Coherent
wenzelm@65549
   540
    Commands
wenzelm@65549
   541
    Computations
traytel@67224
   542
    Conditional_Parametricity_Examples
kleing@59190
   543
    Cubic_Quartic
lars@67611
   544
    Datatype_Record_Examples
wenzelm@65549
   545
    Dedekind_Real
wenzelm@65549
   546
    Erdoes_Szekeres
wenzelm@65549
   547
    Eval_Examples
wenzelm@65549
   548
    Executable_Relation
wenzelm@65549
   549
    Execute_Choice
wenzelm@65549
   550
    Functions
haftmann@66797
   551
    Function_Growth
wenzelm@65549
   552
    Gauge_Integration
wenzelm@65549
   553
    Groebner_Examples
wenzelm@65549
   554
    Guess
wenzelm@65549
   555
    HarmonicSeries
wenzelm@65549
   556
    Hebrew
wenzelm@65549
   557
    Hex_Bin_Examples
wenzelm@65549
   558
    IArray_Examples
wenzelm@65549
   559
    Iff_Oracle
wenzelm@65549
   560
    Induction_Schema
wenzelm@48481
   561
    Intuitionistic
wenzelm@65549
   562
    Lagrange
wenzelm@65549
   563
    List_to_Set_Comprehension_Examples
wenzelm@65549
   564
    LocaleTest2
wenzelm@67215
   565
    "ML"
wenzelm@48481
   566
    MergeSort
wenzelm@65549
   567
    MonoidGroup
wenzelm@65549
   568
    Multiquote
wenzelm@65549
   569
    NatSum
wenzelm@65549
   570
    Normalization_by_Evaluation
wenzelm@65549
   571
    PER
wenzelm@65549
   572
    Parallel_Example
wenzelm@65549
   573
    Peano_Axioms
wenzelm@65549
   574
    Perm_Fragments
wenzelm@65549
   575
    PresburgerEx
wenzelm@48481
   576
    Primrec
wenzelm@65549
   577
    Pythagoras
wenzelm@65549
   578
    Quicksort
nipkow@67612
   579
    Radix_Sort
wenzelm@65549
   580
    Records
wenzelm@65549
   581
    Reflection_Examples
wenzelm@65549
   582
    Refute_Examples
haftmann@68035
   583
    Residue_Ring
wenzelm@65549
   584
    Rewrite_Examples
wenzelm@65549
   585
    SOS
wenzelm@65549
   586
    SOS_Cert
wenzelm@65549
   587
    Seq
wenzelm@65549
   588
    Serbian
wenzelm@65549
   589
    Set_Comprehension_Pointfree_Examples
wenzelm@48481
   590
    Set_Theory
wenzelm@65549
   591
    Simproc_Tests
wenzelm@65549
   592
    Simps_Case_Conv_Examples
wenzelm@48481
   593
    Sqrt
wenzelm@48481
   594
    Sqrt_Script
wenzelm@65549
   595
    Sudoku
wenzelm@65549
   596
    Sum_of_Powers
wenzelm@65549
   597
    Tarski
wenzelm@65549
   598
    Termination
wenzelm@65549
   599
    ThreeDivides
kuncar@61368
   600
    Transfer_Debug
wenzelm@48481
   601
    Transfer_Int_Nat
haftmann@56922
   602
    Transitive_Closure_Table_Ex
wenzelm@65549
   603
    Tree23
wenzelm@65549
   604
    Unification
wenzelm@65549
   605
    While_Combinator_Example
haftmann@64015
   606
    Word_Type
blanchet@64978
   607
    veriT_Preprocessing
wenzelm@51558
   608
  theories [skip_proofs = false]
wenzelm@67159
   609
    SAT_Examples
wenzelm@51558
   610
    Meson_Test
wenzelm@48481
   611
wenzelm@65550
   612
session "HOL-Isar_Examples" in Isar_Examples = "HOL-Computational_Algebra" +
wenzelm@51403
   613
  description {*
wenzelm@61935
   614
    Miscellaneous Isabelle/Isar examples.
wenzelm@51403
   615
  *}
wenzelm@61939
   616
  options [quick_and_dirty]
wenzelm@48481
   617
  theories
wenzelm@61939
   618
    Knaster_Tarski
wenzelm@61939
   619
    Peirce
wenzelm@61939
   620
    Drinker
wenzelm@48481
   621
    Cantor
wenzelm@61939
   622
    Structured_Statements
wenzelm@61939
   623
    Basic_Logic
wenzelm@48481
   624
    Expr_Compiler
wenzelm@48481
   625
    Fibonacci
wenzelm@48481
   626
    Group
wenzelm@48481
   627
    Group_Context
wenzelm@48481
   628
    Group_Notepad
wenzelm@48481
   629
    Hoare_Ex
wenzelm@48481
   630
    Mutilated_Checkerboard
wenzelm@48481
   631
    Puzzle
wenzelm@48481
   632
    Summation
wenzelm@61935
   633
    First_Order_Logic
wenzelm@61935
   634
    Higher_Order_Logic
wenzelm@56781
   635
  document_files
wenzelm@56781
   636
    "root.bib"
wenzelm@56781
   637
    "root.tex"
wenzelm@48481
   638
wenzelm@60119
   639
session "HOL-Eisbach" in Eisbach = HOL +
wenzelm@60119
   640
  description {*
wenzelm@60119
   641
    The Eisbach proof method language and "match" method.
wenzelm@60119
   642
  *}
wenzelm@66444
   643
  sessions
wenzelm@66444
   644
    FOL
wenzelm@60119
   645
  theories
wenzelm@60119
   646
    Eisbach
wenzelm@60119
   647
    Tests
wenzelm@60119
   648
    Examples
wenzelm@62168
   649
    Examples_FOL
wenzelm@60119
   650
wenzelm@65550
   651
session "HOL-SET_Protocol" (timing) in SET_Protocol = "HOL-Library" +
wenzelm@51403
   652
  description {*
wenzelm@51403
   653
    Verification of the SET Protocol.
wenzelm@51403
   654
  *}
wenzelm@65543
   655
  theories
wenzelm@65543
   656
    SET_Protocol
wenzelm@56781
   657
  document_files "root.tex"
wenzelm@48481
   658
wenzelm@65569
   659
session "HOL-Matrix_LP" in Matrix_LP = "HOL-Library" +
wenzelm@51403
   660
  description {*
wenzelm@51403
   661
    Two-dimensional matrices and linear programming.
wenzelm@51403
   662
  *}
wenzelm@48481
   663
  theories Cplex
wenzelm@56781
   664
  document_files "root.tex"
wenzelm@48481
   665
wenzelm@48738
   666
session "HOL-TLA" in TLA = HOL +
wenzelm@51403
   667
  description {*
wenzelm@51403
   668
    Lamport's Temporal Logic of Actions.
wenzelm@51403
   669
  *}
wenzelm@48481
   670
  theories TLA
wenzelm@48481
   671
wenzelm@48738
   672
session "HOL-TLA-Inc" in "TLA/Inc" = "HOL-TLA" +
wenzelm@48481
   673
  theories Inc
wenzelm@48481
   674
wenzelm@48738
   675
session "HOL-TLA-Buffer" in "TLA/Buffer" = "HOL-TLA" +
wenzelm@48481
   676
  theories DBuffer
wenzelm@48481
   677
wenzelm@48738
   678
session "HOL-TLA-Memory" in "TLA/Memory" = "HOL-TLA" +
wenzelm@48481
   679
  theories MemoryImplementation
wenzelm@48481
   680
wenzelm@65573
   681
session "HOL-TPTP" in TPTP = "HOL-Library" +
wenzelm@48481
   682
  description {*
wenzelm@48481
   683
    Author:     Jasmin Blanchette, TU Muenchen
wenzelm@48481
   684
    Author:     Nik Sultana, University of Cambridge
wenzelm@48481
   685
    Copyright   2011
wenzelm@48481
   686
wenzelm@48481
   687
    TPTP-related extensions.
wenzelm@48481
   688
  *}
wenzelm@48481
   689
  theories
wenzelm@48481
   690
    ATP_Theory_Export
wenzelm@48481
   691
    MaSh_Eval
wenzelm@48481
   692
    TPTP_Interpret
wenzelm@48481
   693
    THF_Arith
sultana@55596
   694
    TPTP_Proof_Reconstruction
wenzelm@52488
   695
  theories
wenzelm@48481
   696
    ATP_Problem_Import
wenzelm@48481
   697
wenzelm@65382
   698
session "HOL-Probability" (main timing) in "Probability" = "HOL-Analysis" +
wenzelm@48481
   699
  theories
wenzelm@66992
   700
    Probability
wenzelm@56781
   701
  document_files "root.tex"
wenzelm@48481
   702
wenzelm@63888
   703
session "HOL-Probability-ex" (timing) in "Probability/ex" = "HOL-Probability" +
haftmann@61946
   704
  theories
wenzelm@65538
   705
    Dining_Cryptographers
wenzelm@65538
   706
    Koepf_Duermuth_Countermeasure
wenzelm@65538
   707
    Measure_Not_CCC
haftmann@61946
   708
wenzelm@66752
   709
session "HOL-Nominal" in Nominal = "HOL-Library" +
wenzelm@65573
   710
  theories
wenzelm@65573
   711
    Nominal
wenzelm@48481
   712
wenzelm@63888
   713
session "HOL-Nominal-Examples" (timing) in "Nominal/Examples" = "HOL-Nominal" +
blanchet@58329
   714
  theories
wenzelm@59162
   715
    Class3
wenzelm@59162
   716
    CK_Machine
wenzelm@59162
   717
    Compile
wenzelm@59162
   718
    Contexts
wenzelm@59162
   719
    Crary
wenzelm@59162
   720
    CR_Takahashi
wenzelm@59162
   721
    CR
wenzelm@59162
   722
    Fsub
wenzelm@59162
   723
    Height
wenzelm@59162
   724
    Lambda_mu
wenzelm@59162
   725
    Lam_Funs
wenzelm@59162
   726
    LocalWeakening
wenzelm@59162
   727
    Pattern
wenzelm@59162
   728
    SN
wenzelm@59162
   729
    SOS
wenzelm@59162
   730
    Standardization
wenzelm@59162
   731
    Support
wenzelm@59162
   732
    Type_Preservation
wenzelm@59162
   733
    Weakening
wenzelm@59162
   734
    W
blanchet@58329
   735
  theories [quick_and_dirty]
blanchet@58329
   736
    VC_Condition
wenzelm@48481
   737
wenzelm@65573
   738
session "HOL-Cardinals" (timing) in Cardinals = "HOL-Library" +
wenzelm@51421
   739
  description {*
wenzelm@51421
   740
    Ordinals and Cardinals, Full Theories.
wenzelm@51421
   741
  *}
traytel@59747
   742
  theories
traytel@59747
   743
    Cardinals
traytel@59747
   744
    Bounded_Set
wenzelm@56781
   745
  document_files
wenzelm@56781
   746
    "intro.tex"
wenzelm@56781
   747
    "root.tex"
wenzelm@56781
   748
    "root.bib"
blanchet@48975
   749
wenzelm@65574
   750
session "HOL-Datatype_Examples" (timing) in Datatype_Examples = "HOL-Library" +
wenzelm@51421
   751
  description {*
wenzelm@62286
   752
    (Co)datatype Examples.
wenzelm@51421
   753
  *}
blanchet@48975
   754
  theories
blanchet@56454
   755
    Compat
blanchet@48975
   756
    Lambda_Term
blanchet@48975
   757
    Process
blanchet@48975
   758
    TreeFsetI
popescua@49872
   759
    "Derivation_Trees/Gram_Lang"
wenzelm@65574
   760
    "Derivation_Trees/Parallel_Composition"
traytel@50517
   761
    Koenig
traytel@60921
   762
    Lift_BNF
traytel@61745
   763
    Milner_Tofte
traytel@54961
   764
    Stream_Processor
blanchet@53122
   765
    Misc_Codatatype
blanchet@53122
   766
    Misc_Datatype
blanchet@54193
   767
    Misc_Primcorec
blanchet@53306
   768
    Misc_Primrec
blanchet@48975
   769
wenzelm@65573
   770
session "HOL-Corec_Examples" (timing) in Corec_Examples = "HOL-Library" +
blanchet@62694
   771
  description {*
blanchet@62694
   772
    Corecursion Examples.
blanchet@62694
   773
  *}
blanchet@62694
   774
  theories
blanchet@62694
   775
    LFilter
blanchet@62734
   776
    Paper_Examples
blanchet@62694
   777
    Stream_Processor
blanchet@62696
   778
    "Tests/Simple_Nesting"
traytel@64379
   779
    "Tests/Iterate_GPV"
blanchet@62696
   780
  theories [quick_and_dirty]
blanchet@62696
   781
    "Tests/GPV_Bare_Bones"
blanchet@62696
   782
    "Tests/Merge_D"
blanchet@62696
   783
    "Tests/Merge_Poly"
blanchet@62696
   784
    "Tests/Misc_Mono"
blanchet@62696
   785
    "Tests/Misc_Poly"
blanchet@62696
   786
    "Tests/Small_Concrete"
blanchet@62725
   787
    "Tests/Stream_Friends"
blanchet@62696
   788
    "Tests/TLList_Friends"
blanchet@63190
   789
    "Tests/Type_Class"
blanchet@62694
   790
wenzelm@63888
   791
session "HOL-Word" (main timing) in Word = HOL +
wenzelm@65573
   792
  sessions
wenzelm@65573
   793
    "HOL-Library"
wenzelm@65382
   794
  theories
wenzelm@65462
   795
    Word
wenzelm@66445
   796
    WordBitwise
wenzelm@66443
   797
    Bit_Comparison
wenzelm@67122
   798
    WordExamples
wenzelm@56781
   799
  document_files "root.bib" "root.tex"
wenzelm@48481
   800
wenzelm@48738
   801
session "HOL-Statespace" in Statespace = HOL +
wenzelm@51558
   802
  theories [skip_proofs = false]
wenzelm@51558
   803
    StateSpaceEx
wenzelm@56781
   804
  document_files "root.tex"
wenzelm@48481
   805
wenzelm@65550
   806
session "HOL-Nonstandard_Analysis" (timing) in Nonstandard_Analysis = "HOL-Computational_Algebra" +
wenzelm@51421
   807
  description {*
wenzelm@51421
   808
    Nonstandard analysis.
wenzelm@51421
   809
  *}
wenzelm@62479
   810
  theories
wenzelm@62479
   811
    Nonstandard_Analysis
wenzelm@56781
   812
  document_files "root.tex"
wenzelm@48481
   813
wenzelm@63888
   814
session "HOL-Nonstandard_Analysis-Examples" (timing) in "Nonstandard_Analysis/Examples" = "HOL-Nonstandard_Analysis" +
wenzelm@65543
   815
  theories
wenzelm@65543
   816
    NSPrimes
wenzelm@48481
   817
wenzelm@48738
   818
session "HOL-Mirabelle" in Mirabelle = HOL +
wenzelm@48481
   819
  theories Mirabelle_Test
wenzelm@48589
   820
wenzelm@48738
   821
session "HOL-Mirabelle-ex" in "Mirabelle/ex" = "HOL-Mirabelle" +
wenzelm@66946
   822
  options [timeout = 60]
wenzelm@49448
   823
  theories Ex
wenzelm@48481
   824
wenzelm@63888
   825
session "HOL-Word-SMT_Examples" (timing) in SMT_Examples = "HOL-Word" +
wenzelm@66946
   826
  options [quick_and_dirty]
wenzelm@48481
   827
  theories
boehmes@52722
   828
    Boogie
wenzelm@48481
   829
    SMT_Examples
wenzelm@48481
   830
    SMT_Word_Examples
blanchet@50666
   831
    SMT_Tests
wenzelm@48481
   832
wenzelm@50844
   833
session "HOL-SPARK" (main) in "SPARK" = "HOL-Word" +
wenzelm@65382
   834
  theories
wenzelm@66992
   835
    SPARK
wenzelm@48481
   836
wenzelm@48738
   837
session "HOL-SPARK-Examples" in "SPARK/Examples" = "HOL-SPARK" +
wenzelm@66946
   838
  options [spark_prv = false]
wenzelm@48481
   839
  theories
wenzelm@48481
   840
    "Gcd/Greatest_Common_Divisor"
wenzelm@48481
   841
    "Liseq/Longest_Increasing_Subsequence"
wenzelm@48481
   842
    "RIPEMD-160/F"
wenzelm@48481
   843
    "RIPEMD-160/Hash"
wenzelm@48481
   844
    "RIPEMD-160/K_L"
wenzelm@48481
   845
    "RIPEMD-160/K_R"
wenzelm@48481
   846
    "RIPEMD-160/R_L"
wenzelm@48481
   847
    "RIPEMD-160/Round"
wenzelm@48481
   848
    "RIPEMD-160/R_R"
wenzelm@48481
   849
    "RIPEMD-160/S_L"
wenzelm@48481
   850
    "RIPEMD-160/S_R"
wenzelm@48481
   851
    "Sqrt/Sqrt"
wenzelm@48481
   852
wenzelm@65576
   853
session "HOL-SPARK-Manual" in "SPARK/Manual" = "HOL-SPARK" +
wenzelm@59810
   854
  options [show_question_marks = false, spark_prv = false]
wenzelm@66031
   855
  sessions
wenzelm@66031
   856
    "HOL-SPARK-Examples"
wenzelm@48481
   857
  theories
wenzelm@48481
   858
    Example_Verification
wenzelm@48481
   859
    VC_Principles
wenzelm@48481
   860
    Reference
wenzelm@48481
   861
    Complex_Types
wenzelm@56781
   862
  document_files
wenzelm@56781
   863
    "complex_types.ads"
wenzelm@56781
   864
    "complex_types_app.adb"
wenzelm@56781
   865
    "complex_types_app.ads"
wenzelm@56781
   866
    "Gcd.adb"
wenzelm@56781
   867
    "Gcd.ads"
wenzelm@56781
   868
    "intro.tex"
wenzelm@56781
   869
    "loop_invariant.adb"
wenzelm@56781
   870
    "loop_invariant.ads"
wenzelm@56781
   871
    "root.bib"
wenzelm@56781
   872
    "root.tex"
wenzelm@56781
   873
    "Simple_Gcd.adb"
wenzelm@56781
   874
    "Simple_Gcd.ads"
wenzelm@48481
   875
wenzelm@65573
   876
session "HOL-Mutabelle" in Mutabelle = "HOL-Library" +
wenzelm@48481
   877
  theories MutabelleExtra
wenzelm@48481
   878
wenzelm@65569
   879
session "HOL-Quickcheck_Examples" (timing) in Quickcheck_Examples = "HOL-Library" +
wenzelm@48588
   880
  theories
wenzelm@48690
   881
    Quickcheck_Examples
wenzelm@48690
   882
    Quickcheck_Lattice_Examples
wenzelm@48690
   883
    Completeness
wenzelm@48690
   884
    Quickcheck_Interfaces
blanchet@63731
   885
    Quickcheck_Nesting_Example
wenzelm@57584
   886
  theories [condition = ISABELLE_GHC]
Andreas@57544
   887
    Hotel_Example
wenzelm@48598
   888
    Quickcheck_Narrowing_Examples
wenzelm@48588
   889
wenzelm@65569
   890
session "HOL-Quotient_Examples" (timing) in Quotient_Examples = "HOL-Algebra" +
wenzelm@48481
   891
  description {*
wenzelm@48481
   892
    Author:     Cezary Kaliszyk and Christian Urban
wenzelm@48481
   893
  *}
wenzelm@48481
   894
  theories
wenzelm@48481
   895
    DList
kuncar@63920
   896
    Quotient_FSet
wenzelm@48481
   897
    Quotient_Int
wenzelm@48481
   898
    Quotient_Message
wenzelm@48481
   899
    Lift_FSet
wenzelm@48481
   900
    Lift_Set
wenzelm@48481
   901
    Lift_Fun
wenzelm@48481
   902
    Quotient_Rat
wenzelm@48481
   903
    Lift_DList
kuncar@53682
   904
    Int_Pow
kuncar@60237
   905
    Lifting_Code_Dt_Test
wenzelm@48481
   906
wenzelm@65569
   907
session "HOL-Predicate_Compile_Examples" (timing) in Predicate_Compile_Examples = "HOL-Library" +
wenzelm@62354
   908
  theories
wenzelm@48481
   909
    Examples
wenzelm@48481
   910
    Predicate_Compile_Tests
Andreas@61140
   911
    Predicate_Compile_Quickcheck_Examples
wenzelm@48481
   912
    Specialisation_Examples
wenzelm@48690
   913
    IMP_1
wenzelm@48690
   914
    IMP_2
wenzelm@55450
   915
    (* FIXME since 21-Jul-2011
Andreas@61140
   916
    Hotel_Example_Small_Generator *)
wenzelm@48690
   917
    IMP_3
Andreas@61140
   918
    IMP_4
wenzelm@62242
   919
  theories [condition = ISABELLE_SWIPL]
wenzelm@48690
   920
    Code_Prolog_Examples
wenzelm@48690
   921
    Context_Free_Grammar_Example
wenzelm@48690
   922
    Hotel_Example_Prolog
wenzelm@48690
   923
    Lambda_Example
wenzelm@48690
   924
    List_Examples
wenzelm@62242
   925
  theories [condition = ISABELLE_SWIPL, quick_and_dirty]
wenzelm@48690
   926
    Reg_Exp_Example
wenzelm@48481
   927
wenzelm@64551
   928
session "HOL-Types_To_Sets" in Types_To_Sets = HOL +
wenzelm@64551
   929
  description {*
wenzelm@64551
   930
    Experimental extension of Higher-Order Logic to allow translation of types to sets.
wenzelm@64551
   931
  *}
wenzelm@64551
   932
  theories
wenzelm@64551
   933
    Types_To_Sets
wenzelm@64551
   934
    "Examples/Prerequisites"
wenzelm@64551
   935
    "Examples/Finite"
wenzelm@64551
   936
    "Examples/T2_Spaces"
immler@68522
   937
    "Examples/Linear_Algebra_On"
wenzelm@64551
   938
wenzelm@66982
   939
session HOLCF (main timing) in HOLCF = HOL +
wenzelm@48338
   940
  description {*
wenzelm@48338
   941
    Author:     Franz Regensburger
wenzelm@48338
   942
    Author:     Brian Huffman
wenzelm@48338
   943
wenzelm@48338
   944
    HOLCF -- a semantic extension of HOL by the LCF logic.
wenzelm@48338
   945
  *}
wenzelm@66982
   946
  sessions
wenzelm@66982
   947
    "HOL-Library"
wenzelm@48481
   948
  theories
wenzelm@65382
   949
    HOLCF (global)
wenzelm@56781
   950
  document_files "root.tex"
wenzelm@48481
   951
wenzelm@48738
   952
session "HOLCF-Tutorial" in "HOLCF/Tutorial" = HOLCF +
wenzelm@48481
   953
  theories
wenzelm@48481
   954
    Domain_ex
wenzelm@48481
   955
    Fixrec_ex
wenzelm@48481
   956
    New_Domain
wenzelm@56781
   957
  document_files "root.tex"
wenzelm@48481
   958
wenzelm@48738
   959
session "HOLCF-Library" in "HOLCF/Library" = HOLCF +
wenzelm@65570
   960
  theories
wenzelm@65570
   961
    HOLCF_Library
wenzelm@65570
   962
    HOL_Cpo
wenzelm@48481
   963
wenzelm@48738
   964
session "HOLCF-IMP" in "HOLCF/IMP" = HOLCF +
wenzelm@51403
   965
  description {*
wenzelm@51403
   966
    IMP -- A WHILE-language and its Semantics.
wenzelm@51403
   967
wenzelm@51403
   968
    This is the HOLCF-based denotational semantics of a simple WHILE-language.
wenzelm@51403
   969
  *}
wenzelm@65573
   970
  sessions
wenzelm@65573
   971
    "HOL-IMP"
wenzelm@65573
   972
  theories
wenzelm@65573
   973
    HoareEx
wenzelm@66950
   974
  document_files
wenzelm@66950
   975
    "isaverbatimwrite.sty"
wenzelm@66950
   976
    "root.tex"
wenzelm@66950
   977
    "root.bib"
wenzelm@48338
   978
wenzelm@65573
   979
session "HOLCF-ex" in "HOLCF/ex" = "HOLCF-Library" +
wenzelm@51421
   980
  description {*
wenzelm@51421
   981
    Miscellaneous examples for HOLCF.
wenzelm@51421
   982
  *}
wenzelm@48481
   983
  theories
wenzelm@48481
   984
    Dnat
wenzelm@48481
   985
    Dagstuhl
wenzelm@48481
   986
    Focus_ex
wenzelm@48481
   987
    Fix2
wenzelm@48481
   988
    Hoare
wenzelm@48481
   989
    Concurrency_Monad
wenzelm@48481
   990
    Loop
wenzelm@48481
   991
    Powerdomain_ex
wenzelm@48481
   992
    Domain_Proofs
wenzelm@48481
   993
    Letrec
wenzelm@48481
   994
    Pattern_Match
wenzelm@48481
   995
wenzelm@65573
   996
session "HOLCF-FOCUS" in "HOLCF/FOCUS" = "HOLCF-Library" +
wenzelm@51403
   997
  description {*
wenzelm@51403
   998
    FOCUS: a theory of stream-processing functions Isabelle/HOLCF.
wenzelm@51403
   999
wenzelm@51403
  1000
    For introductions to FOCUS, see
wenzelm@51403
  1001
wenzelm@51403
  1002
    "The Design of Distributed Systems - An Introduction to FOCUS"
wenzelm@51403
  1003
    http://www4.in.tum.de/publ/html.php?e=2
wenzelm@51403
  1004
wenzelm@51403
  1005
    "Specification and Refinement of a Buffer of Length One"
wenzelm@51403
  1006
    http://www4.in.tum.de/publ/html.php?e=15
wenzelm@51403
  1007
wenzelm@51403
  1008
    "Specification and Development of Interactive Systems: Focus on Streams,
wenzelm@51403
  1009
    Interfaces, and Refinement" http://www4.in.tum.de/publ/html.php?e=321
wenzelm@51403
  1010
  *}
wenzelm@48481
  1011
  theories
wenzelm@48481
  1012
    Fstreams
wenzelm@48481
  1013
    FOCUS
wenzelm@48481
  1014
    Buffer_adm
wenzelm@48481
  1015
wenzelm@63888
  1016
session IOA (timing) in "HOLCF/IOA" = HOLCF +
wenzelm@48481
  1017
  description {*
wenzelm@48481
  1018
    Author:     Olaf Mueller
wenzelm@51403
  1019
    Copyright   1997 TU München
wenzelm@48481
  1020
wenzelm@51403
  1021
    A formalization of I/O automata in HOLCF.
wenzelm@51403
  1022
wenzelm@51403
  1023
    The distribution contains simulation relations, temporal logic, and an
wenzelm@51403
  1024
    abstraction theory. Everything is based upon a domain-theoretic model of
wenzelm@51403
  1025
    finite and infinite sequences.
wenzelm@48481
  1026
  *}
wenzelm@65538
  1027
  theories Abstraction
wenzelm@48481
  1028
wenzelm@48738
  1029
session "IOA-ABP" in "HOLCF/IOA/ABP" = IOA +
wenzelm@48481
  1030
  description {*
wenzelm@48481
  1031
    Author:     Olaf Mueller
wenzelm@48481
  1032
wenzelm@48481
  1033
    The Alternating Bit Protocol performed in I/O-Automata.
wenzelm@48481
  1034
  *}
wenzelm@59503
  1035
  theories
wenzelm@59503
  1036
    Correctness
wenzelm@59503
  1037
    Spec
wenzelm@48481
  1038
wenzelm@48738
  1039
session "IOA-NTP" in "HOLCF/IOA/NTP" = IOA +
wenzelm@48481
  1040
  description {*
wenzelm@48481
  1041
    Author:     Tobias Nipkow & Konrad Slind
wenzelm@48481
  1042
wenzelm@48481
  1043
    A network transmission protocol, performed in the
wenzelm@48481
  1044
    I/O automata formalization by Olaf Mueller.
wenzelm@48481
  1045
  *}
wenzelm@48481
  1046
  theories Correctness
wenzelm@48481
  1047
wenzelm@48738
  1048
session "IOA-Storage" in "HOLCF/IOA/Storage" = IOA +
wenzelm@48481
  1049
  description {*
wenzelm@48481
  1050
    Author:     Olaf Mueller
wenzelm@48481
  1051
wenzelm@48481
  1052
    Memory storage case study.
wenzelm@48481
  1053
  *}
wenzelm@48481
  1054
  theories Correctness
wenzelm@48481
  1055
wenzelm@48738
  1056
session "IOA-ex" in "HOLCF/IOA/ex" = IOA +
wenzelm@48481
  1057
  description {*
wenzelm@48481
  1058
    Author:     Olaf Mueller
wenzelm@48481
  1059
  *}
wenzelm@48481
  1060
  theories
wenzelm@48481
  1061
    TrivEx
wenzelm@48481
  1062
    TrivEx2