clasohm@923
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(* Title: HOL/HOL.thy
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wenzelm@11750
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
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wenzelm@11750
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*)
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clasohm@923
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wenzelm@11750
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header {* The basis of Higher-Order Logic *}
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clasohm@923
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nipkow@15131
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theory HOL
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wenzelm@26957
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imports Pure
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wenzelm@23163
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uses
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haftmann@28952
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("Tools/hologic.ML")
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wenzelm@23171
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"~~/src/Tools/IsaPlanner/zipper.ML"
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wenzelm@23171
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"~~/src/Tools/IsaPlanner/isand.ML"
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wenzelm@23171
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"~~/src/Tools/IsaPlanner/rw_tools.ML"
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wenzelm@23171
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"~~/src/Tools/IsaPlanner/rw_inst.ML"
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haftmann@23263
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"~~/src/Provers/project_rule.ML"
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haftmann@23263
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"~~/src/Provers/hypsubst.ML"
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haftmann@23263
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"~~/src/Provers/splitter.ML"
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wenzelm@23163
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"~~/src/Provers/classical.ML"
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wenzelm@23163
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"~~/src/Provers/blast.ML"
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wenzelm@23163
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"~~/src/Provers/clasimp.ML"
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berghofe@28325
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"~~/src/Provers/coherent.ML"
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haftmann@23263
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"~~/src/Provers/eqsubst.ML"
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wenzelm@23163
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"~~/src/Provers/quantifier1.ML"
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haftmann@28952
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("Tools/simpdata.ML")
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wenzelm@25741
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"~~/src/Tools/random_word.ML"
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krauss@26580
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"~~/src/Tools/atomize_elim.ML"
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haftmann@24901
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"~~/src/Tools/induct.ML"
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wenzelm@27326
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("~~/src/Tools/induct_tacs.ML")
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haftmann@29105
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"~~/src/Tools/value.ML"
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haftmann@24280
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"~~/src/Tools/code/code_name.ML"
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haftmann@24280
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"~~/src/Tools/code/code_funcgr.ML"
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haftmann@24280
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"~~/src/Tools/code/code_thingol.ML"
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haftmann@28054
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"~~/src/Tools/code/code_printer.ML"
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haftmann@24280
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"~~/src/Tools/code/code_target.ML"
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haftmann@28054
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"~~/src/Tools/code/code_ml.ML"
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haftmann@28054
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"~~/src/Tools/code/code_haskell.ML"
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haftmann@24166
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"~~/src/Tools/nbe.ML"
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haftmann@29505
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("Tools/recfun_codegen.ML")
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nipkow@15131
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begin
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wenzelm@2260
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wenzelm@11750
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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wenzelm@2260
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wenzelm@14854
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classes type
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wenzelm@12338
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defaultsort type
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wenzelm@25494
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setup {* ObjectLogic.add_base_sort @{sort type} *}
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haftmann@25460
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haftmann@25460
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arities
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haftmann@25460
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"fun" :: (type, type) type
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haftmann@25460
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itself :: (type) type
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haftmann@25460
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wenzelm@3947
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global
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wenzelm@3947
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wenzelm@7357
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typedecl bool
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wenzelm@11750
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judgment
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wenzelm@11750
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Trueprop :: "bool => prop" ("(_)" 5)
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wenzelm@11750
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clasohm@923
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consts
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wenzelm@7357
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Not :: "bool => bool" ("~ _" [40] 40)
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wenzelm@7357
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True :: bool
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wenzelm@7357
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False :: bool
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wenzelm@11432
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The :: "('a => bool) => 'a"
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wenzelm@7357
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All :: "('a => bool) => bool" (binder "ALL " 10)
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wenzelm@7357
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Ex :: "('a => bool) => bool" (binder "EX " 10)
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wenzelm@7357
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Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
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wenzelm@7357
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Let :: "['a, 'a => 'b] => 'b"
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clasohm@923
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haftmann@22839
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"op =" :: "['a, 'a] => bool" (infixl "=" 50)
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haftmann@22839
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"op &" :: "[bool, bool] => bool" (infixr "&" 35)
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haftmann@22839
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"op |" :: "[bool, bool] => bool" (infixr "|" 30)
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haftmann@22839
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"op -->" :: "[bool, bool] => bool" (infixr "-->" 25)
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clasohm@923
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wenzelm@10432
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local
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wenzelm@10432
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paulson@16587
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consts
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paulson@16587
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If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
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clasohm@923
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wenzelm@19656
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wenzelm@11750
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subsubsection {* Additional concrete syntax *}
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wenzelm@2260
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wenzelm@21210
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notation (output)
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wenzelm@19656
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"op =" (infix "=" 50)
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wenzelm@19656
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wenzelm@19656
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abbreviation
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wenzelm@21404
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not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
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wenzelm@19656
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"x ~= y == ~ (x = y)"
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wenzelm@19656
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wenzelm@21210
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notation (output)
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wenzelm@19656
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not_equal (infix "~=" 50)
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wenzelm@19656
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wenzelm@21210
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notation (xsymbols)
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wenzelm@21404
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Not ("\<not> _" [40] 40) and
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wenzelm@21404
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"op &" (infixr "\<and>" 35) and
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wenzelm@21404
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"op |" (infixr "\<or>" 30) and
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wenzelm@21404
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"op -->" (infixr "\<longrightarrow>" 25) and
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wenzelm@19656
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not_equal (infix "\<noteq>" 50)
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wenzelm@19656
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wenzelm@21210
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notation (HTML output)
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wenzelm@21404
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Not ("\<not> _" [40] 40) and
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wenzelm@21404
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"op &" (infixr "\<and>" 35) and
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wenzelm@21404
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"op |" (infixr "\<or>" 30) and
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wenzelm@19656
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not_equal (infix "\<noteq>" 50)
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wenzelm@19656
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wenzelm@19656
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abbreviation (iff)
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wenzelm@21404
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iff :: "[bool, bool] => bool" (infixr "<->" 25) where
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wenzelm@19656
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"A <-> B == A = B"
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wenzelm@19656
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wenzelm@21210
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notation (xsymbols)
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wenzelm@19656
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iff (infixr "\<longleftrightarrow>" 25)
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wenzelm@19656
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wenzelm@19656
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wenzelm@4868
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nonterminals
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clasohm@923
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letbinds letbind
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clasohm@923
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case_syn cases_syn
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clasohm@923
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clasohm@923
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syntax
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wenzelm@11432
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"_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
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clasohm@923
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wenzelm@7357
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"_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
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wenzelm@7357
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"" :: "letbind => letbinds" ("_")
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wenzelm@7357
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"_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
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wenzelm@7357
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"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
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clasohm@923
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wenzelm@9060
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"_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
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wenzelm@9060
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"_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
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wenzelm@7357
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"" :: "case_syn => cases_syn" ("_")
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wenzelm@9060
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"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
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clasohm@923
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clasohm@923
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translations
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nipkow@13764
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"THE x. P" == "The (%x. P)"
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clasohm@923
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"_Let (_binds b bs) e" == "_Let b (_Let bs e)"
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nipkow@1114
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"let x = a in e" == "Let a (%x. e)"
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clasohm@923
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nipkow@13763
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print_translation {*
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nipkow@13763
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(* To avoid eta-contraction of body: *)
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nipkow@13763
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[("The", fn [Abs abs] =>
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nipkow@13763
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let val (x,t) = atomic_abs_tr' abs
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nipkow@13763
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in Syntax.const "_The" $ x $ t end)]
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nipkow@13763
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*}
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nipkow@13763
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wenzelm@12114
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syntax (xsymbols)
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wenzelm@11687
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"_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
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wenzelm@2372
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wenzelm@21524
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notation (xsymbols)
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wenzelm@21524
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All (binder "\<forall>" 10) and
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wenzelm@21524
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Ex (binder "\<exists>" 10) and
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wenzelm@21524
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Ex1 (binder "\<exists>!" 10)
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wenzelm@6340
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wenzelm@21524
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notation (HTML output)
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wenzelm@21524
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All (binder "\<forall>" 10) and
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wenzelm@21524
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Ex (binder "\<exists>" 10) and
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wenzelm@21524
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Ex1 (binder "\<exists>!" 10)
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wenzelm@21524
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wenzelm@21524
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notation (HOL)
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wenzelm@21524
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All (binder "! " 10) and
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wenzelm@21524
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Ex (binder "? " 10) and
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wenzelm@21524
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Ex1 (binder "?! " 10)
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wenzelm@7238
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wenzelm@7238
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wenzelm@11750
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subsubsection {* Axioms and basic definitions *}
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wenzelm@2260
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wenzelm@7357
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axioms
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paulson@15380
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refl: "t = (t::'a)"
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haftmann@28513
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subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t"
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paulson@15380
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ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
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paulson@15380
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-- {*Extensionality is built into the meta-logic, and this rule expresses
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paulson@15380
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a related property. It is an eta-expanded version of the traditional
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paulson@15380
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rule, and similar to the ABS rule of HOL*}
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paulson@6289
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wenzelm@11432
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the_eq_trivial: "(THE x. x = a) = (a::'a)"
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clasohm@923
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paulson@15380
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impI: "(P ==> Q) ==> P-->Q"
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paulson@15380
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mp: "[| P-->Q; P |] ==> Q"
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paulson@15380
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paulson@15380
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clasohm@923
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defs
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wenzelm@7357
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True_def: "True == ((%x::bool. x) = (%x. x))"
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wenzelm@7357
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All_def: "All(P) == (P = (%x. True))"
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paulson@11451
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Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
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wenzelm@7357
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False_def: "False == (!P. P)"
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wenzelm@7357
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not_def: "~ P == P-->False"
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wenzelm@7357
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and_def: "P & Q == !R. (P-->Q-->R) --> R"
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wenzelm@7357
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or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
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wenzelm@7357
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Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
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clasohm@923
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wenzelm@7357
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axioms
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wenzelm@7357
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iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
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wenzelm@7357
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True_or_False: "(P=True) | (P=False)"
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clasohm@923
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clasohm@923
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defs
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haftmann@24219
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Let_def: "Let s f == f(s)"
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paulson@11451
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if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
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wenzelm@5069
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skalberg@14223
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finalconsts
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skalberg@14223
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"op ="
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skalberg@14223
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"op -->"
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skalberg@14223
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The
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nipkow@3320
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haftmann@22481
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axiomatization
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haftmann@22481
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undefined :: 'a
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wenzelm@19656
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haftmann@28682
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abbreviation (input)
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haftmann@28682
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"arbitrary \<equiv> undefined"
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haftmann@22481
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haftmann@22481
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haftmann@22481
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subsubsection {* Generic classes and algebraic operations *}
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haftmann@22481
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haftmann@29608
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class default =
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haftmann@24901
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fixes default :: 'a
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wenzelm@4793
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haftmann@29608
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class zero =
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haftmann@25062
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fixes zero :: 'a ("0")
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haftmann@20713
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haftmann@29608
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class one =
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haftmann@25062
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fixes one :: 'a ("1")
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haftmann@20713
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haftmann@20713
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hide (open) const zero one
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wenzelm@11750
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haftmann@29608
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class plus =
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haftmann@25062
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fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
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haftmann@20590
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haftmann@29608
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class minus =
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haftmann@25762
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fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
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haftmann@25762
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haftmann@29608
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class uminus =
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haftmann@25062
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fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
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haftmann@20590
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haftmann@29608
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class times =
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haftmann@25062
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fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
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haftmann@20590
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haftmann@29608
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class inverse =
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haftmann@20590
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fixes inverse :: "'a \<Rightarrow> 'a"
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haftmann@25062
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and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
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wenzelm@21524
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haftmann@29608
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class abs =
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haftmann@23878
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fixes abs :: "'a \<Rightarrow> 'a"
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wenzelm@25388
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begin
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haftmann@23878
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wenzelm@21524
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notation (xsymbols)
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wenzelm@21524
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abs ("\<bar>_\<bar>")
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wenzelm@25388
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wenzelm@21524
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notation (HTML output)
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wenzelm@21524
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abs ("\<bar>_\<bar>")
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wenzelm@11750
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wenzelm@25388
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end
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wenzelm@25388
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haftmann@29608
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250 |
class sgn =
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haftmann@25062
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fixes sgn :: "'a \<Rightarrow> 'a"
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haftmann@25062
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haftmann@29608
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class ord =
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haftmann@24748
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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haftmann@24748
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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haftmann@23878
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begin
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haftmann@23878
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haftmann@23878
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notation
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haftmann@23878
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less_eq ("op <=") and
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haftmann@23878
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less_eq ("(_/ <= _)" [51, 51] 50) and
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haftmann@23878
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less ("op <") and
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haftmann@23878
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less ("(_/ < _)" [51, 51] 50)
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haftmann@23878
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haftmann@23878
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notation (xsymbols)
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haftmann@23878
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less_eq ("op \<le>") and
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haftmann@23878
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less_eq ("(_/ \<le> _)" [51, 51] 50)
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haftmann@23878
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haftmann@23878
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notation (HTML output)
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haftmann@23878
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less_eq ("op \<le>") and
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haftmann@23878
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less_eq ("(_/ \<le> _)" [51, 51] 50)
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haftmann@23878
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wenzelm@25388
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abbreviation (input)
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wenzelm@25388
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273 |
greater_eq (infix ">=" 50) where
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wenzelm@25388
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"x >= y \<equiv> y <= x"
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wenzelm@25388
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275 |
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haftmann@24842
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276 |
notation (input)
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haftmann@23878
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277 |
greater_eq (infix "\<ge>" 50)
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haftmann@23878
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wenzelm@25388
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279 |
abbreviation (input)
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wenzelm@25388
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greater (infix ">" 50) where
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wenzelm@25388
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"x > y \<equiv> y < x"
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wenzelm@25388
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wenzelm@25388
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end
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wenzelm@25388
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wenzelm@13456
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syntax
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wenzelm@13456
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"_index1" :: index ("\<^sub>1")
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wenzelm@13456
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translations
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wenzelm@14690
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(index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
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wenzelm@13456
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wenzelm@11750
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typed_print_translation {*
|
haftmann@20713
|
291 |
let
|
haftmann@20713
|
292 |
fun tr' c = (c, fn show_sorts => fn T => fn ts =>
|
haftmann@20713
|
293 |
if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
|
haftmann@20713
|
294 |
else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
|
haftmann@22993
|
295 |
in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
|
wenzelm@11750
|
296 |
*} -- {* show types that are presumably too general *}
|
wenzelm@11750
|
297 |
|
wenzelm@11750
|
298 |
|
haftmann@20944
|
299 |
subsection {* Fundamental rules *}
|
haftmann@20944
|
300 |
|
haftmann@20973
|
301 |
subsubsection {* Equality *}
|
haftmann@20944
|
302 |
|
wenzelm@18457
|
303 |
lemma sym: "s = t ==> t = s"
|
wenzelm@18457
|
304 |
by (erule subst) (rule refl)
|
paulson@15411
|
305 |
|
wenzelm@18457
|
306 |
lemma ssubst: "t = s ==> P s ==> P t"
|
wenzelm@18457
|
307 |
by (drule sym) (erule subst)
|
paulson@15411
|
308 |
|
paulson@15411
|
309 |
lemma trans: "[| r=s; s=t |] ==> r=t"
|
wenzelm@18457
|
310 |
by (erule subst)
|
paulson@15411
|
311 |
|
haftmann@20944
|
312 |
lemma meta_eq_to_obj_eq:
|
haftmann@20944
|
313 |
assumes meq: "A == B"
|
haftmann@20944
|
314 |
shows "A = B"
|
haftmann@20944
|
315 |
by (unfold meq) (rule refl)
|
paulson@15411
|
316 |
|
wenzelm@21502
|
317 |
text {* Useful with @{text erule} for proving equalities from known equalities. *}
|
haftmann@20944
|
318 |
(* a = b
|
paulson@15411
|
319 |
| |
|
paulson@15411
|
320 |
c = d *)
|
paulson@15411
|
321 |
lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
|
paulson@15411
|
322 |
apply (rule trans)
|
paulson@15411
|
323 |
apply (rule trans)
|
paulson@15411
|
324 |
apply (rule sym)
|
paulson@15411
|
325 |
apply assumption+
|
paulson@15411
|
326 |
done
|
paulson@15411
|
327 |
|
nipkow@15524
|
328 |
text {* For calculational reasoning: *}
|
nipkow@15524
|
329 |
|
nipkow@15524
|
330 |
lemma forw_subst: "a = b ==> P b ==> P a"
|
nipkow@15524
|
331 |
by (rule ssubst)
|
nipkow@15524
|
332 |
|
nipkow@15524
|
333 |
lemma back_subst: "P a ==> a = b ==> P b"
|
nipkow@15524
|
334 |
by (rule subst)
|
nipkow@15524
|
335 |
|
paulson@15411
|
336 |
|
haftmann@20944
|
337 |
subsubsection {*Congruence rules for application*}
|
paulson@15411
|
338 |
|
paulson@15411
|
339 |
(*similar to AP_THM in Gordon's HOL*)
|
paulson@15411
|
340 |
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
|
paulson@15411
|
341 |
apply (erule subst)
|
paulson@15411
|
342 |
apply (rule refl)
|
paulson@15411
|
343 |
done
|
paulson@15411
|
344 |
|
paulson@15411
|
345 |
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
|
paulson@15411
|
346 |
lemma arg_cong: "x=y ==> f(x)=f(y)"
|
paulson@15411
|
347 |
apply (erule subst)
|
paulson@15411
|
348 |
apply (rule refl)
|
paulson@15411
|
349 |
done
|
paulson@15411
|
350 |
|
paulson@15655
|
351 |
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
|
paulson@15655
|
352 |
apply (erule ssubst)+
|
paulson@15655
|
353 |
apply (rule refl)
|
paulson@15655
|
354 |
done
|
paulson@15655
|
355 |
|
paulson@15411
|
356 |
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
|
paulson@15411
|
357 |
apply (erule subst)+
|
paulson@15411
|
358 |
apply (rule refl)
|
paulson@15411
|
359 |
done
|
paulson@15411
|
360 |
|
paulson@15411
|
361 |
|
haftmann@20944
|
362 |
subsubsection {*Equality of booleans -- iff*}
|
paulson@15411
|
363 |
|
wenzelm@21504
|
364 |
lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
|
wenzelm@21504
|
365 |
by (iprover intro: iff [THEN mp, THEN mp] impI assms)
|
paulson@15411
|
366 |
|
paulson@15411
|
367 |
lemma iffD2: "[| P=Q; Q |] ==> P"
|
wenzelm@18457
|
368 |
by (erule ssubst)
|
paulson@15411
|
369 |
|
paulson@15411
|
370 |
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
|
wenzelm@18457
|
371 |
by (erule iffD2)
|
paulson@15411
|
372 |
|
wenzelm@21504
|
373 |
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
|
wenzelm@21504
|
374 |
by (drule sym) (rule iffD2)
|
wenzelm@21504
|
375 |
|
wenzelm@21504
|
376 |
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
|
wenzelm@21504
|
377 |
by (drule sym) (rule rev_iffD2)
|
paulson@15411
|
378 |
|
paulson@15411
|
379 |
lemma iffE:
|
paulson@15411
|
380 |
assumes major: "P=Q"
|
wenzelm@21504
|
381 |
and minor: "[| P --> Q; Q --> P |] ==> R"
|
wenzelm@18457
|
382 |
shows R
|
wenzelm@18457
|
383 |
by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
|
paulson@15411
|
384 |
|
paulson@15411
|
385 |
|
haftmann@20944
|
386 |
subsubsection {*True*}
|
paulson@15411
|
387 |
|
paulson@15411
|
388 |
lemma TrueI: "True"
|
wenzelm@21504
|
389 |
unfolding True_def by (rule refl)
|
paulson@15411
|
390 |
|
wenzelm@21504
|
391 |
lemma eqTrueI: "P ==> P = True"
|
wenzelm@18457
|
392 |
by (iprover intro: iffI TrueI)
|
paulson@15411
|
393 |
|
wenzelm@21504
|
394 |
lemma eqTrueE: "P = True ==> P"
|
wenzelm@21504
|
395 |
by (erule iffD2) (rule TrueI)
|
paulson@15411
|
396 |
|
paulson@15411
|
397 |
|
haftmann@20944
|
398 |
subsubsection {*Universal quantifier*}
|
paulson@15411
|
399 |
|
wenzelm@21504
|
400 |
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
|
wenzelm@21504
|
401 |
unfolding All_def by (iprover intro: ext eqTrueI assms)
|
paulson@15411
|
402 |
|
paulson@15411
|
403 |
lemma spec: "ALL x::'a. P(x) ==> P(x)"
|
paulson@15411
|
404 |
apply (unfold All_def)
|
paulson@15411
|
405 |
apply (rule eqTrueE)
|
paulson@15411
|
406 |
apply (erule fun_cong)
|
paulson@15411
|
407 |
done
|
paulson@15411
|
408 |
|
paulson@15411
|
409 |
lemma allE:
|
paulson@15411
|
410 |
assumes major: "ALL x. P(x)"
|
wenzelm@21504
|
411 |
and minor: "P(x) ==> R"
|
wenzelm@21504
|
412 |
shows R
|
wenzelm@21504
|
413 |
by (iprover intro: minor major [THEN spec])
|
paulson@15411
|
414 |
|
paulson@15411
|
415 |
lemma all_dupE:
|
paulson@15411
|
416 |
assumes major: "ALL x. P(x)"
|
wenzelm@21504
|
417 |
and minor: "[| P(x); ALL x. P(x) |] ==> R"
|
wenzelm@21504
|
418 |
shows R
|
wenzelm@21504
|
419 |
by (iprover intro: minor major major [THEN spec])
|
paulson@15411
|
420 |
|
paulson@15411
|
421 |
|
wenzelm@21504
|
422 |
subsubsection {* False *}
|
wenzelm@21504
|
423 |
|
wenzelm@21504
|
424 |
text {*
|
wenzelm@21504
|
425 |
Depends upon @{text spec}; it is impossible to do propositional
|
wenzelm@21504
|
426 |
logic before quantifiers!
|
wenzelm@21504
|
427 |
*}
|
paulson@15411
|
428 |
|
paulson@15411
|
429 |
lemma FalseE: "False ==> P"
|
wenzelm@21504
|
430 |
apply (unfold False_def)
|
wenzelm@21504
|
431 |
apply (erule spec)
|
wenzelm@21504
|
432 |
done
|
paulson@15411
|
433 |
|
wenzelm@21504
|
434 |
lemma False_neq_True: "False = True ==> P"
|
wenzelm@21504
|
435 |
by (erule eqTrueE [THEN FalseE])
|
paulson@15411
|
436 |
|
paulson@15411
|
437 |
|
wenzelm@21504
|
438 |
subsubsection {* Negation *}
|
paulson@15411
|
439 |
|
paulson@15411
|
440 |
lemma notI:
|
wenzelm@21504
|
441 |
assumes "P ==> False"
|
paulson@15411
|
442 |
shows "~P"
|
wenzelm@21504
|
443 |
apply (unfold not_def)
|
wenzelm@21504
|
444 |
apply (iprover intro: impI assms)
|
wenzelm@21504
|
445 |
done
|
paulson@15411
|
446 |
|
paulson@15411
|
447 |
lemma False_not_True: "False ~= True"
|
wenzelm@21504
|
448 |
apply (rule notI)
|
wenzelm@21504
|
449 |
apply (erule False_neq_True)
|
wenzelm@21504
|
450 |
done
|
paulson@15411
|
451 |
|
paulson@15411
|
452 |
lemma True_not_False: "True ~= False"
|
wenzelm@21504
|
453 |
apply (rule notI)
|
wenzelm@21504
|
454 |
apply (drule sym)
|
wenzelm@21504
|
455 |
apply (erule False_neq_True)
|
wenzelm@21504
|
456 |
done
|
paulson@15411
|
457 |
|
paulson@15411
|
458 |
lemma notE: "[| ~P; P |] ==> R"
|
wenzelm@21504
|
459 |
apply (unfold not_def)
|
wenzelm@21504
|
460 |
apply (erule mp [THEN FalseE])
|
wenzelm@21504
|
461 |
apply assumption
|
wenzelm@21504
|
462 |
done
|
paulson@15411
|
463 |
|
wenzelm@21504
|
464 |
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
|
wenzelm@21504
|
465 |
by (erule notE [THEN notI]) (erule meta_mp)
|
paulson@15411
|
466 |
|
paulson@15411
|
467 |
|
haftmann@20944
|
468 |
subsubsection {*Implication*}
|
paulson@15411
|
469 |
|
paulson@15411
|
470 |
lemma impE:
|
paulson@15411
|
471 |
assumes "P-->Q" "P" "Q ==> R"
|
paulson@15411
|
472 |
shows "R"
|
wenzelm@23553
|
473 |
by (iprover intro: assms mp)
|
paulson@15411
|
474 |
|
paulson@15411
|
475 |
(* Reduces Q to P-->Q, allowing substitution in P. *)
|
paulson@15411
|
476 |
lemma rev_mp: "[| P; P --> Q |] ==> Q"
|
nipkow@17589
|
477 |
by (iprover intro: mp)
|
paulson@15411
|
478 |
|
paulson@15411
|
479 |
lemma contrapos_nn:
|
paulson@15411
|
480 |
assumes major: "~Q"
|
paulson@15411
|
481 |
and minor: "P==>Q"
|
paulson@15411
|
482 |
shows "~P"
|
nipkow@17589
|
483 |
by (iprover intro: notI minor major [THEN notE])
|
paulson@15411
|
484 |
|
paulson@15411
|
485 |
(*not used at all, but we already have the other 3 combinations *)
|
paulson@15411
|
486 |
lemma contrapos_pn:
|
paulson@15411
|
487 |
assumes major: "Q"
|
paulson@15411
|
488 |
and minor: "P ==> ~Q"
|
paulson@15411
|
489 |
shows "~P"
|
nipkow@17589
|
490 |
by (iprover intro: notI minor major notE)
|
paulson@15411
|
491 |
|
paulson@15411
|
492 |
lemma not_sym: "t ~= s ==> s ~= t"
|
haftmann@21250
|
493 |
by (erule contrapos_nn) (erule sym)
|
haftmann@21250
|
494 |
|
haftmann@21250
|
495 |
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
|
haftmann@21250
|
496 |
by (erule subst, erule ssubst, assumption)
|
paulson@15411
|
497 |
|
paulson@15411
|
498 |
(*still used in HOLCF*)
|
paulson@15411
|
499 |
lemma rev_contrapos:
|
paulson@15411
|
500 |
assumes pq: "P ==> Q"
|
paulson@15411
|
501 |
and nq: "~Q"
|
paulson@15411
|
502 |
shows "~P"
|
paulson@15411
|
503 |
apply (rule nq [THEN contrapos_nn])
|
paulson@15411
|
504 |
apply (erule pq)
|
paulson@15411
|
505 |
done
|
paulson@15411
|
506 |
|
haftmann@20944
|
507 |
subsubsection {*Existential quantifier*}
|
paulson@15411
|
508 |
|
paulson@15411
|
509 |
lemma exI: "P x ==> EX x::'a. P x"
|
paulson@15411
|
510 |
apply (unfold Ex_def)
|
nipkow@17589
|
511 |
apply (iprover intro: allI allE impI mp)
|
paulson@15411
|
512 |
done
|
paulson@15411
|
513 |
|
paulson@15411
|
514 |
lemma exE:
|
paulson@15411
|
515 |
assumes major: "EX x::'a. P(x)"
|
paulson@15411
|
516 |
and minor: "!!x. P(x) ==> Q"
|
paulson@15411
|
517 |
shows "Q"
|
paulson@15411
|
518 |
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
|
nipkow@17589
|
519 |
apply (iprover intro: impI [THEN allI] minor)
|
paulson@15411
|
520 |
done
|
paulson@15411
|
521 |
|
paulson@15411
|
522 |
|
haftmann@20944
|
523 |
subsubsection {*Conjunction*}
|
paulson@15411
|
524 |
|
paulson@15411
|
525 |
lemma conjI: "[| P; Q |] ==> P&Q"
|
paulson@15411
|
526 |
apply (unfold and_def)
|
nipkow@17589
|
527 |
apply (iprover intro: impI [THEN allI] mp)
|
paulson@15411
|
528 |
done
|
paulson@15411
|
529 |
|
paulson@15411
|
530 |
lemma conjunct1: "[| P & Q |] ==> P"
|
paulson@15411
|
531 |
apply (unfold and_def)
|
nipkow@17589
|
532 |
apply (iprover intro: impI dest: spec mp)
|
paulson@15411
|
533 |
done
|
paulson@15411
|
534 |
|
paulson@15411
|
535 |
lemma conjunct2: "[| P & Q |] ==> Q"
|
paulson@15411
|
536 |
apply (unfold and_def)
|
nipkow@17589
|
537 |
apply (iprover intro: impI dest: spec mp)
|
paulson@15411
|
538 |
done
|
paulson@15411
|
539 |
|
paulson@15411
|
540 |
lemma conjE:
|
paulson@15411
|
541 |
assumes major: "P&Q"
|
paulson@15411
|
542 |
and minor: "[| P; Q |] ==> R"
|
paulson@15411
|
543 |
shows "R"
|
paulson@15411
|
544 |
apply (rule minor)
|
paulson@15411
|
545 |
apply (rule major [THEN conjunct1])
|
paulson@15411
|
546 |
apply (rule major [THEN conjunct2])
|
paulson@15411
|
547 |
done
|
paulson@15411
|
548 |
|
paulson@15411
|
549 |
lemma context_conjI:
|
wenzelm@23553
|
550 |
assumes "P" "P ==> Q" shows "P & Q"
|
wenzelm@23553
|
551 |
by (iprover intro: conjI assms)
|
paulson@15411
|
552 |
|
paulson@15411
|
553 |
|
haftmann@20944
|
554 |
subsubsection {*Disjunction*}
|
paulson@15411
|
555 |
|
paulson@15411
|
556 |
lemma disjI1: "P ==> P|Q"
|
paulson@15411
|
557 |
apply (unfold or_def)
|
nipkow@17589
|
558 |
apply (iprover intro: allI impI mp)
|
paulson@15411
|
559 |
done
|
paulson@15411
|
560 |
|
paulson@15411
|
561 |
lemma disjI2: "Q ==> P|Q"
|
paulson@15411
|
562 |
apply (unfold or_def)
|
nipkow@17589
|
563 |
apply (iprover intro: allI impI mp)
|
paulson@15411
|
564 |
done
|
paulson@15411
|
565 |
|
paulson@15411
|
566 |
lemma disjE:
|
paulson@15411
|
567 |
assumes major: "P|Q"
|
paulson@15411
|
568 |
and minorP: "P ==> R"
|
paulson@15411
|
569 |
and minorQ: "Q ==> R"
|
paulson@15411
|
570 |
shows "R"
|
nipkow@17589
|
571 |
by (iprover intro: minorP minorQ impI
|
paulson@15411
|
572 |
major [unfolded or_def, THEN spec, THEN mp, THEN mp])
|
paulson@15411
|
573 |
|
paulson@15411
|
574 |
|
haftmann@20944
|
575 |
subsubsection {*Classical logic*}
|
paulson@15411
|
576 |
|
paulson@15411
|
577 |
lemma classical:
|
paulson@15411
|
578 |
assumes prem: "~P ==> P"
|
paulson@15411
|
579 |
shows "P"
|
paulson@15411
|
580 |
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
|
paulson@15411
|
581 |
apply assumption
|
paulson@15411
|
582 |
apply (rule notI [THEN prem, THEN eqTrueI])
|
paulson@15411
|
583 |
apply (erule subst)
|
paulson@15411
|
584 |
apply assumption
|
paulson@15411
|
585 |
done
|
paulson@15411
|
586 |
|
paulson@15411
|
587 |
lemmas ccontr = FalseE [THEN classical, standard]
|
paulson@15411
|
588 |
|
paulson@15411
|
589 |
(*notE with premises exchanged; it discharges ~R so that it can be used to
|
paulson@15411
|
590 |
make elimination rules*)
|
paulson@15411
|
591 |
lemma rev_notE:
|
paulson@15411
|
592 |
assumes premp: "P"
|
paulson@15411
|
593 |
and premnot: "~R ==> ~P"
|
paulson@15411
|
594 |
shows "R"
|
paulson@15411
|
595 |
apply (rule ccontr)
|
paulson@15411
|
596 |
apply (erule notE [OF premnot premp])
|
paulson@15411
|
597 |
done
|
paulson@15411
|
598 |
|
paulson@15411
|
599 |
(*Double negation law*)
|
paulson@15411
|
600 |
lemma notnotD: "~~P ==> P"
|
paulson@15411
|
601 |
apply (rule classical)
|
paulson@15411
|
602 |
apply (erule notE)
|
paulson@15411
|
603 |
apply assumption
|
paulson@15411
|
604 |
done
|
paulson@15411
|
605 |
|
paulson@15411
|
606 |
lemma contrapos_pp:
|
paulson@15411
|
607 |
assumes p1: "Q"
|
paulson@15411
|
608 |
and p2: "~P ==> ~Q"
|
paulson@15411
|
609 |
shows "P"
|
nipkow@17589
|
610 |
by (iprover intro: classical p1 p2 notE)
|
paulson@15411
|
611 |
|
paulson@15411
|
612 |
|
haftmann@20944
|
613 |
subsubsection {*Unique existence*}
|
paulson@15411
|
614 |
|
paulson@15411
|
615 |
lemma ex1I:
|
wenzelm@23553
|
616 |
assumes "P a" "!!x. P(x) ==> x=a"
|
paulson@15411
|
617 |
shows "EX! x. P(x)"
|
wenzelm@23553
|
618 |
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
|
paulson@15411
|
619 |
|
paulson@15411
|
620 |
text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
|
paulson@15411
|
621 |
lemma ex_ex1I:
|
paulson@15411
|
622 |
assumes ex_prem: "EX x. P(x)"
|
paulson@15411
|
623 |
and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
|
paulson@15411
|
624 |
shows "EX! x. P(x)"
|
nipkow@17589
|
625 |
by (iprover intro: ex_prem [THEN exE] ex1I eq)
|
paulson@15411
|
626 |
|
paulson@15411
|
627 |
lemma ex1E:
|
paulson@15411
|
628 |
assumes major: "EX! x. P(x)"
|
paulson@15411
|
629 |
and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
|
paulson@15411
|
630 |
shows "R"
|
paulson@15411
|
631 |
apply (rule major [unfolded Ex1_def, THEN exE])
|
paulson@15411
|
632 |
apply (erule conjE)
|
nipkow@17589
|
633 |
apply (iprover intro: minor)
|
paulson@15411
|
634 |
done
|
paulson@15411
|
635 |
|
paulson@15411
|
636 |
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
|
paulson@15411
|
637 |
apply (erule ex1E)
|
paulson@15411
|
638 |
apply (rule exI)
|
paulson@15411
|
639 |
apply assumption
|
paulson@15411
|
640 |
done
|
paulson@15411
|
641 |
|
paulson@15411
|
642 |
|
haftmann@20944
|
643 |
subsubsection {*THE: definite description operator*}
|
paulson@15411
|
644 |
|
paulson@15411
|
645 |
lemma the_equality:
|
paulson@15411
|
646 |
assumes prema: "P a"
|
paulson@15411
|
647 |
and premx: "!!x. P x ==> x=a"
|
paulson@15411
|
648 |
shows "(THE x. P x) = a"
|
paulson@15411
|
649 |
apply (rule trans [OF _ the_eq_trivial])
|
paulson@15411
|
650 |
apply (rule_tac f = "The" in arg_cong)
|
paulson@15411
|
651 |
apply (rule ext)
|
paulson@15411
|
652 |
apply (rule iffI)
|
paulson@15411
|
653 |
apply (erule premx)
|
paulson@15411
|
654 |
apply (erule ssubst, rule prema)
|
paulson@15411
|
655 |
done
|
paulson@15411
|
656 |
|
paulson@15411
|
657 |
lemma theI:
|
paulson@15411
|
658 |
assumes "P a" and "!!x. P x ==> x=a"
|
paulson@15411
|
659 |
shows "P (THE x. P x)"
|
wenzelm@23553
|
660 |
by (iprover intro: assms the_equality [THEN ssubst])
|
paulson@15411
|
661 |
|
paulson@15411
|
662 |
lemma theI': "EX! x. P x ==> P (THE x. P x)"
|
paulson@15411
|
663 |
apply (erule ex1E)
|
paulson@15411
|
664 |
apply (erule theI)
|
paulson@15411
|
665 |
apply (erule allE)
|
paulson@15411
|
666 |
apply (erule mp)
|
paulson@15411
|
667 |
apply assumption
|
paulson@15411
|
668 |
done
|
paulson@15411
|
669 |
|
paulson@15411
|
670 |
(*Easier to apply than theI: only one occurrence of P*)
|
paulson@15411
|
671 |
lemma theI2:
|
paulson@15411
|
672 |
assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
|
paulson@15411
|
673 |
shows "Q (THE x. P x)"
|
wenzelm@23553
|
674 |
by (iprover intro: assms theI)
|
paulson@15411
|
675 |
|
nipkow@24553
|
676 |
lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
|
nipkow@24553
|
677 |
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
|
nipkow@24553
|
678 |
elim:allE impE)
|
nipkow@24553
|
679 |
|
wenzelm@18697
|
680 |
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
|
paulson@15411
|
681 |
apply (rule the_equality)
|
paulson@15411
|
682 |
apply assumption
|
paulson@15411
|
683 |
apply (erule ex1E)
|
paulson@15411
|
684 |
apply (erule all_dupE)
|
paulson@15411
|
685 |
apply (drule mp)
|
paulson@15411
|
686 |
apply assumption
|
paulson@15411
|
687 |
apply (erule ssubst)
|
paulson@15411
|
688 |
apply (erule allE)
|
paulson@15411
|
689 |
apply (erule mp)
|
paulson@15411
|
690 |
apply assumption
|
paulson@15411
|
691 |
done
|
paulson@15411
|
692 |
|
paulson@15411
|
693 |
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
|
paulson@15411
|
694 |
apply (rule the_equality)
|
paulson@15411
|
695 |
apply (rule refl)
|
paulson@15411
|
696 |
apply (erule sym)
|
paulson@15411
|
697 |
done
|
paulson@15411
|
698 |
|
paulson@15411
|
699 |
|
haftmann@20944
|
700 |
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
|
paulson@15411
|
701 |
|
paulson@15411
|
702 |
lemma disjCI:
|
paulson@15411
|
703 |
assumes "~Q ==> P" shows "P|Q"
|
paulson@15411
|
704 |
apply (rule classical)
|
wenzelm@23553
|
705 |
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
|
paulson@15411
|
706 |
done
|
paulson@15411
|
707 |
|
paulson@15411
|
708 |
lemma excluded_middle: "~P | P"
|
nipkow@17589
|
709 |
by (iprover intro: disjCI)
|
paulson@15411
|
710 |
|
haftmann@20944
|
711 |
text {*
|
haftmann@20944
|
712 |
case distinction as a natural deduction rule.
|
haftmann@20944
|
713 |
Note that @{term "~P"} is the second case, not the first
|
haftmann@20944
|
714 |
*}
|
wenzelm@27126
|
715 |
lemma case_split [case_names True False]:
|
paulson@15411
|
716 |
assumes prem1: "P ==> Q"
|
paulson@15411
|
717 |
and prem2: "~P ==> Q"
|
paulson@15411
|
718 |
shows "Q"
|
paulson@15411
|
719 |
apply (rule excluded_middle [THEN disjE])
|
paulson@15411
|
720 |
apply (erule prem2)
|
paulson@15411
|
721 |
apply (erule prem1)
|
paulson@15411
|
722 |
done
|
wenzelm@27126
|
723 |
|
paulson@15411
|
724 |
(*Classical implies (-->) elimination. *)
|
paulson@15411
|
725 |
lemma impCE:
|
paulson@15411
|
726 |
assumes major: "P-->Q"
|
paulson@15411
|
727 |
and minor: "~P ==> R" "Q ==> R"
|
paulson@15411
|
728 |
shows "R"
|
paulson@15411
|
729 |
apply (rule excluded_middle [of P, THEN disjE])
|
nipkow@17589
|
730 |
apply (iprover intro: minor major [THEN mp])+
|
paulson@15411
|
731 |
done
|
paulson@15411
|
732 |
|
paulson@15411
|
733 |
(*This version of --> elimination works on Q before P. It works best for
|
paulson@15411
|
734 |
those cases in which P holds "almost everywhere". Can't install as
|
paulson@15411
|
735 |
default: would break old proofs.*)
|
paulson@15411
|
736 |
lemma impCE':
|
paulson@15411
|
737 |
assumes major: "P-->Q"
|
paulson@15411
|
738 |
and minor: "Q ==> R" "~P ==> R"
|
paulson@15411
|
739 |
shows "R"
|
paulson@15411
|
740 |
apply (rule excluded_middle [of P, THEN disjE])
|
nipkow@17589
|
741 |
apply (iprover intro: minor major [THEN mp])+
|
paulson@15411
|
742 |
done
|
paulson@15411
|
743 |
|
paulson@15411
|
744 |
(*Classical <-> elimination. *)
|
paulson@15411
|
745 |
lemma iffCE:
|
paulson@15411
|
746 |
assumes major: "P=Q"
|
paulson@15411
|
747 |
and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
|
paulson@15411
|
748 |
shows "R"
|
paulson@15411
|
749 |
apply (rule major [THEN iffE])
|
nipkow@17589
|
750 |
apply (iprover intro: minor elim: impCE notE)
|
paulson@15411
|
751 |
done
|
paulson@15411
|
752 |
|
paulson@15411
|
753 |
lemma exCI:
|
paulson@15411
|
754 |
assumes "ALL x. ~P(x) ==> P(a)"
|
paulson@15411
|
755 |
shows "EX x. P(x)"
|
paulson@15411
|
756 |
apply (rule ccontr)
|
wenzelm@23553
|
757 |
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
|
paulson@15411
|
758 |
done
|
paulson@15411
|
759 |
|
paulson@15411
|
760 |
|
wenzelm@12386
|
761 |
subsubsection {* Intuitionistic Reasoning *}
|
wenzelm@12386
|
762 |
|
wenzelm@12386
|
763 |
lemma impE':
|
wenzelm@12937
|
764 |
assumes 1: "P --> Q"
|
wenzelm@12937
|
765 |
and 2: "Q ==> R"
|
wenzelm@12937
|
766 |
and 3: "P --> Q ==> P"
|
wenzelm@12937
|
767 |
shows R
|
wenzelm@12386
|
768 |
proof -
|
wenzelm@12386
|
769 |
from 3 and 1 have P .
|
wenzelm@12386
|
770 |
with 1 have Q by (rule impE)
|
wenzelm@12386
|
771 |
with 2 show R .
|
wenzelm@12386
|
772 |
qed
|
wenzelm@12386
|
773 |
|
wenzelm@12386
|
774 |
lemma allE':
|
wenzelm@12937
|
775 |
assumes 1: "ALL x. P x"
|
wenzelm@12937
|
776 |
and 2: "P x ==> ALL x. P x ==> Q"
|
wenzelm@12937
|
777 |
shows Q
|
wenzelm@12386
|
778 |
proof -
|
wenzelm@12386
|
779 |
from 1 have "P x" by (rule spec)
|
wenzelm@12386
|
780 |
from this and 1 show Q by (rule 2)
|
wenzelm@12386
|
781 |
qed
|
wenzelm@12386
|
782 |
|
wenzelm@12937
|
783 |
lemma notE':
|
wenzelm@12937
|
784 |
assumes 1: "~ P"
|
wenzelm@12937
|
785 |
and 2: "~ P ==> P"
|
wenzelm@12937
|
786 |
shows R
|
wenzelm@12386
|
787 |
proof -
|
wenzelm@12386
|
788 |
from 2 and 1 have P .
|
wenzelm@12386
|
789 |
with 1 show R by (rule notE)
|
wenzelm@12386
|
790 |
qed
|
wenzelm@12386
|
791 |
|
dixon@22444
|
792 |
lemma TrueE: "True ==> P ==> P" .
|
dixon@22444
|
793 |
lemma notFalseE: "~ False ==> P ==> P" .
|
dixon@22444
|
794 |
|
dixon@22467
|
795 |
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
|
wenzelm@15801
|
796 |
and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
|
wenzelm@15801
|
797 |
and [Pure.elim 2] = allE notE' impE'
|
wenzelm@15801
|
798 |
and [Pure.intro] = exI disjI2 disjI1
|
wenzelm@12386
|
799 |
|
wenzelm@12386
|
800 |
lemmas [trans] = trans
|
wenzelm@12386
|
801 |
and [sym] = sym not_sym
|
wenzelm@15801
|
802 |
and [Pure.elim?] = iffD1 iffD2 impE
|
wenzelm@11438
|
803 |
|
haftmann@28952
|
804 |
use "Tools/hologic.ML"
|
wenzelm@23553
|
805 |
|
wenzelm@11750
|
806 |
|
wenzelm@11750
|
807 |
subsubsection {* Atomizing meta-level connectives *}
|
wenzelm@11750
|
808 |
|
haftmann@28513
|
809 |
axiomatization where
|
haftmann@28513
|
810 |
eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
|
haftmann@28513
|
811 |
|
wenzelm@11750
|
812 |
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
|
wenzelm@12003
|
813 |
proof
|
wenzelm@9488
|
814 |
assume "!!x. P x"
|
wenzelm@23389
|
815 |
then show "ALL x. P x" ..
|
wenzelm@9488
|
816 |
next
|
wenzelm@9488
|
817 |
assume "ALL x. P x"
|
wenzelm@23553
|
818 |
then show "!!x. P x" by (rule allE)
|
wenzelm@9488
|
819 |
qed
|
wenzelm@9488
|
820 |
|
wenzelm@11750
|
821 |
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
|
wenzelm@12003
|
822 |
proof
|
wenzelm@9488
|
823 |
assume r: "A ==> B"
|
wenzelm@10383
|
824 |
show "A --> B" by (rule impI) (rule r)
|
wenzelm@9488
|
825 |
next
|
wenzelm@9488
|
826 |
assume "A --> B" and A
|
wenzelm@23553
|
827 |
then show B by (rule mp)
|
wenzelm@9488
|
828 |
qed
|
wenzelm@9488
|
829 |
|
paulson@14749
|
830 |
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
|
paulson@14749
|
831 |
proof
|
paulson@14749
|
832 |
assume r: "A ==> False"
|
paulson@14749
|
833 |
show "~A" by (rule notI) (rule r)
|
paulson@14749
|
834 |
next
|
paulson@14749
|
835 |
assume "~A" and A
|
wenzelm@23553
|
836 |
then show False by (rule notE)
|
paulson@14749
|
837 |
qed
|
paulson@14749
|
838 |
|
wenzelm@11750
|
839 |
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
|
wenzelm@12003
|
840 |
proof
|
wenzelm@10432
|
841 |
assume "x == y"
|
wenzelm@23553
|
842 |
show "x = y" by (unfold `x == y`) (rule refl)
|
wenzelm@10432
|
843 |
next
|
wenzelm@10432
|
844 |
assume "x = y"
|
wenzelm@23553
|
845 |
then show "x == y" by (rule eq_reflection)
|
wenzelm@10432
|
846 |
qed
|
wenzelm@10432
|
847 |
|
wenzelm@28856
|
848 |
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
|
wenzelm@12003
|
849 |
proof
|
wenzelm@28856
|
850 |
assume conj: "A &&& B"
|
wenzelm@19121
|
851 |
show "A & B"
|
wenzelm@19121
|
852 |
proof (rule conjI)
|
wenzelm@19121
|
853 |
from conj show A by (rule conjunctionD1)
|
wenzelm@19121
|
854 |
from conj show B by (rule conjunctionD2)
|
wenzelm@19121
|
855 |
qed
|
wenzelm@11953
|
856 |
next
|
wenzelm@19121
|
857 |
assume conj: "A & B"
|
wenzelm@28856
|
858 |
show "A &&& B"
|
wenzelm@19121
|
859 |
proof -
|
wenzelm@19121
|
860 |
from conj show A ..
|
wenzelm@19121
|
861 |
from conj show B ..
|
wenzelm@11953
|
862 |
qed
|
wenzelm@11953
|
863 |
qed
|
wenzelm@11953
|
864 |
|
wenzelm@12386
|
865 |
lemmas [symmetric, rulify] = atomize_all atomize_imp
|
wenzelm@18832
|
866 |
and [symmetric, defn] = atomize_all atomize_imp atomize_eq
|
wenzelm@12386
|
867 |
|
wenzelm@11750
|
868 |
|
krauss@26580
|
869 |
subsubsection {* Atomizing elimination rules *}
|
krauss@26580
|
870 |
|
krauss@26580
|
871 |
setup AtomizeElim.setup
|
krauss@26580
|
872 |
|
krauss@26580
|
873 |
lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
|
krauss@26580
|
874 |
by rule iprover+
|
krauss@26580
|
875 |
|
krauss@26580
|
876 |
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
|
krauss@26580
|
877 |
by rule iprover+
|
krauss@26580
|
878 |
|
krauss@26580
|
879 |
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
|
krauss@26580
|
880 |
by rule iprover+
|
krauss@26580
|
881 |
|
krauss@26580
|
882 |
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
|
krauss@26580
|
883 |
|
krauss@26580
|
884 |
|
haftmann@20944
|
885 |
subsection {* Package setup *}
|
haftmann@20944
|
886 |
|
wenzelm@11750
|
887 |
subsubsection {* Classical Reasoner setup *}
|
wenzelm@9529
|
888 |
|
wenzelm@26411
|
889 |
lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
|
wenzelm@26411
|
890 |
by (rule classical) iprover
|
wenzelm@26411
|
891 |
|
wenzelm@26411
|
892 |
lemma swap: "~ P ==> (~ R ==> P) ==> R"
|
wenzelm@26411
|
893 |
by (rule classical) iprover
|
wenzelm@26411
|
894 |
|
haftmann@20944
|
895 |
lemma thin_refl:
|
haftmann@20944
|
896 |
"\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
|
haftmann@20944
|
897 |
|
haftmann@21151
|
898 |
ML {*
|
haftmann@21151
|
899 |
structure Hypsubst = HypsubstFun(
|
haftmann@21151
|
900 |
struct
|
haftmann@21151
|
901 |
structure Simplifier = Simplifier
|
wenzelm@21218
|
902 |
val dest_eq = HOLogic.dest_eq
|
haftmann@21151
|
903 |
val dest_Trueprop = HOLogic.dest_Trueprop
|
haftmann@21151
|
904 |
val dest_imp = HOLogic.dest_imp
|
wenzelm@26411
|
905 |
val eq_reflection = @{thm eq_reflection}
|
wenzelm@26411
|
906 |
val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
|
wenzelm@26411
|
907 |
val imp_intr = @{thm impI}
|
wenzelm@26411
|
908 |
val rev_mp = @{thm rev_mp}
|
wenzelm@26411
|
909 |
val subst = @{thm subst}
|
wenzelm@26411
|
910 |
val sym = @{thm sym}
|
wenzelm@22129
|
911 |
val thin_refl = @{thm thin_refl};
|
krauss@27572
|
912 |
val prop_subst = @{lemma "PROP P t ==> PROP prop (x = t ==> PROP P x)"
|
krauss@27572
|
913 |
by (unfold prop_def) (drule eq_reflection, unfold)}
|
haftmann@21151
|
914 |
end);
|
wenzelm@21671
|
915 |
open Hypsubst;
|
haftmann@21151
|
916 |
|
haftmann@21151
|
917 |
structure Classical = ClassicalFun(
|
haftmann@21151
|
918 |
struct
|
wenzelm@26411
|
919 |
val imp_elim = @{thm imp_elim}
|
wenzelm@26411
|
920 |
val not_elim = @{thm notE}
|
wenzelm@26411
|
921 |
val swap = @{thm swap}
|
wenzelm@26411
|
922 |
val classical = @{thm classical}
|
haftmann@21151
|
923 |
val sizef = Drule.size_of_thm
|
haftmann@21151
|
924 |
val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
|
haftmann@21151
|
925 |
end);
|
haftmann@21151
|
926 |
|
haftmann@21151
|
927 |
structure BasicClassical: BASIC_CLASSICAL = Classical;
|
wenzelm@21671
|
928 |
open BasicClassical;
|
wenzelm@22129
|
929 |
|
wenzelm@27338
|
930 |
ML_Antiquote.value "claset"
|
wenzelm@27338
|
931 |
(Scan.succeed "Classical.local_claset_of (ML_Context.the_local_context ())");
|
wenzelm@24035
|
932 |
|
wenzelm@24035
|
933 |
structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
|
paulson@24286
|
934 |
|
paulson@24286
|
935 |
structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
|
haftmann@21151
|
936 |
*}
|
haftmann@21151
|
937 |
|
wenzelm@25388
|
938 |
text {*ResBlacklist holds theorems blacklisted to sledgehammer.
|
paulson@24286
|
939 |
These theorems typically produce clauses that are prolific (match too many equality or
|
wenzelm@25388
|
940 |
membership literals) and relate to seldom-used facts. Some duplicate other rules.*}
|
paulson@24286
|
941 |
|
haftmann@21009
|
942 |
setup {*
|
haftmann@21009
|
943 |
let
|
haftmann@21009
|
944 |
(*prevent substitution on bool*)
|
haftmann@21009
|
945 |
fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
|
haftmann@21009
|
946 |
Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
|
haftmann@21009
|
947 |
(nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
|
haftmann@21009
|
948 |
in
|
haftmann@21151
|
949 |
Hypsubst.hypsubst_setup
|
haftmann@21151
|
950 |
#> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
|
haftmann@21151
|
951 |
#> Classical.setup
|
haftmann@21151
|
952 |
#> ResAtpset.setup
|
paulson@24286
|
953 |
#> ResBlacklist.setup
|
haftmann@21009
|
954 |
end
|
haftmann@21009
|
955 |
*}
|
haftmann@21009
|
956 |
|
haftmann@21009
|
957 |
declare iffI [intro!]
|
haftmann@21009
|
958 |
and notI [intro!]
|
haftmann@21009
|
959 |
and impI [intro!]
|
haftmann@21009
|
960 |
and disjCI [intro!]
|
haftmann@21009
|
961 |
and conjI [intro!]
|
haftmann@21009
|
962 |
and TrueI [intro!]
|
haftmann@21009
|
963 |
and refl [intro!]
|
haftmann@21009
|
964 |
|
haftmann@21009
|
965 |
declare iffCE [elim!]
|
haftmann@21009
|
966 |
and FalseE [elim!]
|
haftmann@21009
|
967 |
and impCE [elim!]
|
haftmann@21009
|
968 |
and disjE [elim!]
|
haftmann@21009
|
969 |
and conjE [elim!]
|
haftmann@21009
|
970 |
and conjE [elim!]
|
haftmann@21009
|
971 |
|
haftmann@21009
|
972 |
declare ex_ex1I [intro!]
|
haftmann@21009
|
973 |
and allI [intro!]
|
haftmann@21009
|
974 |
and the_equality [intro]
|
haftmann@21009
|
975 |
and exI [intro]
|
haftmann@21009
|
976 |
|
haftmann@21009
|
977 |
declare exE [elim!]
|
haftmann@21009
|
978 |
allE [elim]
|
haftmann@21009
|
979 |
|
wenzelm@22377
|
980 |
ML {* val HOL_cs = @{claset} *}
|
wenzelm@11977
|
981 |
|
wenzelm@20223
|
982 |
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
|
wenzelm@20223
|
983 |
apply (erule swap)
|
wenzelm@20223
|
984 |
apply (erule (1) meta_mp)
|
wenzelm@20223
|
985 |
done
|
wenzelm@10383
|
986 |
|
wenzelm@18689
|
987 |
declare ex_ex1I [rule del, intro! 2]
|
wenzelm@18689
|
988 |
and ex1I [intro]
|
wenzelm@18689
|
989 |
|
wenzelm@12386
|
990 |
lemmas [intro?] = ext
|
wenzelm@12386
|
991 |
and [elim?] = ex1_implies_ex
|
wenzelm@11977
|
992 |
|
haftmann@20944
|
993 |
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
|
haftmann@20973
|
994 |
lemma alt_ex1E [elim!]:
|
haftmann@20944
|
995 |
assumes major: "\<exists>!x. P x"
|
haftmann@20944
|
996 |
and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
|
haftmann@20944
|
997 |
shows R
|
haftmann@20944
|
998 |
apply (rule ex1E [OF major])
|
haftmann@20944
|
999 |
apply (rule prem)
|
wenzelm@22129
|
1000 |
apply (tactic {* ares_tac @{thms allI} 1 *})+
|
wenzelm@22129
|
1001 |
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
|
wenzelm@22129
|
1002 |
apply iprover
|
wenzelm@22129
|
1003 |
done
|
haftmann@20944
|
1004 |
|
haftmann@21151
|
1005 |
ML {*
|
wenzelm@25388
|
1006 |
structure Blast = BlastFun
|
wenzelm@25388
|
1007 |
(
|
haftmann@21151
|
1008 |
type claset = Classical.claset
|
haftmann@22744
|
1009 |
val equality_name = @{const_name "op ="}
|
haftmann@22993
|
1010 |
val not_name = @{const_name Not}
|
wenzelm@26411
|
1011 |
val notE = @{thm notE}
|
wenzelm@26411
|
1012 |
val ccontr = @{thm ccontr}
|
haftmann@21151
|
1013 |
val contr_tac = Classical.contr_tac
|
haftmann@21151
|
1014 |
val dup_intr = Classical.dup_intr
|
haftmann@21151
|
1015 |
val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
|
wenzelm@21671
|
1016 |
val claset = Classical.claset
|
haftmann@21151
|
1017 |
val rep_cs = Classical.rep_cs
|
haftmann@21151
|
1018 |
val cla_modifiers = Classical.cla_modifiers
|
haftmann@21151
|
1019 |
val cla_meth' = Classical.cla_meth'
|
wenzelm@25388
|
1020 |
);
|
wenzelm@21671
|
1021 |
val Blast_tac = Blast.Blast_tac;
|
wenzelm@21671
|
1022 |
val blast_tac = Blast.blast_tac;
|
haftmann@20944
|
1023 |
*}
|
haftmann@20944
|
1024 |
|
haftmann@21151
|
1025 |
setup Blast.setup
|
haftmann@21151
|
1026 |
|
haftmann@20944
|
1027 |
|
haftmann@20944
|
1028 |
subsubsection {* Simplifier *}
|
wenzelm@12281
|
1029 |
|
wenzelm@12281
|
1030 |
lemma eta_contract_eq: "(%s. f s) = f" ..
|
wenzelm@12281
|
1031 |
|
wenzelm@12281
|
1032 |
lemma simp_thms:
|
wenzelm@12937
|
1033 |
shows not_not: "(~ ~ P) = P"
|
nipkow@15354
|
1034 |
and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
|
wenzelm@12937
|
1035 |
and
|
berghofe@12436
|
1036 |
"(P ~= Q) = (P = (~Q))"
|
berghofe@12436
|
1037 |
"(P | ~P) = True" "(~P | P) = True"
|
wenzelm@12281
|
1038 |
"(x = x) = True"
|
haftmann@20944
|
1039 |
and not_True_eq_False: "(\<not> True) = False"
|
haftmann@20944
|
1040 |
and not_False_eq_True: "(\<not> False) = True"
|
haftmann@20944
|
1041 |
and
|
berghofe@12436
|
1042 |
"(~P) ~= P" "P ~= (~P)"
|
haftmann@20944
|
1043 |
"(True=P) = P"
|
haftmann@20944
|
1044 |
and eq_True: "(P = True) = P"
|
haftmann@20944
|
1045 |
and "(False=P) = (~P)"
|
haftmann@20944
|
1046 |
and eq_False: "(P = False) = (\<not> P)"
|
haftmann@20944
|
1047 |
and
|
wenzelm@12281
|
1048 |
"(True --> P) = P" "(False --> P) = True"
|
wenzelm@12281
|
1049 |
"(P --> True) = True" "(P --> P) = True"
|
wenzelm@12281
|
1050 |
"(P --> False) = (~P)" "(P --> ~P) = (~P)"
|
wenzelm@12281
|
1051 |
"(P & True) = P" "(True & P) = P"
|
wenzelm@12281
|
1052 |
"(P & False) = False" "(False & P) = False"
|
wenzelm@12281
|
1053 |
"(P & P) = P" "(P & (P & Q)) = (P & Q)"
|
wenzelm@12281
|
1054 |
"(P & ~P) = False" "(~P & P) = False"
|
wenzelm@12281
|
1055 |
"(P | True) = True" "(True | P) = True"
|
wenzelm@12281
|
1056 |
"(P | False) = P" "(False | P) = P"
|
berghofe@12436
|
1057 |
"(P | P) = P" "(P | (P | Q)) = (P | Q)" and
|
wenzelm@12281
|
1058 |
"(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
|
wenzelm@12281
|
1059 |
-- {* needed for the one-point-rule quantifier simplification procs *}
|
wenzelm@12281
|
1060 |
-- {* essential for termination!! *} and
|
wenzelm@12281
|
1061 |
"!!P. (EX x. x=t & P(x)) = P(t)"
|
wenzelm@12281
|
1062 |
"!!P. (EX x. t=x & P(x)) = P(t)"
|
wenzelm@12281
|
1063 |
"!!P. (ALL x. x=t --> P(x)) = P(t)"
|
wenzelm@12937
|
1064 |
"!!P. (ALL x. t=x --> P(x)) = P(t)"
|
nipkow@17589
|
1065 |
by (blast, blast, blast, blast, blast, iprover+)
|
wenzelm@13421
|
1066 |
|
paulson@14201
|
1067 |
lemma disj_absorb: "(A | A) = A"
|
paulson@14201
|
1068 |
by blast
|
paulson@14201
|
1069 |
|
paulson@14201
|
1070 |
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
|
paulson@14201
|
1071 |
by blast
|
paulson@14201
|
1072 |
|
paulson@14201
|
1073 |
lemma conj_absorb: "(A & A) = A"
|
paulson@14201
|
1074 |
by blast
|
paulson@14201
|
1075 |
|
paulson@14201
|
1076 |
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
|
paulson@14201
|
1077 |
by blast
|
paulson@14201
|
1078 |
|
wenzelm@12281
|
1079 |
lemma eq_ac:
|
wenzelm@12937
|
1080 |
shows eq_commute: "(a=b) = (b=a)"
|
wenzelm@12937
|
1081 |
and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
|
nipkow@17589
|
1082 |
and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
|
nipkow@17589
|
1083 |
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
|
wenzelm@12281
|
1084 |
|
wenzelm@12281
|
1085 |
lemma conj_comms:
|
wenzelm@12937
|
1086 |
shows conj_commute: "(P&Q) = (Q&P)"
|
nipkow@17589
|
1087 |
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
|
nipkow@17589
|
1088 |
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
|
wenzelm@12281
|
1089 |
|
paulson@19174
|
1090 |
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
|
paulson@19174
|
1091 |
|
wenzelm@12281
|
1092 |
lemma disj_comms:
|
wenzelm@12937
|
1093 |
shows disj_commute: "(P|Q) = (Q|P)"
|
nipkow@17589
|
1094 |
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
|
nipkow@17589
|
1095 |
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
|
wenzelm@12281
|
1096 |
|
paulson@19174
|
1097 |
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
|
paulson@19174
|
1098 |
|
nipkow@17589
|
1099 |
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
|
nipkow@17589
|
1100 |
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
|
wenzelm@12281
|
1101 |
|
nipkow@17589
|
1102 |
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
|
nipkow@17589
|
1103 |
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
|
wenzelm@12281
|
1104 |
|
nipkow@17589
|
1105 |
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
|
nipkow@17589
|
1106 |
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
|
nipkow@17589
|
1107 |
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
|
wenzelm@12281
|
1108 |
|
wenzelm@12281
|
1109 |
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
|
wenzelm@12281
|
1110 |
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
|
wenzelm@12281
|
1111 |
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
|
wenzelm@12281
|
1112 |
|
wenzelm@12281
|
1113 |
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
|
wenzelm@12281
|
1114 |
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
|
wenzelm@12281
|
1115 |
|
haftmann@21151
|
1116 |
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
|
haftmann@21151
|
1117 |
by iprover
|
haftmann@21151
|
1118 |
|
nipkow@17589
|
1119 |
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
|
wenzelm@12281
|
1120 |
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
|
wenzelm@12281
|
1121 |
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
|
wenzelm@12281
|
1122 |
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
|
wenzelm@12281
|
1123 |
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
|
wenzelm@12281
|
1124 |
lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
|
wenzelm@12281
|
1125 |
by blast
|
wenzelm@12281
|
1126 |
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
|
wenzelm@12281
|
1127 |
|
nipkow@17589
|
1128 |
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
|
wenzelm@12281
|
1129 |
|
wenzelm@12281
|
1130 |
|
wenzelm@12281
|
1131 |
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
|
wenzelm@12281
|
1132 |
-- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
|
wenzelm@12281
|
1133 |
-- {* cases boil down to the same thing. *}
|
wenzelm@12281
|
1134 |
by blast
|
wenzelm@12281
|
1135 |
|
wenzelm@12281
|
1136 |
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
|
wenzelm@12281
|
1137 |
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
|
nipkow@17589
|
1138 |
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
|
nipkow@17589
|
1139 |
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
|
chaieb@23403
|
1140 |
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
|
wenzelm@12281
|
1141 |
|
paulson@24286
|
1142 |
declare All_def [noatp]
|
paulson@24286
|
1143 |
|
nipkow@17589
|
1144 |
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
|
nipkow@17589
|
1145 |
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
|
wenzelm@12281
|
1146 |
|
wenzelm@12281
|
1147 |
text {*
|
wenzelm@12281
|
1148 |
\medskip The @{text "&"} congruence rule: not included by default!
|
wenzelm@12281
|
1149 |
May slow rewrite proofs down by as much as 50\% *}
|
wenzelm@12281
|
1150 |
|
wenzelm@12281
|
1151 |
lemma conj_cong:
|
wenzelm@12281
|
1152 |
"(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
|
nipkow@17589
|
1153 |
by iprover
|
wenzelm@12281
|
1154 |
|
wenzelm@12281
|
1155 |
lemma rev_conj_cong:
|
wenzelm@12281
|
1156 |
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
|
nipkow@17589
|
1157 |
by iprover
|
wenzelm@12281
|
1158 |
|
wenzelm@12281
|
1159 |
text {* The @{text "|"} congruence rule: not included by default! *}
|
wenzelm@12281
|
1160 |
|
wenzelm@12281
|
1161 |
lemma disj_cong:
|
wenzelm@12281
|
1162 |
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
|
wenzelm@12281
|
1163 |
by blast
|
wenzelm@12281
|
1164 |
|
wenzelm@12281
|
1165 |
|
wenzelm@12281
|
1166 |
text {* \medskip if-then-else rules *}
|
wenzelm@12281
|
1167 |
|
wenzelm@12281
|
1168 |
lemma if_True: "(if True then x else y) = x"
|
wenzelm@12281
|
1169 |
by (unfold if_def) blast
|
wenzelm@12281
|
1170 |
|
wenzelm@12281
|
1171 |
lemma if_False: "(if False then x else y) = y"
|
wenzelm@12281
|
1172 |
by (unfold if_def) blast
|
wenzelm@12281
|
1173 |
|
wenzelm@12281
|
1174 |
lemma if_P: "P ==> (if P then x else y) = x"
|
wenzelm@12281
|
1175 |
by (unfold if_def) blast
|
wenzelm@12281
|
1176 |
|
wenzelm@12281
|
1177 |
lemma if_not_P: "~P ==> (if P then x else y) = y"
|
wenzelm@12281
|
1178 |
by (unfold if_def) blast
|
wenzelm@12281
|
1179 |
|
wenzelm@12281
|
1180 |
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
|
wenzelm@12281
|
1181 |
apply (rule case_split [of Q])
|
paulson@15481
|
1182 |
apply (simplesubst if_P)
|
paulson@15481
|
1183 |
prefer 3 apply (simplesubst if_not_P, blast+)
|
wenzelm@12281
|
1184 |
done
|
wenzelm@12281
|
1185 |
|
wenzelm@12281
|
1186 |
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
|
paulson@15481
|
1187 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1188 |
|
paulson@24286
|
1189 |
lemmas if_splits [noatp] = split_if split_if_asm
|
wenzelm@12281
|
1190 |
|
wenzelm@12281
|
1191 |
lemma if_cancel: "(if c then x else x) = x"
|
paulson@15481
|
1192 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1193 |
|
wenzelm@12281
|
1194 |
lemma if_eq_cancel: "(if x = y then y else x) = x"
|
paulson@15481
|
1195 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1196 |
|
wenzelm@12281
|
1197 |
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
|
wenzelm@19796
|
1198 |
-- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
|
wenzelm@12281
|
1199 |
by (rule split_if)
|
wenzelm@12281
|
1200 |
|
wenzelm@12281
|
1201 |
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
|
wenzelm@19796
|
1202 |
-- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
|
paulson@15481
|
1203 |
apply (simplesubst split_if, blast)
|
wenzelm@12281
|
1204 |
done
|
wenzelm@12281
|
1205 |
|
nipkow@17589
|
1206 |
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
|
nipkow@17589
|
1207 |
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
|
wenzelm@12281
|
1208 |
|
schirmer@15423
|
1209 |
text {* \medskip let rules for simproc *}
|
schirmer@15423
|
1210 |
|
schirmer@15423
|
1211 |
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
|
schirmer@15423
|
1212 |
by (unfold Let_def)
|
schirmer@15423
|
1213 |
|
schirmer@15423
|
1214 |
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
|
schirmer@15423
|
1215 |
by (unfold Let_def)
|
schirmer@15423
|
1216 |
|
berghofe@16633
|
1217 |
text {*
|
ballarin@16999
|
1218 |
The following copy of the implication operator is useful for
|
ballarin@16999
|
1219 |
fine-tuning congruence rules. It instructs the simplifier to simplify
|
ballarin@16999
|
1220 |
its premise.
|
berghofe@16633
|
1221 |
*}
|
berghofe@16633
|
1222 |
|
wenzelm@17197
|
1223 |
constdefs
|
wenzelm@17197
|
1224 |
simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
|
haftmann@28562
|
1225 |
[code del]: "simp_implies \<equiv> op ==>"
|
berghofe@16633
|
1226 |
|
wenzelm@18457
|
1227 |
lemma simp_impliesI:
|
berghofe@16633
|
1228 |
assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
|
berghofe@16633
|
1229 |
shows "PROP P =simp=> PROP Q"
|
berghofe@16633
|
1230 |
apply (unfold simp_implies_def)
|
berghofe@16633
|
1231 |
apply (rule PQ)
|
berghofe@16633
|
1232 |
apply assumption
|
berghofe@16633
|
1233 |
done
|
berghofe@16633
|
1234 |
|
berghofe@16633
|
1235 |
lemma simp_impliesE:
|
wenzelm@25388
|
1236 |
assumes PQ: "PROP P =simp=> PROP Q"
|
berghofe@16633
|
1237 |
and P: "PROP P"
|
berghofe@16633
|
1238 |
and QR: "PROP Q \<Longrightarrow> PROP R"
|
berghofe@16633
|
1239 |
shows "PROP R"
|
berghofe@16633
|
1240 |
apply (rule QR)
|
berghofe@16633
|
1241 |
apply (rule PQ [unfolded simp_implies_def])
|
berghofe@16633
|
1242 |
apply (rule P)
|
berghofe@16633
|
1243 |
done
|
berghofe@16633
|
1244 |
|
berghofe@16633
|
1245 |
lemma simp_implies_cong:
|
berghofe@16633
|
1246 |
assumes PP' :"PROP P == PROP P'"
|
berghofe@16633
|
1247 |
and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
|
berghofe@16633
|
1248 |
shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
|
berghofe@16633
|
1249 |
proof (unfold simp_implies_def, rule equal_intr_rule)
|
berghofe@16633
|
1250 |
assume PQ: "PROP P \<Longrightarrow> PROP Q"
|
berghofe@16633
|
1251 |
and P': "PROP P'"
|
berghofe@16633
|
1252 |
from PP' [symmetric] and P' have "PROP P"
|
berghofe@16633
|
1253 |
by (rule equal_elim_rule1)
|
wenzelm@23553
|
1254 |
then have "PROP Q" by (rule PQ)
|
berghofe@16633
|
1255 |
with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
|
berghofe@16633
|
1256 |
next
|
berghofe@16633
|
1257 |
assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
|
berghofe@16633
|
1258 |
and P: "PROP P"
|
berghofe@16633
|
1259 |
from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
|
wenzelm@23553
|
1260 |
then have "PROP Q'" by (rule P'Q')
|
berghofe@16633
|
1261 |
with P'QQ' [OF P', symmetric] show "PROP Q"
|
berghofe@16633
|
1262 |
by (rule equal_elim_rule1)
|
berghofe@16633
|
1263 |
qed
|
berghofe@16633
|
1264 |
|
haftmann@20944
|
1265 |
lemma uncurry:
|
haftmann@20944
|
1266 |
assumes "P \<longrightarrow> Q \<longrightarrow> R"
|
haftmann@20944
|
1267 |
shows "P \<and> Q \<longrightarrow> R"
|
wenzelm@23553
|
1268 |
using assms by blast
|
haftmann@20944
|
1269 |
|
haftmann@20944
|
1270 |
lemma iff_allI:
|
haftmann@20944
|
1271 |
assumes "\<And>x. P x = Q x"
|
haftmann@20944
|
1272 |
shows "(\<forall>x. P x) = (\<forall>x. Q x)"
|
wenzelm@23553
|
1273 |
using assms by blast
|
haftmann@20944
|
1274 |
|
haftmann@20944
|
1275 |
lemma iff_exI:
|
haftmann@20944
|
1276 |
assumes "\<And>x. P x = Q x"
|
haftmann@20944
|
1277 |
shows "(\<exists>x. P x) = (\<exists>x. Q x)"
|
wenzelm@23553
|
1278 |
using assms by blast
|
haftmann@20944
|
1279 |
|
haftmann@20944
|
1280 |
lemma all_comm:
|
haftmann@20944
|
1281 |
"(\<forall>x y. P x y) = (\<forall>y x. P x y)"
|
haftmann@20944
|
1282 |
by blast
|
haftmann@20944
|
1283 |
|
haftmann@20944
|
1284 |
lemma ex_comm:
|
haftmann@20944
|
1285 |
"(\<exists>x y. P x y) = (\<exists>y x. P x y)"
|
haftmann@20944
|
1286 |
by blast
|
haftmann@20944
|
1287 |
|
haftmann@28952
|
1288 |
use "Tools/simpdata.ML"
|
wenzelm@21671
|
1289 |
ML {* open Simpdata *}
|
wenzelm@21671
|
1290 |
|
haftmann@21151
|
1291 |
setup {*
|
haftmann@21151
|
1292 |
Simplifier.method_setup Splitter.split_modifiers
|
wenzelm@26496
|
1293 |
#> Simplifier.map_simpset (K Simpdata.simpset_simprocs)
|
haftmann@21151
|
1294 |
#> Splitter.setup
|
wenzelm@26496
|
1295 |
#> clasimp_setup
|
haftmann@21151
|
1296 |
#> EqSubst.setup
|
haftmann@21151
|
1297 |
*}
|
haftmann@21151
|
1298 |
|
wenzelm@24035
|
1299 |
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
|
wenzelm@24035
|
1300 |
|
wenzelm@24035
|
1301 |
simproc_setup neq ("x = y") = {* fn _ =>
|
wenzelm@24035
|
1302 |
let
|
wenzelm@24035
|
1303 |
val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
|
wenzelm@24035
|
1304 |
fun is_neq eq lhs rhs thm =
|
wenzelm@24035
|
1305 |
(case Thm.prop_of thm of
|
wenzelm@24035
|
1306 |
_ $ (Not $ (eq' $ l' $ r')) =>
|
wenzelm@24035
|
1307 |
Not = HOLogic.Not andalso eq' = eq andalso
|
wenzelm@24035
|
1308 |
r' aconv lhs andalso l' aconv rhs
|
wenzelm@24035
|
1309 |
| _ => false);
|
wenzelm@24035
|
1310 |
fun proc ss ct =
|
wenzelm@24035
|
1311 |
(case Thm.term_of ct of
|
wenzelm@24035
|
1312 |
eq $ lhs $ rhs =>
|
wenzelm@24035
|
1313 |
(case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
|
wenzelm@24035
|
1314 |
SOME thm => SOME (thm RS neq_to_EQ_False)
|
wenzelm@24035
|
1315 |
| NONE => NONE)
|
wenzelm@24035
|
1316 |
| _ => NONE);
|
wenzelm@24035
|
1317 |
in proc end;
|
wenzelm@24035
|
1318 |
*}
|
wenzelm@24035
|
1319 |
|
wenzelm@24035
|
1320 |
simproc_setup let_simp ("Let x f") = {*
|
wenzelm@24035
|
1321 |
let
|
wenzelm@24035
|
1322 |
val (f_Let_unfold, x_Let_unfold) =
|
haftmann@28741
|
1323 |
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}
|
wenzelm@24035
|
1324 |
in (cterm_of @{theory} f, cterm_of @{theory} x) end
|
wenzelm@24035
|
1325 |
val (f_Let_folded, x_Let_folded) =
|
haftmann@28741
|
1326 |
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}
|
wenzelm@24035
|
1327 |
in (cterm_of @{theory} f, cterm_of @{theory} x) end;
|
wenzelm@24035
|
1328 |
val g_Let_folded =
|
haftmann@28741
|
1329 |
let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}
|
haftmann@28741
|
1330 |
in cterm_of @{theory} g end;
|
haftmann@28741
|
1331 |
fun count_loose (Bound i) k = if i >= k then 1 else 0
|
haftmann@28741
|
1332 |
| count_loose (s $ t) k = count_loose s k + count_loose t k
|
haftmann@28741
|
1333 |
| count_loose (Abs (_, _, t)) k = count_loose t (k + 1)
|
haftmann@28741
|
1334 |
| count_loose _ _ = 0;
|
haftmann@28741
|
1335 |
fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
|
haftmann@28741
|
1336 |
case t
|
haftmann@28741
|
1337 |
of Abs (_, _, t') => count_loose t' 0 <= 1
|
haftmann@28741
|
1338 |
| _ => true;
|
haftmann@28741
|
1339 |
in fn _ => fn ss => fn ct => if is_trivial_let (Thm.term_of ct)
|
haftmann@28741
|
1340 |
then SOME @{thm Let_def} (*no or one ocurrenc of bound variable*)
|
haftmann@28741
|
1341 |
else let (*Norbert Schirmer's case*)
|
haftmann@28741
|
1342 |
val ctxt = Simplifier.the_context ss;
|
haftmann@28741
|
1343 |
val thy = ProofContext.theory_of ctxt;
|
haftmann@28741
|
1344 |
val t = Thm.term_of ct;
|
haftmann@28741
|
1345 |
val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
|
haftmann@28741
|
1346 |
in Option.map (hd o Variable.export ctxt' ctxt o single)
|
haftmann@28741
|
1347 |
(case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
|
haftmann@28741
|
1348 |
if is_Free x orelse is_Bound x orelse is_Const x
|
haftmann@28741
|
1349 |
then SOME @{thm Let_def}
|
haftmann@28741
|
1350 |
else
|
haftmann@28741
|
1351 |
let
|
haftmann@28741
|
1352 |
val n = case f of (Abs (x, _, _)) => x | _ => "x";
|
haftmann@28741
|
1353 |
val cx = cterm_of thy x;
|
haftmann@28741
|
1354 |
val {T = xT, ...} = rep_cterm cx;
|
haftmann@28741
|
1355 |
val cf = cterm_of thy f;
|
haftmann@28741
|
1356 |
val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
|
haftmann@28741
|
1357 |
val (_ $ _ $ g) = prop_of fx_g;
|
haftmann@28741
|
1358 |
val g' = abstract_over (x,g);
|
haftmann@28741
|
1359 |
in (if (g aconv g')
|
haftmann@28741
|
1360 |
then
|
haftmann@28741
|
1361 |
let
|
haftmann@28741
|
1362 |
val rl =
|
haftmann@28741
|
1363 |
cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
|
haftmann@28741
|
1364 |
in SOME (rl OF [fx_g]) end
|
haftmann@28741
|
1365 |
else if Term.betapply (f, x) aconv g then NONE (*avoid identity conversion*)
|
haftmann@28741
|
1366 |
else let
|
haftmann@28741
|
1367 |
val abs_g'= Abs (n,xT,g');
|
haftmann@28741
|
1368 |
val g'x = abs_g'$x;
|
haftmann@28741
|
1369 |
val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
|
haftmann@28741
|
1370 |
val rl = cterm_instantiate
|
haftmann@28741
|
1371 |
[(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),
|
haftmann@28741
|
1372 |
(g_Let_folded, cterm_of thy abs_g')]
|
haftmann@28741
|
1373 |
@{thm Let_folded};
|
haftmann@28741
|
1374 |
in SOME (rl OF [transitive fx_g g_g'x])
|
haftmann@28741
|
1375 |
end)
|
haftmann@28741
|
1376 |
end
|
haftmann@28741
|
1377 |
| _ => NONE)
|
haftmann@28741
|
1378 |
end
|
haftmann@28741
|
1379 |
end *}
|
wenzelm@24035
|
1380 |
|
haftmann@21151
|
1381 |
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
|
haftmann@21151
|
1382 |
proof
|
wenzelm@23389
|
1383 |
assume "True \<Longrightarrow> PROP P"
|
wenzelm@23389
|
1384 |
from this [OF TrueI] show "PROP P" .
|
haftmann@21151
|
1385 |
next
|
haftmann@21151
|
1386 |
assume "PROP P"
|
wenzelm@23389
|
1387 |
then show "PROP P" .
|
haftmann@21151
|
1388 |
qed
|
haftmann@21151
|
1389 |
|
haftmann@21151
|
1390 |
lemma ex_simps:
|
haftmann@21151
|
1391 |
"!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
|
haftmann@21151
|
1392 |
"!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
|
haftmann@21151
|
1393 |
"!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
|
haftmann@21151
|
1394 |
"!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
|
haftmann@21151
|
1395 |
"!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
|
haftmann@21151
|
1396 |
"!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
|
haftmann@21151
|
1397 |
-- {* Miniscoping: pushing in existential quantifiers. *}
|
haftmann@21151
|
1398 |
by (iprover | blast)+
|
haftmann@21151
|
1399 |
|
haftmann@21151
|
1400 |
lemma all_simps:
|
haftmann@21151
|
1401 |
"!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
|
haftmann@21151
|
1402 |
"!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
|
haftmann@21151
|
1403 |
"!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
|
haftmann@21151
|
1404 |
"!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
|
haftmann@21151
|
1405 |
"!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
|
haftmann@21151
|
1406 |
"!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
|
haftmann@21151
|
1407 |
-- {* Miniscoping: pushing in universal quantifiers. *}
|
haftmann@21151
|
1408 |
by (iprover | blast)+
|
paulson@15481
|
1409 |
|
wenzelm@21671
|
1410 |
lemmas [simp] =
|
wenzelm@21671
|
1411 |
triv_forall_equality (*prunes params*)
|
wenzelm@21671
|
1412 |
True_implies_equals (*prune asms `True'*)
|
wenzelm@21671
|
1413 |
if_True
|
wenzelm@21671
|
1414 |
if_False
|
wenzelm@21671
|
1415 |
if_cancel
|
wenzelm@21671
|
1416 |
if_eq_cancel
|
wenzelm@21671
|
1417 |
imp_disjL
|
haftmann@20973
|
1418 |
(*In general it seems wrong to add distributive laws by default: they
|
haftmann@20973
|
1419 |
might cause exponential blow-up. But imp_disjL has been in for a while
|
haftmann@20973
|
1420 |
and cannot be removed without affecting existing proofs. Moreover,
|
haftmann@20973
|
1421 |
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
|
haftmann@20973
|
1422 |
grounds that it allows simplification of R in the two cases.*)
|
wenzelm@21671
|
1423 |
conj_assoc
|
wenzelm@21671
|
1424 |
disj_assoc
|
wenzelm@21671
|
1425 |
de_Morgan_conj
|
wenzelm@21671
|
1426 |
de_Morgan_disj
|
wenzelm@21671
|
1427 |
imp_disj1
|
wenzelm@21671
|
1428 |
imp_disj2
|
wenzelm@21671
|
1429 |
not_imp
|
wenzelm@21671
|
1430 |
disj_not1
|
wenzelm@21671
|
1431 |
not_all
|
wenzelm@21671
|
1432 |
not_ex
|
wenzelm@21671
|
1433 |
cases_simp
|
wenzelm@21671
|
1434 |
the_eq_trivial
|
wenzelm@21671
|
1435 |
the_sym_eq_trivial
|
wenzelm@21671
|
1436 |
ex_simps
|
wenzelm@21671
|
1437 |
all_simps
|
wenzelm@21671
|
1438 |
simp_thms
|
wenzelm@21671
|
1439 |
|
wenzelm@21671
|
1440 |
lemmas [cong] = imp_cong simp_implies_cong
|
wenzelm@21671
|
1441 |
lemmas [split] = split_if
|
haftmann@20973
|
1442 |
|
wenzelm@22377
|
1443 |
ML {* val HOL_ss = @{simpset} *}
|
haftmann@20973
|
1444 |
|
haftmann@20944
|
1445 |
text {* Simplifies x assuming c and y assuming ~c *}
|
haftmann@20944
|
1446 |
lemma if_cong:
|
haftmann@20944
|
1447 |
assumes "b = c"
|
haftmann@20944
|
1448 |
and "c \<Longrightarrow> x = u"
|
haftmann@20944
|
1449 |
and "\<not> c \<Longrightarrow> y = v"
|
haftmann@20944
|
1450 |
shows "(if b then x else y) = (if c then u else v)"
|
wenzelm@23553
|
1451 |
unfolding if_def using assms by simp
|
haftmann@20944
|
1452 |
|
haftmann@20944
|
1453 |
text {* Prevents simplification of x and y:
|
haftmann@20944
|
1454 |
faster and allows the execution of functional programs. *}
|
haftmann@20944
|
1455 |
lemma if_weak_cong [cong]:
|
haftmann@20944
|
1456 |
assumes "b = c"
|
haftmann@20944
|
1457 |
shows "(if b then x else y) = (if c then x else y)"
|
wenzelm@23553
|
1458 |
using assms by (rule arg_cong)
|
haftmann@20944
|
1459 |
|
haftmann@20944
|
1460 |
text {* Prevents simplification of t: much faster *}
|
haftmann@20944
|
1461 |
lemma let_weak_cong:
|
haftmann@20944
|
1462 |
assumes "a = b"
|
haftmann@20944
|
1463 |
shows "(let x = a in t x) = (let x = b in t x)"
|
wenzelm@23553
|
1464 |
using assms by (rule arg_cong)
|
haftmann@20944
|
1465 |
|
haftmann@20944
|
1466 |
text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
|
haftmann@20944
|
1467 |
lemma eq_cong2:
|
haftmann@20944
|
1468 |
assumes "u = u'"
|
haftmann@20944
|
1469 |
shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
|
wenzelm@23553
|
1470 |
using assms by simp
|
haftmann@20944
|
1471 |
|
haftmann@20944
|
1472 |
lemma if_distrib:
|
haftmann@20944
|
1473 |
"f (if c then x else y) = (if c then f x else f y)"
|
haftmann@20944
|
1474 |
by simp
|
haftmann@20944
|
1475 |
|
haftmann@20944
|
1476 |
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
|
wenzelm@21502
|
1477 |
side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *}
|
haftmann@20944
|
1478 |
lemma restrict_to_left:
|
haftmann@20944
|
1479 |
assumes "x = y"
|
haftmann@20944
|
1480 |
shows "(x = z) = (y = z)"
|
wenzelm@23553
|
1481 |
using assms by simp
|
haftmann@20944
|
1482 |
|
haftmann@20944
|
1483 |
|
haftmann@20944
|
1484 |
subsubsection {* Generic cases and induction *}
|
haftmann@20944
|
1485 |
|
haftmann@20944
|
1486 |
text {* Rule projections: *}
|
haftmann@20944
|
1487 |
|
haftmann@20944
|
1488 |
ML {*
|
haftmann@20944
|
1489 |
structure ProjectRule = ProjectRuleFun
|
wenzelm@25388
|
1490 |
(
|
wenzelm@27126
|
1491 |
val conjunct1 = @{thm conjunct1}
|
wenzelm@27126
|
1492 |
val conjunct2 = @{thm conjunct2}
|
wenzelm@27126
|
1493 |
val mp = @{thm mp}
|
wenzelm@25388
|
1494 |
)
|
haftmann@20944
|
1495 |
*}
|
haftmann@20944
|
1496 |
|
haftmann@20944
|
1497 |
constdefs
|
haftmann@20944
|
1498 |
induct_forall where "induct_forall P == \<forall>x. P x"
|
haftmann@20944
|
1499 |
induct_implies where "induct_implies A B == A \<longrightarrow> B"
|
haftmann@20944
|
1500 |
induct_equal where "induct_equal x y == x = y"
|
haftmann@20944
|
1501 |
induct_conj where "induct_conj A B == A \<and> B"
|
haftmann@20944
|
1502 |
|
haftmann@20944
|
1503 |
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
|
haftmann@20944
|
1504 |
by (unfold atomize_all induct_forall_def)
|
haftmann@20944
|
1505 |
|
haftmann@20944
|
1506 |
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
|
haftmann@20944
|
1507 |
by (unfold atomize_imp induct_implies_def)
|
haftmann@20944
|
1508 |
|
haftmann@20944
|
1509 |
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
|
haftmann@20944
|
1510 |
by (unfold atomize_eq induct_equal_def)
|
haftmann@20944
|
1511 |
|
wenzelm@28856
|
1512 |
lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"
|
haftmann@20944
|
1513 |
by (unfold atomize_conj induct_conj_def)
|
haftmann@20944
|
1514 |
|
haftmann@20944
|
1515 |
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
|
haftmann@20944
|
1516 |
lemmas induct_rulify [symmetric, standard] = induct_atomize
|
haftmann@20944
|
1517 |
lemmas induct_rulify_fallback =
|
haftmann@20944
|
1518 |
induct_forall_def induct_implies_def induct_equal_def induct_conj_def
|
haftmann@20944
|
1519 |
|
haftmann@20944
|
1520 |
|
haftmann@20944
|
1521 |
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
|
haftmann@20944
|
1522 |
induct_conj (induct_forall A) (induct_forall B)"
|
haftmann@20944
|
1523 |
by (unfold induct_forall_def induct_conj_def) iprover
|
haftmann@20944
|
1524 |
|
haftmann@20944
|
1525 |
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
|
haftmann@20944
|
1526 |
induct_conj (induct_implies C A) (induct_implies C B)"
|
haftmann@20944
|
1527 |
by (unfold induct_implies_def induct_conj_def) iprover
|
haftmann@20944
|
1528 |
|
haftmann@20944
|
1529 |
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
|
haftmann@20944
|
1530 |
proof
|
haftmann@20944
|
1531 |
assume r: "induct_conj A B ==> PROP C" and A B
|
haftmann@20944
|
1532 |
show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
|
haftmann@20944
|
1533 |
next
|
haftmann@20944
|
1534 |
assume r: "A ==> B ==> PROP C" and "induct_conj A B"
|
haftmann@20944
|
1535 |
show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
|
haftmann@20944
|
1536 |
qed
|
haftmann@20944
|
1537 |
|
haftmann@20944
|
1538 |
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
|
haftmann@20944
|
1539 |
|
haftmann@20944
|
1540 |
hide const induct_forall induct_implies induct_equal induct_conj
|
haftmann@20944
|
1541 |
|
haftmann@20944
|
1542 |
text {* Method setup. *}
|
haftmann@20944
|
1543 |
|
haftmann@20944
|
1544 |
ML {*
|
wenzelm@27126
|
1545 |
structure Induct = InductFun
|
wenzelm@27126
|
1546 |
(
|
wenzelm@27126
|
1547 |
val cases_default = @{thm case_split}
|
wenzelm@27126
|
1548 |
val atomize = @{thms induct_atomize}
|
wenzelm@27126
|
1549 |
val rulify = @{thms induct_rulify}
|
wenzelm@27126
|
1550 |
val rulify_fallback = @{thms induct_rulify_fallback}
|
wenzelm@27126
|
1551 |
)
|
haftmann@20944
|
1552 |
*}
|
haftmann@20944
|
1553 |
|
wenzelm@24830
|
1554 |
setup Induct.setup
|
wenzelm@17459
|
1555 |
|
wenzelm@27326
|
1556 |
use "~~/src/Tools/induct_tacs.ML"
|
wenzelm@27126
|
1557 |
setup InductTacs.setup
|
wenzelm@27126
|
1558 |
|
haftmann@20944
|
1559 |
|
berghofe@28325
|
1560 |
subsubsection {* Coherent logic *}
|
berghofe@28325
|
1561 |
|
berghofe@28325
|
1562 |
ML {*
|
berghofe@28325
|
1563 |
structure Coherent = CoherentFun
|
berghofe@28325
|
1564 |
(
|
berghofe@28325
|
1565 |
val atomize_elimL = @{thm atomize_elimL}
|
berghofe@28325
|
1566 |
val atomize_exL = @{thm atomize_exL}
|
berghofe@28325
|
1567 |
val atomize_conjL = @{thm atomize_conjL}
|
berghofe@28325
|
1568 |
val atomize_disjL = @{thm atomize_disjL}
|
berghofe@28325
|
1569 |
val operator_names =
|
berghofe@28325
|
1570 |
[@{const_name "op |"}, @{const_name "op &"}, @{const_name "Ex"}]
|
berghofe@28325
|
1571 |
);
|
berghofe@28325
|
1572 |
*}
|
berghofe@28325
|
1573 |
|
berghofe@28325
|
1574 |
setup Coherent.setup
|
berghofe@28325
|
1575 |
|
berghofe@28325
|
1576 |
|
haftmann@20944
|
1577 |
subsection {* Other simple lemmas and lemma duplicates *}
|
haftmann@20944
|
1578 |
|
haftmann@24166
|
1579 |
lemma Let_0 [simp]: "Let 0 f = f 0"
|
haftmann@24166
|
1580 |
unfolding Let_def ..
|
haftmann@24166
|
1581 |
|
haftmann@24166
|
1582 |
lemma Let_1 [simp]: "Let 1 f = f 1"
|
haftmann@24166
|
1583 |
unfolding Let_def ..
|
haftmann@24166
|
1584 |
|
haftmann@20944
|
1585 |
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
|
haftmann@20944
|
1586 |
by blast+
|
haftmann@20944
|
1587 |
|
haftmann@20944
|
1588 |
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
|
haftmann@20944
|
1589 |
apply (rule iffI)
|
haftmann@20944
|
1590 |
apply (rule_tac a = "%x. THE y. P x y" in ex1I)
|
haftmann@20944
|
1591 |
apply (fast dest!: theI')
|
haftmann@20944
|
1592 |
apply (fast intro: ext the1_equality [symmetric])
|
haftmann@20944
|
1593 |
apply (erule ex1E)
|
haftmann@20944
|
1594 |
apply (rule allI)
|
haftmann@20944
|
1595 |
apply (rule ex1I)
|
haftmann@20944
|
1596 |
apply (erule spec)
|
haftmann@20944
|
1597 |
apply (erule_tac x = "%z. if z = x then y else f z" in allE)
|
haftmann@20944
|
1598 |
apply (erule impE)
|
haftmann@20944
|
1599 |
apply (rule allI)
|
wenzelm@27126
|
1600 |
apply (case_tac "xa = x")
|
haftmann@20944
|
1601 |
apply (drule_tac [3] x = x in fun_cong, simp_all)
|
haftmann@20944
|
1602 |
done
|
haftmann@20944
|
1603 |
|
haftmann@20944
|
1604 |
lemma mk_left_commute:
|
haftmann@21547
|
1605 |
fixes f (infix "\<otimes>" 60)
|
haftmann@21547
|
1606 |
assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
|
haftmann@21547
|
1607 |
c: "\<And>x y. x \<otimes> y = y \<otimes> x"
|
haftmann@21547
|
1608 |
shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
|
haftmann@20944
|
1609 |
by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
|
haftmann@20944
|
1610 |
|
haftmann@22218
|
1611 |
lemmas eq_sym_conv = eq_commute
|
haftmann@22218
|
1612 |
|
chaieb@23037
|
1613 |
lemma nnf_simps:
|
chaieb@23037
|
1614 |
"(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
|
chaieb@23037
|
1615 |
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
|
chaieb@23037
|
1616 |
"(\<not> \<not>(P)) = P"
|
chaieb@23037
|
1617 |
by blast+
|
chaieb@23037
|
1618 |
|
wenzelm@21671
|
1619 |
|
wenzelm@21671
|
1620 |
subsection {* Basic ML bindings *}
|
wenzelm@21671
|
1621 |
|
wenzelm@21671
|
1622 |
ML {*
|
wenzelm@22129
|
1623 |
val FalseE = @{thm FalseE}
|
wenzelm@22129
|
1624 |
val Let_def = @{thm Let_def}
|
wenzelm@22129
|
1625 |
val TrueI = @{thm TrueI}
|
wenzelm@22129
|
1626 |
val allE = @{thm allE}
|
wenzelm@22129
|
1627 |
val allI = @{thm allI}
|
wenzelm@22129
|
1628 |
val all_dupE = @{thm all_dupE}
|
wenzelm@22129
|
1629 |
val arg_cong = @{thm arg_cong}
|
wenzelm@22129
|
1630 |
val box_equals = @{thm box_equals}
|
wenzelm@22129
|
1631 |
val ccontr = @{thm ccontr}
|
wenzelm@22129
|
1632 |
val classical = @{thm classical}
|
wenzelm@22129
|
1633 |
val conjE = @{thm conjE}
|
wenzelm@22129
|
1634 |
val conjI = @{thm conjI}
|
wenzelm@22129
|
1635 |
val conjunct1 = @{thm conjunct1}
|
wenzelm@22129
|
1636 |
val conjunct2 = @{thm conjunct2}
|
wenzelm@22129
|
1637 |
val disjCI = @{thm disjCI}
|
wenzelm@22129
|
1638 |
val disjE = @{thm disjE}
|
wenzelm@22129
|
1639 |
val disjI1 = @{thm disjI1}
|
wenzelm@22129
|
1640 |
val disjI2 = @{thm disjI2}
|
wenzelm@22129
|
1641 |
val eq_reflection = @{thm eq_reflection}
|
wenzelm@22129
|
1642 |
val ex1E = @{thm ex1E}
|
wenzelm@22129
|
1643 |
val ex1I = @{thm ex1I}
|
wenzelm@22129
|
1644 |
val ex1_implies_ex = @{thm ex1_implies_ex}
|
wenzelm@22129
|
1645 |
val exE = @{thm exE}
|
wenzelm@22129
|
1646 |
val exI = @{thm exI}
|
wenzelm@22129
|
1647 |
val excluded_middle = @{thm excluded_middle}
|
wenzelm@22129
|
1648 |
val ext = @{thm ext}
|
wenzelm@22129
|
1649 |
val fun_cong = @{thm fun_cong}
|
wenzelm@22129
|
1650 |
val iffD1 = @{thm iffD1}
|
wenzelm@22129
|
1651 |
val iffD2 = @{thm iffD2}
|
wenzelm@22129
|
1652 |
val iffI = @{thm iffI}
|
wenzelm@22129
|
1653 |
val impE = @{thm impE}
|
wenzelm@22129
|
1654 |
val impI = @{thm impI}
|
wenzelm@22129
|
1655 |
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
|
wenzelm@22129
|
1656 |
val mp = @{thm mp}
|
wenzelm@22129
|
1657 |
val notE = @{thm notE}
|
wenzelm@22129
|
1658 |
val notI = @{thm notI}
|
wenzelm@22129
|
1659 |
val not_all = @{thm not_all}
|
wenzelm@22129
|
1660 |
val not_ex = @{thm not_ex}
|
wenzelm@22129
|
1661 |
val not_iff = @{thm not_iff}
|
wenzelm@22129
|
1662 |
val not_not = @{thm not_not}
|
wenzelm@22129
|
1663 |
val not_sym = @{thm not_sym}
|
wenzelm@22129
|
1664 |
val refl = @{thm refl}
|
wenzelm@22129
|
1665 |
val rev_mp = @{thm rev_mp}
|
wenzelm@22129
|
1666 |
val spec = @{thm spec}
|
wenzelm@22129
|
1667 |
val ssubst = @{thm ssubst}
|
wenzelm@22129
|
1668 |
val subst = @{thm subst}
|
wenzelm@22129
|
1669 |
val sym = @{thm sym}
|
wenzelm@22129
|
1670 |
val trans = @{thm trans}
|
wenzelm@21671
|
1671 |
*}
|
wenzelm@21671
|
1672 |
|
wenzelm@21671
|
1673 |
|
haftmann@28400
|
1674 |
subsection {* Code generator basics -- see further theory @{text "Code_Setup"} *}
|
haftmann@28400
|
1675 |
|
haftmann@28400
|
1676 |
text {* Equality *}
|
haftmann@24844
|
1677 |
|
haftmann@29608
|
1678 |
class eq =
|
haftmann@26513
|
1679 |
fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
|
haftmann@28400
|
1680 |
assumes eq_equals: "eq x y \<longleftrightarrow> x = y"
|
haftmann@26513
|
1681 |
begin
|
haftmann@26513
|
1682 |
|
haftmann@28346
|
1683 |
lemma eq: "eq = (op =)"
|
haftmann@28346
|
1684 |
by (rule ext eq_equals)+
|
haftmann@28346
|
1685 |
|
haftmann@28346
|
1686 |
lemma eq_refl: "eq x x \<longleftrightarrow> True"
|
haftmann@28346
|
1687 |
unfolding eq by rule+
|
haftmann@28346
|
1688 |
|
haftmann@26513
|
1689 |
end
|
haftmann@26513
|
1690 |
|
haftmann@28513
|
1691 |
text {* Module setup *}
|
haftmann@28513
|
1692 |
|
haftmann@29505
|
1693 |
use "Tools/recfun_codegen.ML"
|
haftmann@28513
|
1694 |
|
haftmann@28513
|
1695 |
setup {*
|
haftmann@28663
|
1696 |
Code_ML.setup
|
haftmann@28513
|
1697 |
#> Code_Haskell.setup
|
haftmann@28513
|
1698 |
#> Nbe.setup
|
haftmann@28513
|
1699 |
#> Codegen.setup
|
haftmann@28513
|
1700 |
#> RecfunCodegen.setup
|
haftmann@28513
|
1701 |
*}
|
haftmann@28513
|
1702 |
|
haftmann@23247
|
1703 |
|
blanchet@29800
|
1704 |
subsection {* Nitpick theorem store *}
|
blanchet@29800
|
1705 |
|
blanchet@29800
|
1706 |
ML {*
|
blanchet@29800
|
1707 |
structure Nitpick_Const_Simps_Thms = NamedThmsFun
|
blanchet@29800
|
1708 |
(
|
blanchet@29800
|
1709 |
val name = "nitpick_const_simps"
|
blanchet@29800
|
1710 |
val description = "Equational specification of constants as needed by Nitpick"
|
blanchet@29800
|
1711 |
)
|
blanchet@29800
|
1712 |
structure Nitpick_Ind_Intros_Thms = NamedThmsFun
|
blanchet@29800
|
1713 |
(
|
blanchet@29800
|
1714 |
val name = "nitpick_ind_intros"
|
blanchet@29800
|
1715 |
val description = "Introduction rules for inductive predicate as needed by Nitpick"
|
blanchet@29800
|
1716 |
)
|
blanchet@29800
|
1717 |
*}
|
blanchet@29800
|
1718 |
setup {* Nitpick_Const_Simps_Thms.setup
|
blanchet@29800
|
1719 |
o Nitpick_Ind_Intros_Thms.setup *}
|
blanchet@29800
|
1720 |
|
haftmann@22839
|
1721 |
subsection {* Legacy tactics and ML bindings *}
|
wenzelm@21671
|
1722 |
|
wenzelm@21671
|
1723 |
ML {*
|
wenzelm@21671
|
1724 |
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
|
wenzelm@21671
|
1725 |
|
wenzelm@21671
|
1726 |
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
|
wenzelm@21671
|
1727 |
local
|
wenzelm@21671
|
1728 |
fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
|
wenzelm@21671
|
1729 |
| wrong_prem (Bound _) = true
|
wenzelm@21671
|
1730 |
| wrong_prem _ = false;
|
wenzelm@21671
|
1731 |
val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
|
wenzelm@21671
|
1732 |
in
|
wenzelm@21671
|
1733 |
fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
|
wenzelm@21671
|
1734 |
fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
|
wenzelm@21671
|
1735 |
end;
|
haftmann@22839
|
1736 |
|
haftmann@22839
|
1737 |
val all_conj_distrib = thm "all_conj_distrib";
|
haftmann@22839
|
1738 |
val all_simps = thms "all_simps";
|
haftmann@22839
|
1739 |
val atomize_not = thm "atomize_not";
|
wenzelm@24830
|
1740 |
val case_split = thm "case_split";
|
haftmann@22839
|
1741 |
val cases_simp = thm "cases_simp";
|
haftmann@22839
|
1742 |
val choice_eq = thm "choice_eq"
|
haftmann@22839
|
1743 |
val cong = thm "cong"
|
haftmann@22839
|
1744 |
val conj_comms = thms "conj_comms";
|
haftmann@22839
|
1745 |
val conj_cong = thm "conj_cong";
|
haftmann@22839
|
1746 |
val de_Morgan_conj = thm "de_Morgan_conj";
|
haftmann@22839
|
1747 |
val de_Morgan_disj = thm "de_Morgan_disj";
|
haftmann@22839
|
1748 |
val disj_assoc = thm "disj_assoc";
|
haftmann@22839
|
1749 |
val disj_comms = thms "disj_comms";
|
haftmann@22839
|
1750 |
val disj_cong = thm "disj_cong";
|
haftmann@22839
|
1751 |
val eq_ac = thms "eq_ac";
|
haftmann@22839
|
1752 |
val eq_cong2 = thm "eq_cong2"
|
haftmann@22839
|
1753 |
val Eq_FalseI = thm "Eq_FalseI";
|
haftmann@22839
|
1754 |
val Eq_TrueI = thm "Eq_TrueI";
|
haftmann@22839
|
1755 |
val Ex1_def = thm "Ex1_def"
|
haftmann@22839
|
1756 |
val ex_disj_distrib = thm "ex_disj_distrib";
|
haftmann@22839
|
1757 |
val ex_simps = thms "ex_simps";
|
haftmann@22839
|
1758 |
val if_cancel = thm "if_cancel";
|
haftmann@22839
|
1759 |
val if_eq_cancel = thm "if_eq_cancel";
|
haftmann@22839
|
1760 |
val if_False = thm "if_False";
|
haftmann@22839
|
1761 |
val iff_conv_conj_imp = thm "iff_conv_conj_imp";
|
haftmann@22839
|
1762 |
val iff = thm "iff"
|
haftmann@22839
|
1763 |
val if_splits = thms "if_splits";
|
haftmann@22839
|
1764 |
val if_True = thm "if_True";
|
haftmann@22839
|
1765 |
val if_weak_cong = thm "if_weak_cong"
|
haftmann@22839
|
1766 |
val imp_all = thm "imp_all";
|
haftmann@22839
|
1767 |
val imp_cong = thm "imp_cong";
|
haftmann@22839
|
1768 |
val imp_conjL = thm "imp_conjL";
|
haftmann@22839
|
1769 |
val imp_conjR = thm "imp_conjR";
|
haftmann@22839
|
1770 |
val imp_conv_disj = thm "imp_conv_disj";
|
haftmann@22839
|
1771 |
val simp_implies_def = thm "simp_implies_def";
|
haftmann@22839
|
1772 |
val simp_thms = thms "simp_thms";
|
haftmann@22839
|
1773 |
val split_if = thm "split_if";
|
haftmann@22839
|
1774 |
val the1_equality = thm "the1_equality"
|
haftmann@22839
|
1775 |
val theI = thm "theI"
|
haftmann@22839
|
1776 |
val theI' = thm "theI'"
|
haftmann@22839
|
1777 |
val True_implies_equals = thm "True_implies_equals";
|
chaieb@23037
|
1778 |
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})
|
chaieb@23037
|
1779 |
|
wenzelm@21671
|
1780 |
*}
|
wenzelm@21671
|
1781 |
|
kleing@14357
|
1782 |
end
|