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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haftmann@30929
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imports Pure "~~/src/Tools/Code_Generator"
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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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wenzelm@30165
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"~~/src/Tools/intuitionistic.ML"
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wenzelm@30160
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"~~/src/Tools/project_rule.ML"
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wenzelm@32733
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"~~/src/Tools/cong_tac.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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wenzelm@30160
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"~~/src/Tools/coherent.ML"
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wenzelm@30160
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"~~/src/Tools/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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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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nipkow@45885
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("~~/src/Tools/induction.ML")
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wenzelm@27326
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("~~/src/Tools/induct_tacs.ML")
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haftmann@29505
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("Tools/recfun_codegen.ML")
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blanchet@39280
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("Tools/cnf_funcs.ML")
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wenzelm@41187
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"~~/src/Tools/subtyping.ML"
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noschinl@42698
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"~~/src/Tools/case_product.ML"
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nipkow@15131
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begin
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wenzelm@2260
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wenzelm@31299
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setup {* Intuitionistic.method_setup @{binding iprover} *}
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wenzelm@41187
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setup Subtyping.setup
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noschinl@42698
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setup Case_Product.setup
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wenzelm@33324
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wenzelm@30165
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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@36452
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default_sort type
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wenzelm@35625
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setup {* Object_Logic.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@7357
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typedecl bool
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clasohm@923
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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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clasohm@923
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consts
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True :: bool
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wenzelm@7357
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False :: bool
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haftmann@38772
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Not :: "bool => bool" ("~ _" [40] 40)
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haftmann@39028
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haftmann@39028
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conj :: "[bool, bool] => bool" (infixr "&" 35)
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haftmann@39028
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disj :: "[bool, bool] => bool" (infixr "|" 30)
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haftmann@39019
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implies :: "[bool, bool] => bool" (infixr "-->" 25)
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haftmann@38780
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haftmann@39093
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eq :: "['a, 'a] => bool" (infixl "=" 50)
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wenzelm@38956
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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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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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haftmann@39093
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eq (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@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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haftmann@39093
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conj (infixr "\<and>" 35) and
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haftmann@39093
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disj (infixr "\<or>" 30) and
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haftmann@39093
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implies (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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haftmann@39093
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conj (infixr "\<and>" 35) and
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haftmann@39093
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disj (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@41495
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nonterminal letbinds and letbind
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wenzelm@42938
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nonterminal case_pat and case_syn and 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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huffman@36359
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"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" [0, 10] 10)
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clasohm@923
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wenzelm@42938
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"_case_syntax" :: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
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wenzelm@42938
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"_case1" :: "[case_pat, 'b] => case_syn" ("(2_ =>/ _)" 10)
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wenzelm@42938
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"" :: "case_syn => cases_syn" ("_")
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wenzelm@42938
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"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
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wenzelm@42938
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"_strip_positions" :: "'a => case_pat" ("_")
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wenzelm@42938
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wenzelm@42938
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syntax (xsymbols)
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wenzelm@42938
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"_case1" :: "[case_pat, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
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clasohm@923
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clasohm@923
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translations
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wenzelm@35118
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"THE x. P" == "CONST The (%x. P)"
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clasohm@923
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nipkow@13763
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print_translation {*
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wenzelm@35118
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[(@{const_syntax The}, fn [Abs abs] =>
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wenzelm@43156
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let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
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wenzelm@35118
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in Syntax.const @{syntax_const "_The"} $ x $ t end)]
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wenzelm@35118
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*} -- {* To avoid eta-contraction of body *}
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nipkow@13763
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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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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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skalberg@14223
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finalconsts
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haftmann@39093
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eq
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haftmann@39093
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implies
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skalberg@14223
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The
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nipkow@3320
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haftmann@38750
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definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10) where
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haftmann@38750
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"If P x y \<equiv> (THE z::'a. (P=True --> z=x) & (P=False --> z=y))"
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haftmann@38750
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haftmann@38750
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definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" where
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haftmann@38750
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"Let s f \<equiv> f s"
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haftmann@38750
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haftmann@38750
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translations
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haftmann@38750
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"_Let (_binds b bs) e" == "_Let b (_Let bs e)"
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haftmann@38750
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"let x = a in e" == "CONST Let a (%x. e)"
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haftmann@38750
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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@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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wenzelm@11750
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haftmann@20944
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subsection {* Fundamental rules *}
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haftmann@20944
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haftmann@20973
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subsubsection {* Equality *}
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haftmann@20944
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wenzelm@18457
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lemma sym: "s = t ==> t = s"
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wenzelm@18457
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by (erule subst) (rule refl)
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paulson@15411
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wenzelm@18457
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lemma ssubst: "t = s ==> P s ==> P t"
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wenzelm@18457
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by (drule sym) (erule subst)
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paulson@15411
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paulson@15411
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lemma trans: "[| r=s; s=t |] ==> r=t"
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wenzelm@18457
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by (erule subst)
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paulson@15411
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wenzelm@40963
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lemma trans_sym [Pure.elim?]: "r = s ==> t = s ==> r = t"
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wenzelm@40963
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by (rule trans [OF _ sym])
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wenzelm@40963
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haftmann@20944
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lemma meta_eq_to_obj_eq:
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haftmann@20944
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assumes meq: "A == B"
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haftmann@20944
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shows "A = B"
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haftmann@20944
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by (unfold meq) (rule refl)
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paulson@15411
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wenzelm@21502
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text {* Useful with @{text erule} for proving equalities from known equalities. *}
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haftmann@20944
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(* a = b
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paulson@15411
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| |
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paulson@15411
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c = d *)
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paulson@15411
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lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
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paulson@15411
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apply (rule trans)
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paulson@15411
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apply (rule trans)
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paulson@15411
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apply (rule sym)
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paulson@15411
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apply assumption+
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paulson@15411
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done
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paulson@15411
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nipkow@15524
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text {* For calculational reasoning: *}
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nipkow@15524
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nipkow@15524
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lemma forw_subst: "a = b ==> P b ==> P a"
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nipkow@15524
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by (rule ssubst)
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nipkow@15524
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nipkow@15524
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lemma back_subst: "P a ==> a = b ==> P b"
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nipkow@15524
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by (rule subst)
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nipkow@15524
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paulson@15411
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wenzelm@32733
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subsubsection {* Congruence rules for application *}
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paulson@15411
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wenzelm@32733
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text {* Similar to @{text AP_THM} in Gordon's HOL. *}
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paulson@15411
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lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
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paulson@15411
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apply (erule subst)
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paulson@15411
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apply (rule refl)
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paulson@15411
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done
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paulson@15411
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wenzelm@32733
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text {* Similar to @{text AP_TERM} in Gordon's HOL and FOL's @{text subst_context}. *}
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paulson@15411
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lemma arg_cong: "x=y ==> f(x)=f(y)"
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paulson@15411
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apply (erule subst)
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paulson@15411
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apply (rule refl)
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paulson@15411
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done
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paulson@15411
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paulson@15655
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lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
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paulson@15655
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apply (erule ssubst)+
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paulson@15655
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apply (rule refl)
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paulson@15655
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done
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paulson@15655
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wenzelm@32733
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lemma cong: "[| f = g; (x::'a) = y |] ==> f x = g y"
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paulson@15411
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apply (erule subst)+
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paulson@15411
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apply (rule refl)
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paulson@15411
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done
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paulson@15411
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wenzelm@32733
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ML {* val cong_tac = Cong_Tac.cong_tac @{thm cong} *}
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paulson@15411
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wenzelm@32733
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wenzelm@32733
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subsubsection {* Equality of booleans -- iff *}
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paulson@15411
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wenzelm@21504
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lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
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wenzelm@21504
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by (iprover intro: iff [THEN mp, THEN mp] impI assms)
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paulson@15411
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paulson@15411
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lemma iffD2: "[| P=Q; Q |] ==> P"
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wenzelm@18457
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by (erule ssubst)
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paulson@15411
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paulson@15411
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lemma rev_iffD2: "[| Q; P=Q |] ==> P"
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wenzelm@18457
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by (erule iffD2)
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paulson@15411
|
283 |
|
wenzelm@21504
|
284 |
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
|
wenzelm@21504
|
285 |
by (drule sym) (rule iffD2)
|
wenzelm@21504
|
286 |
|
wenzelm@21504
|
287 |
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
|
wenzelm@21504
|
288 |
by (drule sym) (rule rev_iffD2)
|
paulson@15411
|
289 |
|
paulson@15411
|
290 |
lemma iffE:
|
paulson@15411
|
291 |
assumes major: "P=Q"
|
wenzelm@21504
|
292 |
and minor: "[| P --> Q; Q --> P |] ==> R"
|
wenzelm@18457
|
293 |
shows R
|
wenzelm@18457
|
294 |
by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
|
paulson@15411
|
295 |
|
paulson@15411
|
296 |
|
haftmann@20944
|
297 |
subsubsection {*True*}
|
paulson@15411
|
298 |
|
paulson@15411
|
299 |
lemma TrueI: "True"
|
wenzelm@21504
|
300 |
unfolding True_def by (rule refl)
|
paulson@15411
|
301 |
|
wenzelm@21504
|
302 |
lemma eqTrueI: "P ==> P = True"
|
wenzelm@18457
|
303 |
by (iprover intro: iffI TrueI)
|
paulson@15411
|
304 |
|
wenzelm@21504
|
305 |
lemma eqTrueE: "P = True ==> P"
|
wenzelm@21504
|
306 |
by (erule iffD2) (rule TrueI)
|
paulson@15411
|
307 |
|
paulson@15411
|
308 |
|
haftmann@20944
|
309 |
subsubsection {*Universal quantifier*}
|
paulson@15411
|
310 |
|
wenzelm@21504
|
311 |
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
|
wenzelm@21504
|
312 |
unfolding All_def by (iprover intro: ext eqTrueI assms)
|
paulson@15411
|
313 |
|
paulson@15411
|
314 |
lemma spec: "ALL x::'a. P(x) ==> P(x)"
|
paulson@15411
|
315 |
apply (unfold All_def)
|
paulson@15411
|
316 |
apply (rule eqTrueE)
|
paulson@15411
|
317 |
apply (erule fun_cong)
|
paulson@15411
|
318 |
done
|
paulson@15411
|
319 |
|
paulson@15411
|
320 |
lemma allE:
|
paulson@15411
|
321 |
assumes major: "ALL x. P(x)"
|
wenzelm@21504
|
322 |
and minor: "P(x) ==> R"
|
wenzelm@21504
|
323 |
shows R
|
wenzelm@21504
|
324 |
by (iprover intro: minor major [THEN spec])
|
paulson@15411
|
325 |
|
paulson@15411
|
326 |
lemma all_dupE:
|
paulson@15411
|
327 |
assumes major: "ALL x. P(x)"
|
wenzelm@21504
|
328 |
and minor: "[| P(x); ALL x. P(x) |] ==> R"
|
wenzelm@21504
|
329 |
shows R
|
wenzelm@21504
|
330 |
by (iprover intro: minor major major [THEN spec])
|
paulson@15411
|
331 |
|
paulson@15411
|
332 |
|
wenzelm@21504
|
333 |
subsubsection {* False *}
|
wenzelm@21504
|
334 |
|
wenzelm@21504
|
335 |
text {*
|
wenzelm@21504
|
336 |
Depends upon @{text spec}; it is impossible to do propositional
|
wenzelm@21504
|
337 |
logic before quantifiers!
|
wenzelm@21504
|
338 |
*}
|
paulson@15411
|
339 |
|
paulson@15411
|
340 |
lemma FalseE: "False ==> P"
|
wenzelm@21504
|
341 |
apply (unfold False_def)
|
wenzelm@21504
|
342 |
apply (erule spec)
|
wenzelm@21504
|
343 |
done
|
paulson@15411
|
344 |
|
wenzelm@21504
|
345 |
lemma False_neq_True: "False = True ==> P"
|
wenzelm@21504
|
346 |
by (erule eqTrueE [THEN FalseE])
|
paulson@15411
|
347 |
|
paulson@15411
|
348 |
|
wenzelm@21504
|
349 |
subsubsection {* Negation *}
|
paulson@15411
|
350 |
|
paulson@15411
|
351 |
lemma notI:
|
wenzelm@21504
|
352 |
assumes "P ==> False"
|
paulson@15411
|
353 |
shows "~P"
|
wenzelm@21504
|
354 |
apply (unfold not_def)
|
wenzelm@21504
|
355 |
apply (iprover intro: impI assms)
|
wenzelm@21504
|
356 |
done
|
paulson@15411
|
357 |
|
paulson@15411
|
358 |
lemma False_not_True: "False ~= True"
|
wenzelm@21504
|
359 |
apply (rule notI)
|
wenzelm@21504
|
360 |
apply (erule False_neq_True)
|
wenzelm@21504
|
361 |
done
|
paulson@15411
|
362 |
|
paulson@15411
|
363 |
lemma True_not_False: "True ~= False"
|
wenzelm@21504
|
364 |
apply (rule notI)
|
wenzelm@21504
|
365 |
apply (drule sym)
|
wenzelm@21504
|
366 |
apply (erule False_neq_True)
|
wenzelm@21504
|
367 |
done
|
paulson@15411
|
368 |
|
paulson@15411
|
369 |
lemma notE: "[| ~P; P |] ==> R"
|
wenzelm@21504
|
370 |
apply (unfold not_def)
|
wenzelm@21504
|
371 |
apply (erule mp [THEN FalseE])
|
wenzelm@21504
|
372 |
apply assumption
|
wenzelm@21504
|
373 |
done
|
paulson@15411
|
374 |
|
wenzelm@21504
|
375 |
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
|
wenzelm@21504
|
376 |
by (erule notE [THEN notI]) (erule meta_mp)
|
paulson@15411
|
377 |
|
paulson@15411
|
378 |
|
haftmann@20944
|
379 |
subsubsection {*Implication*}
|
paulson@15411
|
380 |
|
paulson@15411
|
381 |
lemma impE:
|
paulson@15411
|
382 |
assumes "P-->Q" "P" "Q ==> R"
|
paulson@15411
|
383 |
shows "R"
|
wenzelm@23553
|
384 |
by (iprover intro: assms mp)
|
paulson@15411
|
385 |
|
paulson@15411
|
386 |
(* Reduces Q to P-->Q, allowing substitution in P. *)
|
paulson@15411
|
387 |
lemma rev_mp: "[| P; P --> Q |] ==> Q"
|
nipkow@17589
|
388 |
by (iprover intro: mp)
|
paulson@15411
|
389 |
|
paulson@15411
|
390 |
lemma contrapos_nn:
|
paulson@15411
|
391 |
assumes major: "~Q"
|
paulson@15411
|
392 |
and minor: "P==>Q"
|
paulson@15411
|
393 |
shows "~P"
|
nipkow@17589
|
394 |
by (iprover intro: notI minor major [THEN notE])
|
paulson@15411
|
395 |
|
paulson@15411
|
396 |
(*not used at all, but we already have the other 3 combinations *)
|
paulson@15411
|
397 |
lemma contrapos_pn:
|
paulson@15411
|
398 |
assumes major: "Q"
|
paulson@15411
|
399 |
and minor: "P ==> ~Q"
|
paulson@15411
|
400 |
shows "~P"
|
nipkow@17589
|
401 |
by (iprover intro: notI minor major notE)
|
paulson@15411
|
402 |
|
paulson@15411
|
403 |
lemma not_sym: "t ~= s ==> s ~= t"
|
haftmann@21250
|
404 |
by (erule contrapos_nn) (erule sym)
|
haftmann@21250
|
405 |
|
haftmann@21250
|
406 |
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
|
haftmann@21250
|
407 |
by (erule subst, erule ssubst, assumption)
|
paulson@15411
|
408 |
|
paulson@15411
|
409 |
(*still used in HOLCF*)
|
paulson@15411
|
410 |
lemma rev_contrapos:
|
paulson@15411
|
411 |
assumes pq: "P ==> Q"
|
paulson@15411
|
412 |
and nq: "~Q"
|
paulson@15411
|
413 |
shows "~P"
|
paulson@15411
|
414 |
apply (rule nq [THEN contrapos_nn])
|
paulson@15411
|
415 |
apply (erule pq)
|
paulson@15411
|
416 |
done
|
paulson@15411
|
417 |
|
haftmann@20944
|
418 |
subsubsection {*Existential quantifier*}
|
paulson@15411
|
419 |
|
paulson@15411
|
420 |
lemma exI: "P x ==> EX x::'a. P x"
|
paulson@15411
|
421 |
apply (unfold Ex_def)
|
nipkow@17589
|
422 |
apply (iprover intro: allI allE impI mp)
|
paulson@15411
|
423 |
done
|
paulson@15411
|
424 |
|
paulson@15411
|
425 |
lemma exE:
|
paulson@15411
|
426 |
assumes major: "EX x::'a. P(x)"
|
paulson@15411
|
427 |
and minor: "!!x. P(x) ==> Q"
|
paulson@15411
|
428 |
shows "Q"
|
paulson@15411
|
429 |
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
|
nipkow@17589
|
430 |
apply (iprover intro: impI [THEN allI] minor)
|
paulson@15411
|
431 |
done
|
paulson@15411
|
432 |
|
paulson@15411
|
433 |
|
haftmann@20944
|
434 |
subsubsection {*Conjunction*}
|
paulson@15411
|
435 |
|
paulson@15411
|
436 |
lemma conjI: "[| P; Q |] ==> P&Q"
|
paulson@15411
|
437 |
apply (unfold and_def)
|
nipkow@17589
|
438 |
apply (iprover intro: impI [THEN allI] mp)
|
paulson@15411
|
439 |
done
|
paulson@15411
|
440 |
|
paulson@15411
|
441 |
lemma conjunct1: "[| P & Q |] ==> P"
|
paulson@15411
|
442 |
apply (unfold and_def)
|
nipkow@17589
|
443 |
apply (iprover intro: impI dest: spec mp)
|
paulson@15411
|
444 |
done
|
paulson@15411
|
445 |
|
paulson@15411
|
446 |
lemma conjunct2: "[| P & Q |] ==> Q"
|
paulson@15411
|
447 |
apply (unfold and_def)
|
nipkow@17589
|
448 |
apply (iprover intro: impI dest: spec mp)
|
paulson@15411
|
449 |
done
|
paulson@15411
|
450 |
|
paulson@15411
|
451 |
lemma conjE:
|
paulson@15411
|
452 |
assumes major: "P&Q"
|
paulson@15411
|
453 |
and minor: "[| P; Q |] ==> R"
|
paulson@15411
|
454 |
shows "R"
|
paulson@15411
|
455 |
apply (rule minor)
|
paulson@15411
|
456 |
apply (rule major [THEN conjunct1])
|
paulson@15411
|
457 |
apply (rule major [THEN conjunct2])
|
paulson@15411
|
458 |
done
|
paulson@15411
|
459 |
|
paulson@15411
|
460 |
lemma context_conjI:
|
wenzelm@23553
|
461 |
assumes "P" "P ==> Q" shows "P & Q"
|
wenzelm@23553
|
462 |
by (iprover intro: conjI assms)
|
paulson@15411
|
463 |
|
paulson@15411
|
464 |
|
haftmann@20944
|
465 |
subsubsection {*Disjunction*}
|
paulson@15411
|
466 |
|
paulson@15411
|
467 |
lemma disjI1: "P ==> P|Q"
|
paulson@15411
|
468 |
apply (unfold or_def)
|
nipkow@17589
|
469 |
apply (iprover intro: allI impI mp)
|
paulson@15411
|
470 |
done
|
paulson@15411
|
471 |
|
paulson@15411
|
472 |
lemma disjI2: "Q ==> P|Q"
|
paulson@15411
|
473 |
apply (unfold or_def)
|
nipkow@17589
|
474 |
apply (iprover intro: allI impI mp)
|
paulson@15411
|
475 |
done
|
paulson@15411
|
476 |
|
paulson@15411
|
477 |
lemma disjE:
|
paulson@15411
|
478 |
assumes major: "P|Q"
|
paulson@15411
|
479 |
and minorP: "P ==> R"
|
paulson@15411
|
480 |
and minorQ: "Q ==> R"
|
paulson@15411
|
481 |
shows "R"
|
nipkow@17589
|
482 |
by (iprover intro: minorP minorQ impI
|
paulson@15411
|
483 |
major [unfolded or_def, THEN spec, THEN mp, THEN mp])
|
paulson@15411
|
484 |
|
paulson@15411
|
485 |
|
haftmann@20944
|
486 |
subsubsection {*Classical logic*}
|
paulson@15411
|
487 |
|
paulson@15411
|
488 |
lemma classical:
|
paulson@15411
|
489 |
assumes prem: "~P ==> P"
|
paulson@15411
|
490 |
shows "P"
|
paulson@15411
|
491 |
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
|
paulson@15411
|
492 |
apply assumption
|
paulson@15411
|
493 |
apply (rule notI [THEN prem, THEN eqTrueI])
|
paulson@15411
|
494 |
apply (erule subst)
|
paulson@15411
|
495 |
apply assumption
|
paulson@15411
|
496 |
done
|
paulson@15411
|
497 |
|
paulson@15411
|
498 |
lemmas ccontr = FalseE [THEN classical, standard]
|
paulson@15411
|
499 |
|
paulson@15411
|
500 |
(*notE with premises exchanged; it discharges ~R so that it can be used to
|
paulson@15411
|
501 |
make elimination rules*)
|
paulson@15411
|
502 |
lemma rev_notE:
|
paulson@15411
|
503 |
assumes premp: "P"
|
paulson@15411
|
504 |
and premnot: "~R ==> ~P"
|
paulson@15411
|
505 |
shows "R"
|
paulson@15411
|
506 |
apply (rule ccontr)
|
paulson@15411
|
507 |
apply (erule notE [OF premnot premp])
|
paulson@15411
|
508 |
done
|
paulson@15411
|
509 |
|
paulson@15411
|
510 |
(*Double negation law*)
|
paulson@15411
|
511 |
lemma notnotD: "~~P ==> P"
|
paulson@15411
|
512 |
apply (rule classical)
|
paulson@15411
|
513 |
apply (erule notE)
|
paulson@15411
|
514 |
apply assumption
|
paulson@15411
|
515 |
done
|
paulson@15411
|
516 |
|
paulson@15411
|
517 |
lemma contrapos_pp:
|
paulson@15411
|
518 |
assumes p1: "Q"
|
paulson@15411
|
519 |
and p2: "~P ==> ~Q"
|
paulson@15411
|
520 |
shows "P"
|
nipkow@17589
|
521 |
by (iprover intro: classical p1 p2 notE)
|
paulson@15411
|
522 |
|
paulson@15411
|
523 |
|
haftmann@20944
|
524 |
subsubsection {*Unique existence*}
|
paulson@15411
|
525 |
|
paulson@15411
|
526 |
lemma ex1I:
|
wenzelm@23553
|
527 |
assumes "P a" "!!x. P(x) ==> x=a"
|
paulson@15411
|
528 |
shows "EX! x. P(x)"
|
wenzelm@23553
|
529 |
by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
|
paulson@15411
|
530 |
|
paulson@15411
|
531 |
text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
|
paulson@15411
|
532 |
lemma ex_ex1I:
|
paulson@15411
|
533 |
assumes ex_prem: "EX x. P(x)"
|
paulson@15411
|
534 |
and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
|
paulson@15411
|
535 |
shows "EX! x. P(x)"
|
nipkow@17589
|
536 |
by (iprover intro: ex_prem [THEN exE] ex1I eq)
|
paulson@15411
|
537 |
|
paulson@15411
|
538 |
lemma ex1E:
|
paulson@15411
|
539 |
assumes major: "EX! x. P(x)"
|
paulson@15411
|
540 |
and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
|
paulson@15411
|
541 |
shows "R"
|
paulson@15411
|
542 |
apply (rule major [unfolded Ex1_def, THEN exE])
|
paulson@15411
|
543 |
apply (erule conjE)
|
nipkow@17589
|
544 |
apply (iprover intro: minor)
|
paulson@15411
|
545 |
done
|
paulson@15411
|
546 |
|
paulson@15411
|
547 |
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
|
paulson@15411
|
548 |
apply (erule ex1E)
|
paulson@15411
|
549 |
apply (rule exI)
|
paulson@15411
|
550 |
apply assumption
|
paulson@15411
|
551 |
done
|
paulson@15411
|
552 |
|
paulson@15411
|
553 |
|
haftmann@20944
|
554 |
subsubsection {*THE: definite description operator*}
|
paulson@15411
|
555 |
|
paulson@15411
|
556 |
lemma the_equality:
|
paulson@15411
|
557 |
assumes prema: "P a"
|
paulson@15411
|
558 |
and premx: "!!x. P x ==> x=a"
|
paulson@15411
|
559 |
shows "(THE x. P x) = a"
|
paulson@15411
|
560 |
apply (rule trans [OF _ the_eq_trivial])
|
paulson@15411
|
561 |
apply (rule_tac f = "The" in arg_cong)
|
paulson@15411
|
562 |
apply (rule ext)
|
paulson@15411
|
563 |
apply (rule iffI)
|
paulson@15411
|
564 |
apply (erule premx)
|
paulson@15411
|
565 |
apply (erule ssubst, rule prema)
|
paulson@15411
|
566 |
done
|
paulson@15411
|
567 |
|
paulson@15411
|
568 |
lemma theI:
|
paulson@15411
|
569 |
assumes "P a" and "!!x. P x ==> x=a"
|
paulson@15411
|
570 |
shows "P (THE x. P x)"
|
wenzelm@23553
|
571 |
by (iprover intro: assms the_equality [THEN ssubst])
|
paulson@15411
|
572 |
|
paulson@15411
|
573 |
lemma theI': "EX! x. P x ==> P (THE x. P x)"
|
paulson@15411
|
574 |
apply (erule ex1E)
|
paulson@15411
|
575 |
apply (erule theI)
|
paulson@15411
|
576 |
apply (erule allE)
|
paulson@15411
|
577 |
apply (erule mp)
|
paulson@15411
|
578 |
apply assumption
|
paulson@15411
|
579 |
done
|
paulson@15411
|
580 |
|
paulson@15411
|
581 |
(*Easier to apply than theI: only one occurrence of P*)
|
paulson@15411
|
582 |
lemma theI2:
|
paulson@15411
|
583 |
assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
|
paulson@15411
|
584 |
shows "Q (THE x. P x)"
|
wenzelm@23553
|
585 |
by (iprover intro: assms theI)
|
paulson@15411
|
586 |
|
nipkow@24553
|
587 |
lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
|
nipkow@24553
|
588 |
by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
|
nipkow@24553
|
589 |
elim:allE impE)
|
nipkow@24553
|
590 |
|
wenzelm@18697
|
591 |
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
|
paulson@15411
|
592 |
apply (rule the_equality)
|
paulson@15411
|
593 |
apply assumption
|
paulson@15411
|
594 |
apply (erule ex1E)
|
paulson@15411
|
595 |
apply (erule all_dupE)
|
paulson@15411
|
596 |
apply (drule mp)
|
paulson@15411
|
597 |
apply assumption
|
paulson@15411
|
598 |
apply (erule ssubst)
|
paulson@15411
|
599 |
apply (erule allE)
|
paulson@15411
|
600 |
apply (erule mp)
|
paulson@15411
|
601 |
apply assumption
|
paulson@15411
|
602 |
done
|
paulson@15411
|
603 |
|
paulson@15411
|
604 |
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
|
paulson@15411
|
605 |
apply (rule the_equality)
|
paulson@15411
|
606 |
apply (rule refl)
|
paulson@15411
|
607 |
apply (erule sym)
|
paulson@15411
|
608 |
done
|
paulson@15411
|
609 |
|
paulson@15411
|
610 |
|
haftmann@20944
|
611 |
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
|
paulson@15411
|
612 |
|
paulson@15411
|
613 |
lemma disjCI:
|
paulson@15411
|
614 |
assumes "~Q ==> P" shows "P|Q"
|
paulson@15411
|
615 |
apply (rule classical)
|
wenzelm@23553
|
616 |
apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
|
paulson@15411
|
617 |
done
|
paulson@15411
|
618 |
|
paulson@15411
|
619 |
lemma excluded_middle: "~P | P"
|
nipkow@17589
|
620 |
by (iprover intro: disjCI)
|
paulson@15411
|
621 |
|
haftmann@20944
|
622 |
text {*
|
haftmann@20944
|
623 |
case distinction as a natural deduction rule.
|
haftmann@20944
|
624 |
Note that @{term "~P"} is the second case, not the first
|
haftmann@20944
|
625 |
*}
|
wenzelm@27126
|
626 |
lemma case_split [case_names True False]:
|
paulson@15411
|
627 |
assumes prem1: "P ==> Q"
|
paulson@15411
|
628 |
and prem2: "~P ==> Q"
|
paulson@15411
|
629 |
shows "Q"
|
paulson@15411
|
630 |
apply (rule excluded_middle [THEN disjE])
|
paulson@15411
|
631 |
apply (erule prem2)
|
paulson@15411
|
632 |
apply (erule prem1)
|
paulson@15411
|
633 |
done
|
wenzelm@27126
|
634 |
|
paulson@15411
|
635 |
(*Classical implies (-->) elimination. *)
|
paulson@15411
|
636 |
lemma impCE:
|
paulson@15411
|
637 |
assumes major: "P-->Q"
|
paulson@15411
|
638 |
and minor: "~P ==> R" "Q ==> R"
|
paulson@15411
|
639 |
shows "R"
|
paulson@15411
|
640 |
apply (rule excluded_middle [of P, THEN disjE])
|
nipkow@17589
|
641 |
apply (iprover intro: minor major [THEN mp])+
|
paulson@15411
|
642 |
done
|
paulson@15411
|
643 |
|
paulson@15411
|
644 |
(*This version of --> elimination works on Q before P. It works best for
|
paulson@15411
|
645 |
those cases in which P holds "almost everywhere". Can't install as
|
paulson@15411
|
646 |
default: would break old proofs.*)
|
paulson@15411
|
647 |
lemma impCE':
|
paulson@15411
|
648 |
assumes major: "P-->Q"
|
paulson@15411
|
649 |
and minor: "Q ==> R" "~P ==> R"
|
paulson@15411
|
650 |
shows "R"
|
paulson@15411
|
651 |
apply (rule excluded_middle [of P, THEN disjE])
|
nipkow@17589
|
652 |
apply (iprover intro: minor major [THEN mp])+
|
paulson@15411
|
653 |
done
|
paulson@15411
|
654 |
|
paulson@15411
|
655 |
(*Classical <-> elimination. *)
|
paulson@15411
|
656 |
lemma iffCE:
|
paulson@15411
|
657 |
assumes major: "P=Q"
|
paulson@15411
|
658 |
and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
|
paulson@15411
|
659 |
shows "R"
|
paulson@15411
|
660 |
apply (rule major [THEN iffE])
|
nipkow@17589
|
661 |
apply (iprover intro: minor elim: impCE notE)
|
paulson@15411
|
662 |
done
|
paulson@15411
|
663 |
|
paulson@15411
|
664 |
lemma exCI:
|
paulson@15411
|
665 |
assumes "ALL x. ~P(x) ==> P(a)"
|
paulson@15411
|
666 |
shows "EX x. P(x)"
|
paulson@15411
|
667 |
apply (rule ccontr)
|
wenzelm@23553
|
668 |
apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
|
paulson@15411
|
669 |
done
|
paulson@15411
|
670 |
|
paulson@15411
|
671 |
|
wenzelm@12386
|
672 |
subsubsection {* Intuitionistic Reasoning *}
|
wenzelm@12386
|
673 |
|
wenzelm@12386
|
674 |
lemma impE':
|
wenzelm@12937
|
675 |
assumes 1: "P --> Q"
|
wenzelm@12937
|
676 |
and 2: "Q ==> R"
|
wenzelm@12937
|
677 |
and 3: "P --> Q ==> P"
|
wenzelm@12937
|
678 |
shows R
|
wenzelm@12386
|
679 |
proof -
|
wenzelm@12386
|
680 |
from 3 and 1 have P .
|
wenzelm@12386
|
681 |
with 1 have Q by (rule impE)
|
wenzelm@12386
|
682 |
with 2 show R .
|
wenzelm@12386
|
683 |
qed
|
wenzelm@12386
|
684 |
|
wenzelm@12386
|
685 |
lemma allE':
|
wenzelm@12937
|
686 |
assumes 1: "ALL x. P x"
|
wenzelm@12937
|
687 |
and 2: "P x ==> ALL x. P x ==> Q"
|
wenzelm@12937
|
688 |
shows Q
|
wenzelm@12386
|
689 |
proof -
|
wenzelm@12386
|
690 |
from 1 have "P x" by (rule spec)
|
wenzelm@12386
|
691 |
from this and 1 show Q by (rule 2)
|
wenzelm@12386
|
692 |
qed
|
wenzelm@12386
|
693 |
|
wenzelm@12937
|
694 |
lemma notE':
|
wenzelm@12937
|
695 |
assumes 1: "~ P"
|
wenzelm@12937
|
696 |
and 2: "~ P ==> P"
|
wenzelm@12937
|
697 |
shows R
|
wenzelm@12386
|
698 |
proof -
|
wenzelm@12386
|
699 |
from 2 and 1 have P .
|
wenzelm@12386
|
700 |
with 1 show R by (rule notE)
|
wenzelm@12386
|
701 |
qed
|
wenzelm@12386
|
702 |
|
dixon@22444
|
703 |
lemma TrueE: "True ==> P ==> P" .
|
dixon@22444
|
704 |
lemma notFalseE: "~ False ==> P ==> P" .
|
dixon@22444
|
705 |
|
dixon@22467
|
706 |
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
|
wenzelm@15801
|
707 |
and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
|
wenzelm@15801
|
708 |
and [Pure.elim 2] = allE notE' impE'
|
wenzelm@15801
|
709 |
and [Pure.intro] = exI disjI2 disjI1
|
wenzelm@12386
|
710 |
|
wenzelm@12386
|
711 |
lemmas [trans] = trans
|
wenzelm@12386
|
712 |
and [sym] = sym not_sym
|
wenzelm@15801
|
713 |
and [Pure.elim?] = iffD1 iffD2 impE
|
wenzelm@11438
|
714 |
|
haftmann@28952
|
715 |
use "Tools/hologic.ML"
|
wenzelm@23553
|
716 |
|
wenzelm@11750
|
717 |
|
wenzelm@11750
|
718 |
subsubsection {* Atomizing meta-level connectives *}
|
wenzelm@11750
|
719 |
|
haftmann@28513
|
720 |
axiomatization where
|
haftmann@28513
|
721 |
eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
|
haftmann@28513
|
722 |
|
wenzelm@11750
|
723 |
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
|
wenzelm@12003
|
724 |
proof
|
wenzelm@9488
|
725 |
assume "!!x. P x"
|
wenzelm@23389
|
726 |
then show "ALL x. P x" ..
|
wenzelm@9488
|
727 |
next
|
wenzelm@9488
|
728 |
assume "ALL x. P x"
|
wenzelm@23553
|
729 |
then show "!!x. P x" by (rule allE)
|
wenzelm@9488
|
730 |
qed
|
wenzelm@9488
|
731 |
|
wenzelm@11750
|
732 |
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
|
wenzelm@12003
|
733 |
proof
|
wenzelm@9488
|
734 |
assume r: "A ==> B"
|
wenzelm@10383
|
735 |
show "A --> B" by (rule impI) (rule r)
|
wenzelm@9488
|
736 |
next
|
wenzelm@9488
|
737 |
assume "A --> B" and A
|
wenzelm@23553
|
738 |
then show B by (rule mp)
|
wenzelm@9488
|
739 |
qed
|
wenzelm@9488
|
740 |
|
paulson@14749
|
741 |
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
|
paulson@14749
|
742 |
proof
|
paulson@14749
|
743 |
assume r: "A ==> False"
|
paulson@14749
|
744 |
show "~A" by (rule notI) (rule r)
|
paulson@14749
|
745 |
next
|
paulson@14749
|
746 |
assume "~A" and A
|
wenzelm@23553
|
747 |
then show False by (rule notE)
|
paulson@14749
|
748 |
qed
|
paulson@14749
|
749 |
|
haftmann@39808
|
750 |
lemma atomize_eq [atomize, code]: "(x == y) == Trueprop (x = y)"
|
wenzelm@12003
|
751 |
proof
|
wenzelm@10432
|
752 |
assume "x == y"
|
wenzelm@23553
|
753 |
show "x = y" by (unfold `x == y`) (rule refl)
|
wenzelm@10432
|
754 |
next
|
wenzelm@10432
|
755 |
assume "x = y"
|
wenzelm@23553
|
756 |
then show "x == y" by (rule eq_reflection)
|
wenzelm@10432
|
757 |
qed
|
wenzelm@10432
|
758 |
|
wenzelm@28856
|
759 |
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
|
wenzelm@12003
|
760 |
proof
|
wenzelm@28856
|
761 |
assume conj: "A &&& B"
|
wenzelm@19121
|
762 |
show "A & B"
|
wenzelm@19121
|
763 |
proof (rule conjI)
|
wenzelm@19121
|
764 |
from conj show A by (rule conjunctionD1)
|
wenzelm@19121
|
765 |
from conj show B by (rule conjunctionD2)
|
wenzelm@19121
|
766 |
qed
|
wenzelm@11953
|
767 |
next
|
wenzelm@19121
|
768 |
assume conj: "A & B"
|
wenzelm@28856
|
769 |
show "A &&& B"
|
wenzelm@19121
|
770 |
proof -
|
wenzelm@19121
|
771 |
from conj show A ..
|
wenzelm@19121
|
772 |
from conj show B ..
|
wenzelm@11953
|
773 |
qed
|
wenzelm@11953
|
774 |
qed
|
wenzelm@11953
|
775 |
|
wenzelm@12386
|
776 |
lemmas [symmetric, rulify] = atomize_all atomize_imp
|
wenzelm@18832
|
777 |
and [symmetric, defn] = atomize_all atomize_imp atomize_eq
|
wenzelm@12386
|
778 |
|
wenzelm@11750
|
779 |
|
krauss@26580
|
780 |
subsubsection {* Atomizing elimination rules *}
|
krauss@26580
|
781 |
|
krauss@26580
|
782 |
setup AtomizeElim.setup
|
krauss@26580
|
783 |
|
krauss@26580
|
784 |
lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
|
krauss@26580
|
785 |
by rule iprover+
|
krauss@26580
|
786 |
|
krauss@26580
|
787 |
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
|
krauss@26580
|
788 |
by rule iprover+
|
krauss@26580
|
789 |
|
krauss@26580
|
790 |
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
|
krauss@26580
|
791 |
by rule iprover+
|
krauss@26580
|
792 |
|
krauss@26580
|
793 |
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
|
krauss@26580
|
794 |
|
krauss@26580
|
795 |
|
haftmann@20944
|
796 |
subsection {* Package setup *}
|
haftmann@20944
|
797 |
|
blanchet@35828
|
798 |
subsubsection {* Sledgehammer setup *}
|
blanchet@35828
|
799 |
|
blanchet@35828
|
800 |
text {*
|
blanchet@35828
|
801 |
Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
|
blanchet@35828
|
802 |
that are prolific (match too many equality or membership literals) and relate to
|
blanchet@35828
|
803 |
seldom-used facts. Some duplicate other rules.
|
blanchet@35828
|
804 |
*}
|
blanchet@35828
|
805 |
|
blanchet@35828
|
806 |
ML {*
|
wenzelm@36313
|
807 |
structure No_ATPs = Named_Thms
|
blanchet@35828
|
808 |
(
|
blanchet@35828
|
809 |
val name = "no_atp"
|
blanchet@36060
|
810 |
val description = "theorems that should be filtered out by Sledgehammer"
|
blanchet@35828
|
811 |
)
|
blanchet@35828
|
812 |
*}
|
blanchet@35828
|
813 |
|
blanchet@35828
|
814 |
setup {* No_ATPs.setup *}
|
blanchet@35828
|
815 |
|
blanchet@35828
|
816 |
|
wenzelm@11750
|
817 |
subsubsection {* Classical Reasoner setup *}
|
wenzelm@9529
|
818 |
|
wenzelm@26411
|
819 |
lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
|
wenzelm@26411
|
820 |
by (rule classical) iprover
|
wenzelm@26411
|
821 |
|
wenzelm@26411
|
822 |
lemma swap: "~ P ==> (~ R ==> P) ==> R"
|
wenzelm@26411
|
823 |
by (rule classical) iprover
|
wenzelm@26411
|
824 |
|
haftmann@20944
|
825 |
lemma thin_refl:
|
haftmann@20944
|
826 |
"\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
|
haftmann@20944
|
827 |
|
haftmann@21151
|
828 |
ML {*
|
wenzelm@43671
|
829 |
structure Hypsubst = Hypsubst
|
wenzelm@43671
|
830 |
(
|
wenzelm@21218
|
831 |
val dest_eq = HOLogic.dest_eq
|
haftmann@21151
|
832 |
val dest_Trueprop = HOLogic.dest_Trueprop
|
haftmann@21151
|
833 |
val dest_imp = HOLogic.dest_imp
|
wenzelm@26411
|
834 |
val eq_reflection = @{thm eq_reflection}
|
wenzelm@26411
|
835 |
val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
|
wenzelm@26411
|
836 |
val imp_intr = @{thm impI}
|
wenzelm@26411
|
837 |
val rev_mp = @{thm rev_mp}
|
wenzelm@26411
|
838 |
val subst = @{thm subst}
|
wenzelm@26411
|
839 |
val sym = @{thm sym}
|
wenzelm@22129
|
840 |
val thin_refl = @{thm thin_refl};
|
wenzelm@43671
|
841 |
);
|
wenzelm@21671
|
842 |
open Hypsubst;
|
haftmann@21151
|
843 |
|
wenzelm@43671
|
844 |
structure Classical = Classical
|
wenzelm@43671
|
845 |
(
|
wenzelm@26411
|
846 |
val imp_elim = @{thm imp_elim}
|
wenzelm@26411
|
847 |
val not_elim = @{thm notE}
|
wenzelm@26411
|
848 |
val swap = @{thm swap}
|
wenzelm@26411
|
849 |
val classical = @{thm classical}
|
haftmann@21151
|
850 |
val sizef = Drule.size_of_thm
|
haftmann@21151
|
851 |
val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
|
wenzelm@43671
|
852 |
);
|
haftmann@21151
|
853 |
|
wenzelm@33316
|
854 |
structure Basic_Classical: BASIC_CLASSICAL = Classical;
|
wenzelm@33316
|
855 |
open Basic_Classical;
|
wenzelm@44446
|
856 |
*}
|
wenzelm@22129
|
857 |
|
wenzelm@44446
|
858 |
setup {*
|
wenzelm@44446
|
859 |
ML_Antiquote.value @{binding claset}
|
wenzelm@44446
|
860 |
(Scan.succeed "Classical.claset_of (ML_Context.the_local_context ())")
|
haftmann@21151
|
861 |
*}
|
haftmann@21151
|
862 |
|
wenzelm@33316
|
863 |
setup Classical.setup
|
paulson@24286
|
864 |
|
haftmann@21009
|
865 |
setup {*
|
haftmann@21009
|
866 |
let
|
haftmann@39093
|
867 |
fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
|
wenzelm@35389
|
868 |
| non_bool_eq _ = false;
|
wenzelm@35389
|
869 |
val hyp_subst_tac' =
|
wenzelm@35389
|
870 |
SUBGOAL (fn (goal, i) =>
|
wenzelm@35389
|
871 |
if Term.exists_Const non_bool_eq goal
|
wenzelm@35389
|
872 |
then Hypsubst.hyp_subst_tac i
|
wenzelm@35389
|
873 |
else no_tac);
|
haftmann@21009
|
874 |
in
|
haftmann@21151
|
875 |
Hypsubst.hypsubst_setup
|
wenzelm@35389
|
876 |
(*prevent substitution on bool*)
|
wenzelm@33369
|
877 |
#> Context_Rules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
|
haftmann@21009
|
878 |
end
|
haftmann@21009
|
879 |
*}
|
haftmann@21009
|
880 |
|
haftmann@21009
|
881 |
declare iffI [intro!]
|
haftmann@21009
|
882 |
and notI [intro!]
|
haftmann@21009
|
883 |
and impI [intro!]
|
haftmann@21009
|
884 |
and disjCI [intro!]
|
haftmann@21009
|
885 |
and conjI [intro!]
|
haftmann@21009
|
886 |
and TrueI [intro!]
|
haftmann@21009
|
887 |
and refl [intro!]
|
haftmann@21009
|
888 |
|
haftmann@21009
|
889 |
declare iffCE [elim!]
|
haftmann@21009
|
890 |
and FalseE [elim!]
|
haftmann@21009
|
891 |
and impCE [elim!]
|
haftmann@21009
|
892 |
and disjE [elim!]
|
haftmann@21009
|
893 |
and conjE [elim!]
|
haftmann@21009
|
894 |
|
haftmann@21009
|
895 |
declare ex_ex1I [intro!]
|
haftmann@21009
|
896 |
and allI [intro!]
|
haftmann@21009
|
897 |
and the_equality [intro]
|
haftmann@21009
|
898 |
and exI [intro]
|
haftmann@21009
|
899 |
|
haftmann@21009
|
900 |
declare exE [elim!]
|
haftmann@21009
|
901 |
allE [elim]
|
haftmann@21009
|
902 |
|
wenzelm@22377
|
903 |
ML {* val HOL_cs = @{claset} *}
|
wenzelm@11977
|
904 |
|
wenzelm@20223
|
905 |
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
|
wenzelm@20223
|
906 |
apply (erule swap)
|
wenzelm@20223
|
907 |
apply (erule (1) meta_mp)
|
wenzelm@20223
|
908 |
done
|
wenzelm@10383
|
909 |
|
wenzelm@18689
|
910 |
declare ex_ex1I [rule del, intro! 2]
|
wenzelm@18689
|
911 |
and ex1I [intro]
|
wenzelm@18689
|
912 |
|
paulson@42736
|
913 |
declare ext [intro]
|
paulson@42736
|
914 |
|
wenzelm@12386
|
915 |
lemmas [intro?] = ext
|
wenzelm@12386
|
916 |
and [elim?] = ex1_implies_ex
|
wenzelm@11977
|
917 |
|
haftmann@20944
|
918 |
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
|
haftmann@20973
|
919 |
lemma alt_ex1E [elim!]:
|
haftmann@20944
|
920 |
assumes major: "\<exists>!x. P x"
|
haftmann@20944
|
921 |
and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
|
haftmann@20944
|
922 |
shows R
|
haftmann@20944
|
923 |
apply (rule ex1E [OF major])
|
haftmann@20944
|
924 |
apply (rule prem)
|
wenzelm@22129
|
925 |
apply (tactic {* ares_tac @{thms allI} 1 *})+
|
wenzelm@22129
|
926 |
apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
|
wenzelm@22129
|
927 |
apply iprover
|
wenzelm@22129
|
928 |
done
|
haftmann@20944
|
929 |
|
haftmann@21151
|
930 |
ML {*
|
wenzelm@43348
|
931 |
structure Blast = Blast
|
wenzelm@43348
|
932 |
(
|
wenzelm@43348
|
933 |
structure Classical = Classical
|
wenzelm@43674
|
934 |
val Trueprop_const = dest_Const @{const Trueprop}
|
wenzelm@43348
|
935 |
val equality_name = @{const_name HOL.eq}
|
wenzelm@43348
|
936 |
val not_name = @{const_name Not}
|
wenzelm@43348
|
937 |
val notE = @{thm notE}
|
wenzelm@43348
|
938 |
val ccontr = @{thm ccontr}
|
wenzelm@43348
|
939 |
val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
|
wenzelm@43348
|
940 |
);
|
wenzelm@43348
|
941 |
val blast_tac = Blast.blast_tac;
|
haftmann@20944
|
942 |
*}
|
haftmann@20944
|
943 |
|
haftmann@21151
|
944 |
setup Blast.setup
|
haftmann@21151
|
945 |
|
haftmann@20944
|
946 |
|
haftmann@20944
|
947 |
subsubsection {* Simplifier *}
|
wenzelm@12281
|
948 |
|
wenzelm@12281
|
949 |
lemma eta_contract_eq: "(%s. f s) = f" ..
|
wenzelm@12281
|
950 |
|
wenzelm@12281
|
951 |
lemma simp_thms:
|
wenzelm@12937
|
952 |
shows not_not: "(~ ~ P) = P"
|
nipkow@15354
|
953 |
and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
|
wenzelm@12937
|
954 |
and
|
berghofe@12436
|
955 |
"(P ~= Q) = (P = (~Q))"
|
berghofe@12436
|
956 |
"(P | ~P) = True" "(~P | P) = True"
|
wenzelm@12281
|
957 |
"(x = x) = True"
|
haftmann@32060
|
958 |
and not_True_eq_False [code]: "(\<not> True) = False"
|
haftmann@32060
|
959 |
and not_False_eq_True [code]: "(\<not> False) = True"
|
haftmann@20944
|
960 |
and
|
berghofe@12436
|
961 |
"(~P) ~= P" "P ~= (~P)"
|
haftmann@20944
|
962 |
"(True=P) = P"
|
haftmann@20944
|
963 |
and eq_True: "(P = True) = P"
|
haftmann@20944
|
964 |
and "(False=P) = (~P)"
|
haftmann@20944
|
965 |
and eq_False: "(P = False) = (\<not> P)"
|
haftmann@20944
|
966 |
and
|
wenzelm@12281
|
967 |
"(True --> P) = P" "(False --> P) = True"
|
wenzelm@12281
|
968 |
"(P --> True) = True" "(P --> P) = True"
|
wenzelm@12281
|
969 |
"(P --> False) = (~P)" "(P --> ~P) = (~P)"
|
wenzelm@12281
|
970 |
"(P & True) = P" "(True & P) = P"
|
wenzelm@12281
|
971 |
"(P & False) = False" "(False & P) = False"
|
wenzelm@12281
|
972 |
"(P & P) = P" "(P & (P & Q)) = (P & Q)"
|
wenzelm@12281
|
973 |
"(P & ~P) = False" "(~P & P) = False"
|
wenzelm@12281
|
974 |
"(P | True) = True" "(True | P) = True"
|
wenzelm@12281
|
975 |
"(P | False) = P" "(False | P) = P"
|
berghofe@12436
|
976 |
"(P | P) = P" "(P | (P | Q)) = (P | Q)" and
|
wenzelm@12281
|
977 |
"(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
|
nipkow@31166
|
978 |
and
|
wenzelm@12281
|
979 |
"!!P. (EX x. x=t & P(x)) = P(t)"
|
wenzelm@12281
|
980 |
"!!P. (EX x. t=x & P(x)) = P(t)"
|
wenzelm@12281
|
981 |
"!!P. (ALL x. x=t --> P(x)) = P(t)"
|
wenzelm@12937
|
982 |
"!!P. (ALL x. t=x --> P(x)) = P(t)"
|
nipkow@17589
|
983 |
by (blast, blast, blast, blast, blast, iprover+)
|
wenzelm@13421
|
984 |
|
paulson@14201
|
985 |
lemma disj_absorb: "(A | A) = A"
|
paulson@14201
|
986 |
by blast
|
paulson@14201
|
987 |
|
paulson@14201
|
988 |
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
|
paulson@14201
|
989 |
by blast
|
paulson@14201
|
990 |
|
paulson@14201
|
991 |
lemma conj_absorb: "(A & A) = A"
|
paulson@14201
|
992 |
by blast
|
paulson@14201
|
993 |
|
paulson@14201
|
994 |
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
|
paulson@14201
|
995 |
by blast
|
paulson@14201
|
996 |
|
wenzelm@12281
|
997 |
lemma eq_ac:
|
wenzelm@12937
|
998 |
shows eq_commute: "(a=b) = (b=a)"
|
wenzelm@12937
|
999 |
and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
|
nipkow@17589
|
1000 |
and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
|
nipkow@17589
|
1001 |
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
|
wenzelm@12281
|
1002 |
|
wenzelm@12281
|
1003 |
lemma conj_comms:
|
wenzelm@12937
|
1004 |
shows conj_commute: "(P&Q) = (Q&P)"
|
nipkow@17589
|
1005 |
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
|
nipkow@17589
|
1006 |
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
|
wenzelm@12281
|
1007 |
|
paulson@19174
|
1008 |
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
|
paulson@19174
|
1009 |
|
wenzelm@12281
|
1010 |
lemma disj_comms:
|
wenzelm@12937
|
1011 |
shows disj_commute: "(P|Q) = (Q|P)"
|
nipkow@17589
|
1012 |
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
|
nipkow@17589
|
1013 |
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
|
wenzelm@12281
|
1014 |
|
paulson@19174
|
1015 |
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
|
paulson@19174
|
1016 |
|
nipkow@17589
|
1017 |
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
|
nipkow@17589
|
1018 |
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
|
wenzelm@12281
|
1019 |
|
nipkow@17589
|
1020 |
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
|
nipkow@17589
|
1021 |
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
|
wenzelm@12281
|
1022 |
|
nipkow@17589
|
1023 |
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
|
nipkow@17589
|
1024 |
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
|
nipkow@17589
|
1025 |
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
|
wenzelm@12281
|
1026 |
|
wenzelm@12281
|
1027 |
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
|
wenzelm@12281
|
1028 |
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
|
wenzelm@12281
|
1029 |
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
|
wenzelm@12281
|
1030 |
|
wenzelm@12281
|
1031 |
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
|
wenzelm@12281
|
1032 |
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
|
wenzelm@12281
|
1033 |
|
haftmann@21151
|
1034 |
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
|
haftmann@21151
|
1035 |
by iprover
|
haftmann@21151
|
1036 |
|
nipkow@17589
|
1037 |
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
|
wenzelm@12281
|
1038 |
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
|
wenzelm@12281
|
1039 |
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
|
wenzelm@12281
|
1040 |
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
|
wenzelm@12281
|
1041 |
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
|
wenzelm@12281
|
1042 |
lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
|
wenzelm@12281
|
1043 |
by blast
|
wenzelm@12281
|
1044 |
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
|
wenzelm@12281
|
1045 |
|
nipkow@17589
|
1046 |
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
|
wenzelm@12281
|
1047 |
|
wenzelm@12281
|
1048 |
|
wenzelm@12281
|
1049 |
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
|
wenzelm@12281
|
1050 |
-- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
|
wenzelm@12281
|
1051 |
-- {* cases boil down to the same thing. *}
|
wenzelm@12281
|
1052 |
by blast
|
wenzelm@12281
|
1053 |
|
wenzelm@12281
|
1054 |
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
|
wenzelm@12281
|
1055 |
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
|
nipkow@17589
|
1056 |
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
|
nipkow@17589
|
1057 |
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
|
chaieb@23403
|
1058 |
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
|
wenzelm@12281
|
1059 |
|
blanchet@35828
|
1060 |
declare All_def [no_atp]
|
paulson@24286
|
1061 |
|
nipkow@17589
|
1062 |
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
|
nipkow@17589
|
1063 |
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
|
wenzelm@12281
|
1064 |
|
wenzelm@12281
|
1065 |
text {*
|
wenzelm@12281
|
1066 |
\medskip The @{text "&"} congruence rule: not included by default!
|
wenzelm@12281
|
1067 |
May slow rewrite proofs down by as much as 50\% *}
|
wenzelm@12281
|
1068 |
|
wenzelm@12281
|
1069 |
lemma conj_cong:
|
wenzelm@12281
|
1070 |
"(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
|
nipkow@17589
|
1071 |
by iprover
|
wenzelm@12281
|
1072 |
|
wenzelm@12281
|
1073 |
lemma rev_conj_cong:
|
wenzelm@12281
|
1074 |
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
|
nipkow@17589
|
1075 |
by iprover
|
wenzelm@12281
|
1076 |
|
wenzelm@12281
|
1077 |
text {* The @{text "|"} congruence rule: not included by default! *}
|
wenzelm@12281
|
1078 |
|
wenzelm@12281
|
1079 |
lemma disj_cong:
|
wenzelm@12281
|
1080 |
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
|
wenzelm@12281
|
1081 |
by blast
|
wenzelm@12281
|
1082 |
|
wenzelm@12281
|
1083 |
|
wenzelm@12281
|
1084 |
text {* \medskip if-then-else rules *}
|
wenzelm@12281
|
1085 |
|
haftmann@32060
|
1086 |
lemma if_True [code]: "(if True then x else y) = x"
|
haftmann@38750
|
1087 |
by (unfold If_def) blast
|
wenzelm@12281
|
1088 |
|
haftmann@32060
|
1089 |
lemma if_False [code]: "(if False then x else y) = y"
|
haftmann@38750
|
1090 |
by (unfold If_def) blast
|
wenzelm@12281
|
1091 |
|
wenzelm@12281
|
1092 |
lemma if_P: "P ==> (if P then x else y) = x"
|
haftmann@38750
|
1093 |
by (unfold If_def) blast
|
wenzelm@12281
|
1094 |
|
wenzelm@12281
|
1095 |
lemma if_not_P: "~P ==> (if P then x else y) = y"
|
haftmann@38750
|
1096 |
by (unfold If_def) blast
|
wenzelm@12281
|
1097 |
|
wenzelm@12281
|
1098 |
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
|
wenzelm@12281
|
1099 |
apply (rule case_split [of Q])
|
paulson@15481
|
1100 |
apply (simplesubst if_P)
|
paulson@15481
|
1101 |
prefer 3 apply (simplesubst if_not_P, blast+)
|
wenzelm@12281
|
1102 |
done
|
wenzelm@12281
|
1103 |
|
wenzelm@12281
|
1104 |
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
|
paulson@15481
|
1105 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1106 |
|
blanchet@35828
|
1107 |
lemmas if_splits [no_atp] = split_if split_if_asm
|
wenzelm@12281
|
1108 |
|
wenzelm@12281
|
1109 |
lemma if_cancel: "(if c then x else x) = x"
|
paulson@15481
|
1110 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1111 |
|
wenzelm@12281
|
1112 |
lemma if_eq_cancel: "(if x = y then y else x) = x"
|
paulson@15481
|
1113 |
by (simplesubst split_if, blast)
|
wenzelm@12281
|
1114 |
|
blanchet@42663
|
1115 |
lemma if_bool_eq_conj:
|
blanchet@42663
|
1116 |
"(if P then Q else R) = ((P-->Q) & (~P-->R))"
|
wenzelm@19796
|
1117 |
-- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
|
wenzelm@12281
|
1118 |
by (rule split_if)
|
wenzelm@12281
|
1119 |
|
wenzelm@12281
|
1120 |
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
|
wenzelm@19796
|
1121 |
-- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
|
paulson@15481
|
1122 |
apply (simplesubst split_if, blast)
|
wenzelm@12281
|
1123 |
done
|
wenzelm@12281
|
1124 |
|
nipkow@17589
|
1125 |
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
|
nipkow@17589
|
1126 |
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
|
wenzelm@12281
|
1127 |
|
schirmer@15423
|
1128 |
text {* \medskip let rules for simproc *}
|
schirmer@15423
|
1129 |
|
schirmer@15423
|
1130 |
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
|
schirmer@15423
|
1131 |
by (unfold Let_def)
|
schirmer@15423
|
1132 |
|
schirmer@15423
|
1133 |
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
|
schirmer@15423
|
1134 |
by (unfold Let_def)
|
schirmer@15423
|
1135 |
|
berghofe@16633
|
1136 |
text {*
|
ballarin@16999
|
1137 |
The following copy of the implication operator is useful for
|
ballarin@16999
|
1138 |
fine-tuning congruence rules. It instructs the simplifier to simplify
|
ballarin@16999
|
1139 |
its premise.
|
berghofe@16633
|
1140 |
*}
|
berghofe@16633
|
1141 |
|
haftmann@35413
|
1142 |
definition simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1) where
|
haftmann@37767
|
1143 |
"simp_implies \<equiv> op ==>"
|
berghofe@16633
|
1144 |
|
wenzelm@18457
|
1145 |
lemma simp_impliesI:
|
berghofe@16633
|
1146 |
assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
|
berghofe@16633
|
1147 |
shows "PROP P =simp=> PROP Q"
|
berghofe@16633
|
1148 |
apply (unfold simp_implies_def)
|
berghofe@16633
|
1149 |
apply (rule PQ)
|
berghofe@16633
|
1150 |
apply assumption
|
berghofe@16633
|
1151 |
done
|
berghofe@16633
|
1152 |
|
berghofe@16633
|
1153 |
lemma simp_impliesE:
|
wenzelm@25388
|
1154 |
assumes PQ: "PROP P =simp=> PROP Q"
|
berghofe@16633
|
1155 |
and P: "PROP P"
|
berghofe@16633
|
1156 |
and QR: "PROP Q \<Longrightarrow> PROP R"
|
berghofe@16633
|
1157 |
shows "PROP R"
|
berghofe@16633
|
1158 |
apply (rule QR)
|
berghofe@16633
|
1159 |
apply (rule PQ [unfolded simp_implies_def])
|
berghofe@16633
|
1160 |
apply (rule P)
|
berghofe@16633
|
1161 |
done
|
berghofe@16633
|
1162 |
|
berghofe@16633
|
1163 |
lemma simp_implies_cong:
|
berghofe@16633
|
1164 |
assumes PP' :"PROP P == PROP P'"
|
berghofe@16633
|
1165 |
and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
|
berghofe@16633
|
1166 |
shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
|
berghofe@16633
|
1167 |
proof (unfold simp_implies_def, rule equal_intr_rule)
|
berghofe@16633
|
1168 |
assume PQ: "PROP P \<Longrightarrow> PROP Q"
|
berghofe@16633
|
1169 |
and P': "PROP P'"
|
berghofe@16633
|
1170 |
from PP' [symmetric] and P' have "PROP P"
|
berghofe@16633
|
1171 |
by (rule equal_elim_rule1)
|
wenzelm@23553
|
1172 |
then have "PROP Q" by (rule PQ)
|
berghofe@16633
|
1173 |
with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
|
berghofe@16633
|
1174 |
next
|
berghofe@16633
|
1175 |
assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
|
berghofe@16633
|
1176 |
and P: "PROP P"
|
berghofe@16633
|
1177 |
from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
|
wenzelm@23553
|
1178 |
then have "PROP Q'" by (rule P'Q')
|
berghofe@16633
|
1179 |
with P'QQ' [OF P', symmetric] show "PROP Q"
|
berghofe@16633
|
1180 |
by (rule equal_elim_rule1)
|
berghofe@16633
|
1181 |
qed
|
berghofe@16633
|
1182 |
|
haftmann@20944
|
1183 |
lemma uncurry:
|
haftmann@20944
|
1184 |
assumes "P \<longrightarrow> Q \<longrightarrow> R"
|
haftmann@20944
|
1185 |
shows "P \<and> Q \<longrightarrow> R"
|
wenzelm@23553
|
1186 |
using assms by blast
|
haftmann@20944
|
1187 |
|
haftmann@20944
|
1188 |
lemma iff_allI:
|
haftmann@20944
|
1189 |
assumes "\<And>x. P x = Q x"
|
haftmann@20944
|
1190 |
shows "(\<forall>x. P x) = (\<forall>x. Q x)"
|
wenzelm@23553
|
1191 |
using assms by blast
|
haftmann@20944
|
1192 |
|
haftmann@20944
|
1193 |
lemma iff_exI:
|
haftmann@20944
|
1194 |
assumes "\<And>x. P x = Q x"
|
haftmann@20944
|
1195 |
shows "(\<exists>x. P x) = (\<exists>x. Q x)"
|
wenzelm@23553
|
1196 |
using assms by blast
|
haftmann@20944
|
1197 |
|
haftmann@20944
|
1198 |
lemma all_comm:
|
haftmann@20944
|
1199 |
"(\<forall>x y. P x y) = (\<forall>y x. P x y)"
|
haftmann@20944
|
1200 |
by blast
|
haftmann@20944
|
1201 |
|
haftmann@20944
|
1202 |
lemma ex_comm:
|
haftmann@20944
|
1203 |
"(\<exists>x y. P x y) = (\<exists>y x. P x y)"
|
haftmann@20944
|
1204 |
by blast
|
haftmann@20944
|
1205 |
|
haftmann@28952
|
1206 |
use "Tools/simpdata.ML"
|
wenzelm@21671
|
1207 |
ML {* open Simpdata *}
|
wenzelm@43326
|
1208 |
|
wenzelm@43667
|
1209 |
setup {* Simplifier.map_simpset_global (K HOL_basic_ss) *}
|
wenzelm@43326
|
1210 |
|
wenzelm@43330
|
1211 |
simproc_setup defined_Ex ("EX x. P x") = {* fn _ => Quantifier1.rearrange_ex *}
|
wenzelm@43330
|
1212 |
simproc_setup defined_All ("ALL x. P x") = {* fn _ => Quantifier1.rearrange_all *}
|
wenzelm@21671
|
1213 |
|
haftmann@21151
|
1214 |
setup {*
|
haftmann@21151
|
1215 |
Simplifier.method_setup Splitter.split_modifiers
|
haftmann@21151
|
1216 |
#> Splitter.setup
|
wenzelm@26496
|
1217 |
#> clasimp_setup
|
haftmann@21151
|
1218 |
#> EqSubst.setup
|
haftmann@21151
|
1219 |
*}
|
haftmann@21151
|
1220 |
|
wenzelm@24035
|
1221 |
text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
|
wenzelm@24035
|
1222 |
|
wenzelm@24035
|
1223 |
simproc_setup neq ("x = y") = {* fn _ =>
|
wenzelm@24035
|
1224 |
let
|
wenzelm@24035
|
1225 |
val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
|
wenzelm@24035
|
1226 |
fun is_neq eq lhs rhs thm =
|
wenzelm@24035
|
1227 |
(case Thm.prop_of thm of
|
wenzelm@24035
|
1228 |
_ $ (Not $ (eq' $ l' $ r')) =>
|
wenzelm@24035
|
1229 |
Not = HOLogic.Not andalso eq' = eq andalso
|
wenzelm@24035
|
1230 |
r' aconv lhs andalso l' aconv rhs
|
wenzelm@24035
|
1231 |
| _ => false);
|
wenzelm@24035
|
1232 |
fun proc ss ct =
|
wenzelm@24035
|
1233 |
(case Thm.term_of ct of
|
wenzelm@24035
|
1234 |
eq $ lhs $ rhs =>
|
wenzelm@44470
|
1235 |
(case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
|
wenzelm@24035
|
1236 |
SOME thm => SOME (thm RS neq_to_EQ_False)
|
wenzelm@24035
|
1237 |
| NONE => NONE)
|
wenzelm@24035
|
1238 |
| _ => NONE);
|
wenzelm@24035
|
1239 |
in proc end;
|
wenzelm@24035
|
1240 |
*}
|
wenzelm@24035
|
1241 |
|
wenzelm@24035
|
1242 |
simproc_setup let_simp ("Let x f") = {*
|
wenzelm@24035
|
1243 |
let
|
wenzelm@24035
|
1244 |
val (f_Let_unfold, x_Let_unfold) =
|
haftmann@28741
|
1245 |
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}
|
wenzelm@24035
|
1246 |
in (cterm_of @{theory} f, cterm_of @{theory} x) end
|
wenzelm@24035
|
1247 |
val (f_Let_folded, x_Let_folded) =
|
haftmann@28741
|
1248 |
let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}
|
wenzelm@24035
|
1249 |
in (cterm_of @{theory} f, cterm_of @{theory} x) end;
|
wenzelm@24035
|
1250 |
val g_Let_folded =
|
haftmann@28741
|
1251 |
let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}
|
haftmann@28741
|
1252 |
in cterm_of @{theory} g end;
|
haftmann@28741
|
1253 |
fun count_loose (Bound i) k = if i >= k then 1 else 0
|
haftmann@28741
|
1254 |
| count_loose (s $ t) k = count_loose s k + count_loose t k
|
haftmann@28741
|
1255 |
| count_loose (Abs (_, _, t)) k = count_loose t (k + 1)
|
haftmann@28741
|
1256 |
| count_loose _ _ = 0;
|
haftmann@28741
|
1257 |
fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
|
haftmann@28741
|
1258 |
case t
|
haftmann@28741
|
1259 |
of Abs (_, _, t') => count_loose t' 0 <= 1
|
haftmann@28741
|
1260 |
| _ => true;
|
haftmann@28741
|
1261 |
in fn _ => fn ss => fn ct => if is_trivial_let (Thm.term_of ct)
|
haftmann@31151
|
1262 |
then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
|
haftmann@28741
|
1263 |
else let (*Norbert Schirmer's case*)
|
haftmann@28741
|
1264 |
val ctxt = Simplifier.the_context ss;
|
wenzelm@43232
|
1265 |
val thy = Proof_Context.theory_of ctxt;
|
haftmann@28741
|
1266 |
val t = Thm.term_of ct;
|
haftmann@28741
|
1267 |
val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
|
haftmann@28741
|
1268 |
in Option.map (hd o Variable.export ctxt' ctxt o single)
|
haftmann@28741
|
1269 |
(case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
|
haftmann@28741
|
1270 |
if is_Free x orelse is_Bound x orelse is_Const x
|
haftmann@28741
|
1271 |
then SOME @{thm Let_def}
|
haftmann@28741
|
1272 |
else
|
haftmann@28741
|
1273 |
let
|
haftmann@28741
|
1274 |
val n = case f of (Abs (x, _, _)) => x | _ => "x";
|
haftmann@28741
|
1275 |
val cx = cterm_of thy x;
|
haftmann@28741
|
1276 |
val {T = xT, ...} = rep_cterm cx;
|
haftmann@28741
|
1277 |
val cf = cterm_of thy f;
|
haftmann@28741
|
1278 |
val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
|
haftmann@28741
|
1279 |
val (_ $ _ $ g) = prop_of fx_g;
|
haftmann@28741
|
1280 |
val g' = abstract_over (x,g);
|
haftmann@28741
|
1281 |
in (if (g aconv g')
|
haftmann@28741
|
1282 |
then
|
haftmann@28741
|
1283 |
let
|
haftmann@28741
|
1284 |
val rl =
|
haftmann@28741
|
1285 |
cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
|
haftmann@28741
|
1286 |
in SOME (rl OF [fx_g]) end
|
haftmann@28741
|
1287 |
else if Term.betapply (f, x) aconv g then NONE (*avoid identity conversion*)
|
haftmann@28741
|
1288 |
else let
|
haftmann@28741
|
1289 |
val abs_g'= Abs (n,xT,g');
|
haftmann@28741
|
1290 |
val g'x = abs_g'$x;
|
wenzelm@36945
|
1291 |
val g_g'x = Thm.symmetric (Thm.beta_conversion false (cterm_of thy g'x));
|
haftmann@28741
|
1292 |
val rl = cterm_instantiate
|
haftmann@28741
|
1293 |
[(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),
|
haftmann@28741
|
1294 |
(g_Let_folded, cterm_of thy abs_g')]
|
haftmann@28741
|
1295 |
@{thm Let_folded};
|
wenzelm@36945
|
1296 |
in SOME (rl OF [Thm.transitive fx_g g_g'x])
|
haftmann@28741
|
1297 |
end)
|
haftmann@28741
|
1298 |
end
|
haftmann@28741
|
1299 |
| _ => NONE)
|
haftmann@28741
|
1300 |
end
|
haftmann@28741
|
1301 |
end *}
|
wenzelm@24035
|
1302 |
|
haftmann@21151
|
1303 |
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
|
haftmann@21151
|
1304 |
proof
|
wenzelm@23389
|
1305 |
assume "True \<Longrightarrow> PROP P"
|
wenzelm@23389
|
1306 |
from this [OF TrueI] show "PROP P" .
|
haftmann@21151
|
1307 |
next
|
haftmann@21151
|
1308 |
assume "PROP P"
|
wenzelm@23389
|
1309 |
then show "PROP P" .
|
haftmann@21151
|
1310 |
qed
|
haftmann@21151
|
1311 |
|
haftmann@21151
|
1312 |
lemma ex_simps:
|
haftmann@21151
|
1313 |
"!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
|
haftmann@21151
|
1314 |
"!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
|
haftmann@21151
|
1315 |
"!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
|
haftmann@21151
|
1316 |
"!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
|
haftmann@21151
|
1317 |
"!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
|
haftmann@21151
|
1318 |
"!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
|
haftmann@21151
|
1319 |
-- {* Miniscoping: pushing in existential quantifiers. *}
|
haftmann@21151
|
1320 |
by (iprover | blast)+
|
haftmann@21151
|
1321 |
|
haftmann@21151
|
1322 |
lemma all_simps:
|
haftmann@21151
|
1323 |
"!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
|
haftmann@21151
|
1324 |
"!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
|
haftmann@21151
|
1325 |
"!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
|
haftmann@21151
|
1326 |
"!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
|
haftmann@21151
|
1327 |
"!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
|
haftmann@21151
|
1328 |
"!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
|
haftmann@21151
|
1329 |
-- {* Miniscoping: pushing in universal quantifiers. *}
|
haftmann@21151
|
1330 |
by (iprover | blast)+
|
paulson@15481
|
1331 |
|
wenzelm@21671
|
1332 |
lemmas [simp] =
|
wenzelm@21671
|
1333 |
triv_forall_equality (*prunes params*)
|
wenzelm@21671
|
1334 |
True_implies_equals (*prune asms `True'*)
|
wenzelm@21671
|
1335 |
if_True
|
wenzelm@21671
|
1336 |
if_False
|
wenzelm@21671
|
1337 |
if_cancel
|
wenzelm@21671
|
1338 |
if_eq_cancel
|
wenzelm@21671
|
1339 |
imp_disjL
|
haftmann@20973
|
1340 |
(*In general it seems wrong to add distributive laws by default: they
|
haftmann@20973
|
1341 |
might cause exponential blow-up. But imp_disjL has been in for a while
|
haftmann@20973
|
1342 |
and cannot be removed without affecting existing proofs. Moreover,
|
haftmann@20973
|
1343 |
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
|
haftmann@20973
|
1344 |
grounds that it allows simplification of R in the two cases.*)
|
wenzelm@21671
|
1345 |
conj_assoc
|
wenzelm@21671
|
1346 |
disj_assoc
|
wenzelm@21671
|
1347 |
de_Morgan_conj
|
wenzelm@21671
|
1348 |
de_Morgan_disj
|
wenzelm@21671
|
1349 |
imp_disj1
|
wenzelm@21671
|
1350 |
imp_disj2
|
wenzelm@21671
|
1351 |
not_imp
|
wenzelm@21671
|
1352 |
disj_not1
|
wenzelm@21671
|
1353 |
not_all
|
wenzelm@21671
|
1354 |
not_ex
|
wenzelm@21671
|
1355 |
cases_simp
|
wenzelm@21671
|
1356 |
the_eq_trivial
|
wenzelm@21671
|
1357 |
the_sym_eq_trivial
|
wenzelm@21671
|
1358 |
ex_simps
|
wenzelm@21671
|
1359 |
all_simps
|
wenzelm@21671
|
1360 |
simp_thms
|
wenzelm@21671
|
1361 |
|
wenzelm@21671
|
1362 |
lemmas [cong] = imp_cong simp_implies_cong
|
wenzelm@21671
|
1363 |
lemmas [split] = split_if
|
haftmann@20973
|
1364 |
|
wenzelm@22377
|
1365 |
ML {* val HOL_ss = @{simpset} *}
|
haftmann@20973
|
1366 |
|
haftmann@20944
|
1367 |
text {* Simplifies x assuming c and y assuming ~c *}
|
haftmann@20944
|
1368 |
lemma if_cong:
|
haftmann@20944
|
1369 |
assumes "b = c"
|
haftmann@20944
|
1370 |
and "c \<Longrightarrow> x = u"
|
haftmann@20944
|
1371 |
and "\<not> c \<Longrightarrow> y = v"
|
haftmann@20944
|
1372 |
shows "(if b then x else y) = (if c then u else v)"
|
haftmann@38750
|
1373 |
using assms by simp
|
haftmann@20944
|
1374 |
|
haftmann@20944
|
1375 |
text {* Prevents simplification of x and y:
|
haftmann@20944
|
1376 |
faster and allows the execution of functional programs. *}
|
haftmann@20944
|
1377 |
lemma if_weak_cong [cong]:
|
haftmann@20944
|
1378 |
assumes "b = c"
|
haftmann@20944
|
1379 |
shows "(if b then x else y) = (if c then x else y)"
|
wenzelm@23553
|
1380 |
using assms by (rule arg_cong)
|
haftmann@20944
|
1381 |
|
haftmann@20944
|
1382 |
text {* Prevents simplification of t: much faster *}
|
haftmann@20944
|
1383 |
lemma let_weak_cong:
|
haftmann@20944
|
1384 |
assumes "a = b"
|
haftmann@20944
|
1385 |
shows "(let x = a in t x) = (let x = b in t x)"
|
wenzelm@23553
|
1386 |
using assms by (rule arg_cong)
|
haftmann@20944
|
1387 |
|
haftmann@20944
|
1388 |
text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
|
haftmann@20944
|
1389 |
lemma eq_cong2:
|
haftmann@20944
|
1390 |
assumes "u = u'"
|
haftmann@20944
|
1391 |
shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
|
wenzelm@23553
|
1392 |
using assms by simp
|
haftmann@20944
|
1393 |
|
haftmann@20944
|
1394 |
lemma if_distrib:
|
haftmann@20944
|
1395 |
"f (if c then x else y) = (if c then f x else f y)"
|
haftmann@20944
|
1396 |
by simp
|
haftmann@20944
|
1397 |
|
haftmann@45140
|
1398 |
text{*As a simplification rule, it replaces all function equalities by
|
haftmann@45140
|
1399 |
first-order equalities.*}
|
haftmann@45140
|
1400 |
lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
|
haftmann@45140
|
1401 |
by auto
|
haftmann@45140
|
1402 |
|
haftmann@20944
|
1403 |
|
haftmann@20944
|
1404 |
subsubsection {* Generic cases and induction *}
|
haftmann@20944
|
1405 |
|
haftmann@20944
|
1406 |
text {* Rule projections: *}
|
haftmann@20944
|
1407 |
|
haftmann@20944
|
1408 |
ML {*
|
wenzelm@32172
|
1409 |
structure Project_Rule = Project_Rule
|
wenzelm@25388
|
1410 |
(
|
wenzelm@27126
|
1411 |
val conjunct1 = @{thm conjunct1}
|
wenzelm@27126
|
1412 |
val conjunct2 = @{thm conjunct2}
|
wenzelm@27126
|
1413 |
val mp = @{thm mp}
|
wenzelm@25388
|
1414 |
)
|
haftmann@20944
|
1415 |
*}
|
haftmann@20944
|
1416 |
|
haftmann@35413
|
1417 |
definition induct_forall where
|
haftmann@35413
|
1418 |
"induct_forall P == \<forall>x. P x"
|
haftmann@35413
|
1419 |
|
haftmann@35413
|
1420 |
definition induct_implies where
|
haftmann@35413
|
1421 |
"induct_implies A B == A \<longrightarrow> B"
|
haftmann@35413
|
1422 |
|
haftmann@35413
|
1423 |
definition induct_equal where
|
haftmann@35413
|
1424 |
"induct_equal x y == x = y"
|
haftmann@35413
|
1425 |
|
haftmann@35413
|
1426 |
definition induct_conj where
|
haftmann@35413
|
1427 |
"induct_conj A B == A \<and> B"
|
haftmann@35413
|
1428 |
|
haftmann@35413
|
1429 |
definition induct_true where
|
haftmann@35413
|
1430 |
"induct_true == True"
|
haftmann@35413
|
1431 |
|
haftmann@35413
|
1432 |
definition induct_false where
|
haftmann@35413
|
1433 |
"induct_false == False"
|
haftmann@20944
|
1434 |
|
haftmann@20944
|
1435 |
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
|
haftmann@20944
|
1436 |
by (unfold atomize_all induct_forall_def)
|
haftmann@20944
|
1437 |
|
haftmann@20944
|
1438 |
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
|
haftmann@20944
|
1439 |
by (unfold atomize_imp induct_implies_def)
|
haftmann@20944
|
1440 |
|
haftmann@20944
|
1441 |
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
|
haftmann@20944
|
1442 |
by (unfold atomize_eq induct_equal_def)
|
haftmann@20944
|
1443 |
|
wenzelm@28856
|
1444 |
lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"
|
haftmann@20944
|
1445 |
by (unfold atomize_conj induct_conj_def)
|
haftmann@20944
|
1446 |
|
berghofe@34908
|
1447 |
lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
|
berghofe@34908
|
1448 |
lemmas induct_atomize = induct_atomize' induct_equal_eq
|
berghofe@34908
|
1449 |
lemmas induct_rulify' [symmetric, standard] = induct_atomize'
|
haftmann@20944
|
1450 |
lemmas induct_rulify [symmetric, standard] = induct_atomize
|
haftmann@20944
|
1451 |
lemmas induct_rulify_fallback =
|
haftmann@20944
|
1452 |
induct_forall_def induct_implies_def induct_equal_def induct_conj_def
|
berghofe@34908
|
1453 |
induct_true_def induct_false_def
|
haftmann@20944
|
1454 |
|
haftmann@20944
|
1455 |
|
haftmann@20944
|
1456 |
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
|
haftmann@20944
|
1457 |
induct_conj (induct_forall A) (induct_forall B)"
|
haftmann@20944
|
1458 |
by (unfold induct_forall_def induct_conj_def) iprover
|
haftmann@20944
|
1459 |
|
haftmann@20944
|
1460 |
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
|
haftmann@20944
|
1461 |
induct_conj (induct_implies C A) (induct_implies C B)"
|
haftmann@20944
|
1462 |
by (unfold induct_implies_def induct_conj_def) iprover
|
haftmann@20944
|
1463 |
|
haftmann@20944
|
1464 |
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
|
haftmann@20944
|
1465 |
proof
|
haftmann@20944
|
1466 |
assume r: "induct_conj A B ==> PROP C" and A B
|
haftmann@20944
|
1467 |
show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
|
haftmann@20944
|
1468 |
next
|
haftmann@20944
|
1469 |
assume r: "A ==> B ==> PROP C" and "induct_conj A B"
|
haftmann@20944
|
1470 |
show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
|
haftmann@20944
|
1471 |
qed
|
haftmann@20944
|
1472 |
|
haftmann@20944
|
1473 |
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
|
haftmann@20944
|
1474 |
|
berghofe@34908
|
1475 |
lemma induct_trueI: "induct_true"
|
berghofe@34908
|
1476 |
by (simp add: induct_true_def)
|
haftmann@20944
|
1477 |
|
haftmann@20944
|
1478 |
text {* Method setup. *}
|
haftmann@20944
|
1479 |
|
haftmann@20944
|
1480 |
ML {*
|
wenzelm@32171
|
1481 |
structure Induct = Induct
|
wenzelm@27126
|
1482 |
(
|
wenzelm@27126
|
1483 |
val cases_default = @{thm case_split}
|
wenzelm@27126
|
1484 |
val atomize = @{thms induct_atomize}
|
berghofe@34908
|
1485 |
val rulify = @{thms induct_rulify'}
|
wenzelm@27126
|
1486 |
val rulify_fallback = @{thms induct_rulify_fallback}
|
berghofe@34975
|
1487 |
val equal_def = @{thm induct_equal_def}
|
berghofe@34908
|
1488 |
fun dest_def (Const (@{const_name induct_equal}, _) $ t $ u) = SOME (t, u)
|
berghofe@34908
|
1489 |
| dest_def _ = NONE
|
berghofe@34908
|
1490 |
val trivial_tac = match_tac @{thms induct_trueI}
|
wenzelm@27126
|
1491 |
)
|
haftmann@20944
|
1492 |
*}
|
haftmann@20944
|
1493 |
|
nipkow@45885
|
1494 |
use "~~/src/Tools/induction.ML"
|
nipkow@45885
|
1495 |
|
berghofe@34908
|
1496 |
setup {*
|
nipkow@45885
|
1497 |
Induct.setup #> Induction.setup #>
|
berghofe@34908
|
1498 |
Context.theory_map (Induct.map_simpset (fn ss => ss
|
wenzelm@36543
|
1499 |
setmksimps (fn ss => Simpdata.mksimps Simpdata.mksimps_pairs ss #>
|
berghofe@34908
|
1500 |
map (Simplifier.rewrite_rule (map Thm.symmetric
|
berghofe@36629
|
1501 |
@{thms induct_rulify_fallback})))
|
berghofe@34908
|
1502 |
addsimprocs
|
wenzelm@38963
|
1503 |
[Simplifier.simproc_global @{theory} "swap_induct_false"
|
berghofe@34908
|
1504 |
["induct_false ==> PROP P ==> PROP Q"]
|
berghofe@34908
|
1505 |
(fn _ => fn _ =>
|
berghofe@34908
|
1506 |
(fn _ $ (P as _ $ @{const induct_false}) $ (_ $ Q $ _) =>
|
berghofe@34908
|
1507 |
if P <> Q then SOME Drule.swap_prems_eq else NONE
|
berghofe@34908
|
1508 |
| _ => NONE)),
|
wenzelm@38963
|
1509 |
Simplifier.simproc_global @{theory} "induct_equal_conj_curry"
|
berghofe@34908
|
1510 |
["induct_conj P Q ==> PROP R"]
|
berghofe@34908
|
1511 |
(fn _ => fn _ =>
|
berghofe@34908
|
1512 |
(fn _ $ (_ $ P) $ _ =>
|
berghofe@34908
|
1513 |
let
|
berghofe@34908
|
1514 |
fun is_conj (@{const induct_conj} $ P $ Q) =
|
berghofe@34908
|
1515 |
is_conj P andalso is_conj Q
|
berghofe@34908
|
1516 |
| is_conj (Const (@{const_name induct_equal}, _) $ _ $ _) = true
|
berghofe@34908
|
1517 |
| is_conj @{const induct_true} = true
|
berghofe@34908
|
1518 |
| is_conj @{const induct_false} = true
|
berghofe@34908
|
1519 |
| is_conj _ = false
|
berghofe@34908
|
1520 |
in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
|
berghofe@34908
|
1521 |
| _ => NONE))]))
|
berghofe@34908
|
1522 |
*}
|
berghofe@34908
|
1523 |
|
berghofe@34908
|
1524 |
text {* Pre-simplification of induction and cases rules *}
|
berghofe@34908
|
1525 |
|
berghofe@34908
|
1526 |
lemma [induct_simp]: "(!!x. induct_equal x t ==> PROP P x) == PROP P t"
|
berghofe@34908
|
1527 |
unfolding induct_equal_def
|
berghofe@34908
|
1528 |
proof
|
berghofe@34908
|
1529 |
assume R: "!!x. x = t ==> PROP P x"
|
berghofe@34908
|
1530 |
show "PROP P t" by (rule R [OF refl])
|
berghofe@34908
|
1531 |
next
|
berghofe@34908
|
1532 |
fix x assume "PROP P t" "x = t"
|
berghofe@34908
|
1533 |
then show "PROP P x" by simp
|
berghofe@34908
|
1534 |
qed
|
berghofe@34908
|
1535 |
|
berghofe@34908
|
1536 |
lemma [induct_simp]: "(!!x. induct_equal t x ==> PROP P x) == PROP P t"
|
berghofe@34908
|
1537 |
unfolding induct_equal_def
|
berghofe@34908
|
1538 |
proof
|
berghofe@34908
|
1539 |
assume R: "!!x. t = x ==> PROP P x"
|
berghofe@34908
|
1540 |
show "PROP P t" by (rule R [OF refl])
|
berghofe@34908
|
1541 |
next
|
berghofe@34908
|
1542 |
fix x assume "PROP P t" "t = x"
|
berghofe@34908
|
1543 |
then show "PROP P x" by simp
|
berghofe@34908
|
1544 |
qed
|
berghofe@34908
|
1545 |
|
berghofe@34908
|
1546 |
lemma [induct_simp]: "(induct_false ==> P) == Trueprop induct_true"
|
berghofe@34908
|
1547 |
unfolding induct_false_def induct_true_def
|
berghofe@34908
|
1548 |
by (iprover intro: equal_intr_rule)
|
berghofe@34908
|
1549 |
|
berghofe@34908
|
1550 |
lemma [induct_simp]: "(induct_true ==> PROP P) == PROP P"
|
berghofe@34908
|
1551 |
unfolding induct_true_def
|
berghofe@34908
|
1552 |
proof
|
berghofe@34908
|
1553 |
assume R: "True \<Longrightarrow> PROP P"
|
berghofe@34908
|
1554 |
from TrueI show "PROP P" by (rule R)
|
berghofe@34908
|
1555 |
next
|
berghofe@34908
|
1556 |
assume "PROP P"
|
berghofe@34908
|
1557 |
then show "PROP P" .
|
berghofe@34908
|
1558 |
qed
|
berghofe@34908
|
1559 |
|
berghofe@34908
|
1560 |
lemma [induct_simp]: "(PROP P ==> induct_true) == Trueprop induct_true"
|
berghofe@34908
|
1561 |
unfolding induct_true_def
|
berghofe@34908
|
1562 |
by (iprover intro: equal_intr_rule)
|
berghofe@34908
|
1563 |
|
berghofe@34908
|
1564 |
lemma [induct_simp]: "(!!x. induct_true) == Trueprop induct_true"
|
berghofe@34908
|
1565 |
unfolding induct_true_def
|
berghofe@34908
|
1566 |
by (iprover intro: equal_intr_rule)
|
berghofe@34908
|
1567 |
|
berghofe@34908
|
1568 |
lemma [induct_simp]: "induct_implies induct_true P == P"
|
berghofe@34908
|
1569 |
by (simp add: induct_implies_def induct_true_def)
|
berghofe@34908
|
1570 |
|
berghofe@34908
|
1571 |
lemma [induct_simp]: "(x = x) = True"
|
berghofe@34908
|
1572 |
by (rule simp_thms)
|
berghofe@34908
|
1573 |
|
wenzelm@36176
|
1574 |
hide_const induct_forall induct_implies induct_equal induct_conj induct_true induct_false
|
wenzelm@17459
|
1575 |
|
wenzelm@27326
|
1576 |
use "~~/src/Tools/induct_tacs.ML"
|
wenzelm@27126
|
1577 |
setup InductTacs.setup
|
wenzelm@27126
|
1578 |
|
haftmann@20944
|
1579 |
|
berghofe@28325
|
1580 |
subsubsection {* Coherent logic *}
|
berghofe@28325
|
1581 |
|
berghofe@28325
|
1582 |
ML {*
|
wenzelm@32734
|
1583 |
structure Coherent = Coherent
|
berghofe@28325
|
1584 |
(
|
berghofe@28325
|
1585 |
val atomize_elimL = @{thm atomize_elimL}
|
berghofe@28325
|
1586 |
val atomize_exL = @{thm atomize_exL}
|
berghofe@28325
|
1587 |
val atomize_conjL = @{thm atomize_conjL}
|
berghofe@28325
|
1588 |
val atomize_disjL = @{thm atomize_disjL}
|
berghofe@28325
|
1589 |
val operator_names =
|
haftmann@39028
|
1590 |
[@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}]
|
berghofe@28325
|
1591 |
);
|
berghofe@28325
|
1592 |
*}
|
berghofe@28325
|
1593 |
|
berghofe@28325
|
1594 |
setup Coherent.setup
|
berghofe@28325
|
1595 |
|
berghofe@28325
|
1596 |
|
huffman@31024
|
1597 |
subsubsection {* Reorienting equalities *}
|
huffman@31024
|
1598 |
|
huffman@31024
|
1599 |
ML {*
|
huffman@31024
|
1600 |
signature REORIENT_PROC =
|
huffman@31024
|
1601 |
sig
|
huffman@31024
|
1602 |
val add : (term -> bool) -> theory -> theory
|
huffman@31024
|
1603 |
val proc : morphism -> simpset -> cterm -> thm option
|
huffman@31024
|
1604 |
end;
|
huffman@31024
|
1605 |
|
wenzelm@33523
|
1606 |
structure Reorient_Proc : REORIENT_PROC =
|
huffman@31024
|
1607 |
struct
|
wenzelm@33523
|
1608 |
structure Data = Theory_Data
|
huffman@31024
|
1609 |
(
|
wenzelm@33523
|
1610 |
type T = ((term -> bool) * stamp) list;
|
wenzelm@33523
|
1611 |
val empty = [];
|
huffman@31024
|
1612 |
val extend = I;
|
wenzelm@33523
|
1613 |
fun merge data : T = Library.merge (eq_snd op =) data;
|
wenzelm@33523
|
1614 |
);
|
wenzelm@33523
|
1615 |
fun add m = Data.map (cons (m, stamp ()));
|
wenzelm@33523
|
1616 |
fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
|
huffman@31024
|
1617 |
|
huffman@31024
|
1618 |
val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
|
huffman@31024
|
1619 |
fun proc phi ss ct =
|
huffman@31024
|
1620 |
let
|
huffman@31024
|
1621 |
val ctxt = Simplifier.the_context ss;
|
wenzelm@43232
|
1622 |
val thy = Proof_Context.theory_of ctxt;
|
huffman@31024
|
1623 |
in
|
huffman@31024
|
1624 |
case Thm.term_of ct of
|
wenzelm@33523
|
1625 |
(_ $ t $ u) => if matches thy u then NONE else SOME meta_reorient
|
huffman@31024
|
1626 |
| _ => NONE
|
huffman@31024
|
1627 |
end;
|
huffman@31024
|
1628 |
end;
|
huffman@31024
|
1629 |
*}
|
huffman@31024
|
1630 |
|
huffman@31024
|
1631 |
|
haftmann@20944
|
1632 |
subsection {* Other simple lemmas and lemma duplicates *}
|
haftmann@20944
|
1633 |
|
haftmann@20944
|
1634 |
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
|
haftmann@20944
|
1635 |
by blast+
|
haftmann@20944
|
1636 |
|
haftmann@20944
|
1637 |
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
|
haftmann@20944
|
1638 |
apply (rule iffI)
|
haftmann@20944
|
1639 |
apply (rule_tac a = "%x. THE y. P x y" in ex1I)
|
haftmann@20944
|
1640 |
apply (fast dest!: theI')
|
huffman@45792
|
1641 |
apply (fast intro: the1_equality [symmetric])
|
haftmann@20944
|
1642 |
apply (erule ex1E)
|
haftmann@20944
|
1643 |
apply (rule allI)
|
haftmann@20944
|
1644 |
apply (rule ex1I)
|
haftmann@20944
|
1645 |
apply (erule spec)
|
haftmann@20944
|
1646 |
apply (erule_tac x = "%z. if z = x then y else f z" in allE)
|
haftmann@20944
|
1647 |
apply (erule impE)
|
haftmann@20944
|
1648 |
apply (rule allI)
|
wenzelm@27126
|
1649 |
apply (case_tac "xa = x")
|
haftmann@20944
|
1650 |
apply (drule_tac [3] x = x in fun_cong, simp_all)
|
haftmann@20944
|
1651 |
done
|
haftmann@20944
|
1652 |
|
haftmann@22218
|
1653 |
lemmas eq_sym_conv = eq_commute
|
haftmann@22218
|
1654 |
|
chaieb@23037
|
1655 |
lemma nnf_simps:
|
chaieb@23037
|
1656 |
"(\<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
|
1657 |
"(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
|
1658 |
"(\<not> \<not>(P)) = P"
|
chaieb@23037
|
1659 |
by blast+
|
chaieb@23037
|
1660 |
|
wenzelm@21671
|
1661 |
subsection {* Basic ML bindings *}
|
wenzelm@21671
|
1662 |
|
wenzelm@21671
|
1663 |
ML {*
|
wenzelm@22129
|
1664 |
val FalseE = @{thm FalseE}
|
wenzelm@22129
|
1665 |
val Let_def = @{thm Let_def}
|
wenzelm@22129
|
1666 |
val TrueI = @{thm TrueI}
|
wenzelm@22129
|
1667 |
val allE = @{thm allE}
|
wenzelm@22129
|
1668 |
val allI = @{thm allI}
|
wenzelm@22129
|
1669 |
val all_dupE = @{thm all_dupE}
|
wenzelm@22129
|
1670 |
val arg_cong = @{thm arg_cong}
|
wenzelm@22129
|
1671 |
val box_equals = @{thm box_equals}
|
wenzelm@22129
|
1672 |
val ccontr = @{thm ccontr}
|
wenzelm@22129
|
1673 |
val classical = @{thm classical}
|
wenzelm@22129
|
1674 |
val conjE = @{thm conjE}
|
wenzelm@22129
|
1675 |
val conjI = @{thm conjI}
|
wenzelm@22129
|
1676 |
val conjunct1 = @{thm conjunct1}
|
wenzelm@22129
|
1677 |
val conjunct2 = @{thm conjunct2}
|
wenzelm@22129
|
1678 |
val disjCI = @{thm disjCI}
|
wenzelm@22129
|
1679 |
val disjE = @{thm disjE}
|
wenzelm@22129
|
1680 |
val disjI1 = @{thm disjI1}
|
wenzelm@22129
|
1681 |
val disjI2 = @{thm disjI2}
|
wenzelm@22129
|
1682 |
val eq_reflection = @{thm eq_reflection}
|
wenzelm@22129
|
1683 |
val ex1E = @{thm ex1E}
|
wenzelm@22129
|
1684 |
val ex1I = @{thm ex1I}
|
wenzelm@22129
|
1685 |
val ex1_implies_ex = @{thm ex1_implies_ex}
|
wenzelm@22129
|
1686 |
val exE = @{thm exE}
|
wenzelm@22129
|
1687 |
val exI = @{thm exI}
|
wenzelm@22129
|
1688 |
val excluded_middle = @{thm excluded_middle}
|
wenzelm@22129
|
1689 |
val ext = @{thm ext}
|
wenzelm@22129
|
1690 |
val fun_cong = @{thm fun_cong}
|
wenzelm@22129
|
1691 |
val iffD1 = @{thm iffD1}
|
wenzelm@22129
|
1692 |
val iffD2 = @{thm iffD2}
|
wenzelm@22129
|
1693 |
val iffI = @{thm iffI}
|
wenzelm@22129
|
1694 |
val impE = @{thm impE}
|
wenzelm@22129
|
1695 |
val impI = @{thm impI}
|
wenzelm@22129
|
1696 |
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
|
wenzelm@22129
|
1697 |
val mp = @{thm mp}
|
wenzelm@22129
|
1698 |
val notE = @{thm notE}
|
wenzelm@22129
|
1699 |
val notI = @{thm notI}
|
wenzelm@22129
|
1700 |
val not_all = @{thm not_all}
|
wenzelm@22129
|
1701 |
val not_ex = @{thm not_ex}
|
wenzelm@22129
|
1702 |
val not_iff = @{thm not_iff}
|
wenzelm@22129
|
1703 |
val not_not = @{thm not_not}
|
wenzelm@22129
|
1704 |
val not_sym = @{thm not_sym}
|
wenzelm@22129
|
1705 |
val refl = @{thm refl}
|
wenzelm@22129
|
1706 |
val rev_mp = @{thm rev_mp}
|
wenzelm@22129
|
1707 |
val spec = @{thm spec}
|
wenzelm@22129
|
1708 |
val ssubst = @{thm ssubst}
|
wenzelm@22129
|
1709 |
val subst = @{thm subst}
|
wenzelm@22129
|
1710 |
val sym = @{thm sym}
|
wenzelm@22129
|
1711 |
val trans = @{thm trans}
|
wenzelm@21671
|
1712 |
*}
|
wenzelm@21671
|
1713 |
|
blanchet@39280
|
1714 |
use "Tools/cnf_funcs.ML"
|
wenzelm@21671
|
1715 |
|
haftmann@30929
|
1716 |
subsection {* Code generator setup *}
|
haftmann@28400
|
1717 |
|
haftmann@30929
|
1718 |
subsubsection {* SML code generator setup *}
|
haftmann@30929
|
1719 |
|
haftmann@30929
|
1720 |
use "Tools/recfun_codegen.ML"
|
haftmann@30929
|
1721 |
|
haftmann@30929
|
1722 |
setup {*
|
haftmann@30929
|
1723 |
Codegen.setup
|
haftmann@30929
|
1724 |
#> RecfunCodegen.setup
|
haftmann@32060
|
1725 |
#> Codegen.map_unfold (K HOL_basic_ss)
|
haftmann@30929
|
1726 |
*}
|
haftmann@30929
|
1727 |
|
haftmann@30929
|
1728 |
types_code
|
haftmann@30929
|
1729 |
"bool" ("bool")
|
haftmann@30929
|
1730 |
attach (term_of) {*
|
haftmann@30929
|
1731 |
fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const;
|
haftmann@30929
|
1732 |
*}
|
haftmann@30929
|
1733 |
attach (test) {*
|
haftmann@30929
|
1734 |
fun gen_bool i =
|
haftmann@30929
|
1735 |
let val b = one_of [false, true]
|
haftmann@30929
|
1736 |
in (b, fn () => term_of_bool b) end;
|
haftmann@30929
|
1737 |
*}
|
haftmann@30929
|
1738 |
"prop" ("bool")
|
haftmann@30929
|
1739 |
attach (term_of) {*
|
haftmann@30929
|
1740 |
fun term_of_prop b =
|
haftmann@30929
|
1741 |
HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const);
|
haftmann@30929
|
1742 |
*}
|
haftmann@30929
|
1743 |
|
haftmann@30929
|
1744 |
consts_code
|
haftmann@30929
|
1745 |
"Trueprop" ("(_)")
|
haftmann@30929
|
1746 |
"True" ("true")
|
haftmann@30929
|
1747 |
"False" ("false")
|
haftmann@30929
|
1748 |
"Not" ("Bool.not")
|
haftmann@39028
|
1749 |
HOL.disj ("(_ orelse/ _)")
|
haftmann@39028
|
1750 |
HOL.conj ("(_ andalso/ _)")
|
haftmann@30929
|
1751 |
"If" ("(if _/ then _/ else _)")
|
haftmann@30929
|
1752 |
|
haftmann@30929
|
1753 |
setup {*
|
haftmann@30929
|
1754 |
let
|
haftmann@30929
|
1755 |
|
wenzelm@43292
|
1756 |
fun eq_codegen thy mode defs dep thyname b t gr =
|
haftmann@30929
|
1757 |
(case strip_comb t of
|
haftmann@39093
|
1758 |
(Const (@{const_name HOL.eq}, Type (_, [Type ("fun", _), _])), _) => NONE
|
haftmann@39093
|
1759 |
| (Const (@{const_name HOL.eq}, _), [t, u]) =>
|
haftmann@30929
|
1760 |
let
|
wenzelm@43292
|
1761 |
val (pt, gr') = Codegen.invoke_codegen thy mode defs dep thyname false t gr;
|
wenzelm@43292
|
1762 |
val (pu, gr'') = Codegen.invoke_codegen thy mode defs dep thyname false u gr';
|
wenzelm@43292
|
1763 |
val (_, gr''') =
|
wenzelm@43292
|
1764 |
Codegen.invoke_tycodegen thy mode defs dep thyname false HOLogic.boolT gr'';
|
haftmann@30929
|
1765 |
in
|
haftmann@30929
|
1766 |
SOME (Codegen.parens
|
haftmann@30929
|
1767 |
(Pretty.block [pt, Codegen.str " =", Pretty.brk 1, pu]), gr''')
|
haftmann@30929
|
1768 |
end
|
haftmann@39093
|
1769 |
| (t as Const (@{const_name HOL.eq}, _), ts) => SOME (Codegen.invoke_codegen
|
wenzelm@43292
|
1770 |
thy mode defs dep thyname b (Codegen.eta_expand t ts 2) gr)
|
haftmann@30929
|
1771 |
| _ => NONE);
|
haftmann@30929
|
1772 |
|
haftmann@30929
|
1773 |
in
|
haftmann@30929
|
1774 |
Codegen.add_codegen "eq_codegen" eq_codegen
|
haftmann@30929
|
1775 |
end
|
haftmann@30929
|
1776 |
*}
|
haftmann@30929
|
1777 |
|
haftmann@31151
|
1778 |
subsubsection {* Generic code generator preprocessor setup *}
|
haftmann@31151
|
1779 |
|
haftmann@31151
|
1780 |
setup {*
|
haftmann@31151
|
1781 |
Code_Preproc.map_pre (K HOL_basic_ss)
|
haftmann@31151
|
1782 |
#> Code_Preproc.map_post (K HOL_basic_ss)
|
haftmann@37417
|
1783 |
#> Code_Simp.map_ss (K HOL_basic_ss)
|
haftmann@31151
|
1784 |
*}
|
haftmann@31151
|
1785 |
|
haftmann@30929
|
1786 |
subsubsection {* Equality *}
|
haftmann@24844
|
1787 |
|
haftmann@39086
|
1788 |
class equal =
|
haftmann@39086
|
1789 |
fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
|
haftmann@39086
|
1790 |
assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
|
haftmann@26513
|
1791 |
begin
|
haftmann@26513
|
1792 |
|
haftmann@39086
|
1793 |
lemma equal [code_unfold, code_inline del]: "equal = (op =)"
|
haftmann@39086
|
1794 |
by (rule ext equal_eq)+
|
haftmann@28346
|
1795 |
|
haftmann@39086
|
1796 |
lemma equal_refl: "equal x x \<longleftrightarrow> True"
|
haftmann@39086
|
1797 |
unfolding equal by rule+
|
haftmann@28346
|
1798 |
|
haftmann@39086
|
1799 |
lemma eq_equal: "(op =) \<equiv> equal"
|
haftmann@39086
|
1800 |
by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
|
haftmann@30929
|
1801 |
|
haftmann@26513
|
1802 |
end
|
haftmann@26513
|
1803 |
|
haftmann@39086
|
1804 |
declare eq_equal [symmetric, code_post]
|
haftmann@39086
|
1805 |
declare eq_equal [code]
|
haftmann@30966
|
1806 |
|
haftmann@31151
|
1807 |
setup {*
|
haftmann@31151
|
1808 |
Code_Preproc.map_pre (fn simpset =>
|
haftmann@39093
|
1809 |
simpset addsimprocs [Simplifier.simproc_global_i @{theory} "equal" [@{term HOL.eq}]
|
wenzelm@41091
|
1810 |
(fn thy => fn _ =>
|
wenzelm@41091
|
1811 |
fn Const (_, Type ("fun", [Type _, _])) => SOME @{thm eq_equal} | _ => NONE)])
|
haftmann@31151
|
1812 |
*}
|
haftmann@31151
|
1813 |
|
haftmann@30966
|
1814 |
|
haftmann@30929
|
1815 |
subsubsection {* Generic code generator foundation *}
|
haftmann@28513
|
1816 |
|
haftmann@39667
|
1817 |
text {* Datatype @{typ bool} *}
|
haftmann@30929
|
1818 |
|
haftmann@30929
|
1819 |
code_datatype True False
|
haftmann@30929
|
1820 |
|
haftmann@30929
|
1821 |
lemma [code]:
|
haftmann@33185
|
1822 |
shows "False \<and> P \<longleftrightarrow> False"
|
haftmann@33185
|
1823 |
and "True \<and> P \<longleftrightarrow> P"
|
haftmann@33185
|
1824 |
and "P \<and> False \<longleftrightarrow> False"
|
haftmann@33185
|
1825 |
and "P \<and> True \<longleftrightarrow> P" by simp_all
|
haftmann@30929
|
1826 |
|
haftmann@30929
|
1827 |
lemma [code]:
|
haftmann@33185
|
1828 |
shows "False \<or> P \<longleftrightarrow> P"
|
haftmann@33185
|
1829 |
and "True \<or> P \<longleftrightarrow> True"
|
haftmann@33185
|
1830 |
and "P \<or> False \<longleftrightarrow> P"
|
haftmann@33185
|
1831 |
and "P \<or> True \<longleftrightarrow> True" by simp_all
|
haftmann@30929
|
1832 |
|
haftmann@33185
|
1833 |
lemma [code]:
|
haftmann@33185
|
1834 |
shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
|
haftmann@33185
|
1835 |
and "(True \<longrightarrow> P) \<longleftrightarrow> P"
|
haftmann@33185
|
1836 |
and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
|
haftmann@33185
|
1837 |
and "(P \<longrightarrow> True) \<longleftrightarrow> True" by simp_all
|
haftmann@30929
|
1838 |
|
haftmann@39667
|
1839 |
text {* More about @{typ prop} *}
|
haftmann@39667
|
1840 |
|
haftmann@39667
|
1841 |
lemma [code nbe]:
|
haftmann@39667
|
1842 |
shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
|
haftmann@39667
|
1843 |
and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
|
haftmann@39667
|
1844 |
and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)" by (auto intro!: equal_intr_rule)
|
haftmann@39667
|
1845 |
|
haftmann@39667
|
1846 |
lemma Trueprop_code [code]:
|
haftmann@39667
|
1847 |
"Trueprop True \<equiv> Code_Generator.holds"
|
haftmann@39667
|
1848 |
by (auto intro!: equal_intr_rule holds)
|
haftmann@39667
|
1849 |
|
haftmann@39667
|
1850 |
declare Trueprop_code [symmetric, code_post]
|
haftmann@39667
|
1851 |
|
haftmann@39667
|
1852 |
text {* Equality *}
|
haftmann@39667
|
1853 |
|
haftmann@39667
|
1854 |
declare simp_thms(6) [code nbe]
|
haftmann@39667
|
1855 |
|
haftmann@39086
|
1856 |
instantiation itself :: (type) equal
|
haftmann@31132
|
1857 |
begin
|
haftmann@31132
|
1858 |
|
haftmann@39086
|
1859 |
definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where
|
haftmann@39086
|
1860 |
"equal_itself x y \<longleftrightarrow> x = y"
|
haftmann@31132
|
1861 |
|
haftmann@31132
|
1862 |
instance proof
|
haftmann@39086
|
1863 |
qed (fact equal_itself_def)
|
haftmann@31132
|
1864 |
|
haftmann@31132
|
1865 |
end
|
haftmann@31132
|
1866 |
|
haftmann@39086
|
1867 |
lemma equal_itself_code [code]:
|
haftmann@39086
|
1868 |
"equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
|
haftmann@39086
|
1869 |
by (simp add: equal)
|
haftmann@31132
|
1870 |
|
haftmann@28513
|
1871 |
setup {*
|
haftmann@39086
|
1872 |
Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>type \<Rightarrow> 'a \<Rightarrow> bool"})
|
haftmann@31956
|
1873 |
*}
|
haftmann@31956
|
1874 |
|
haftmann@39086
|
1875 |
lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)" (is "?ofclass \<equiv> ?equal")
|
haftmann@31956
|
1876 |
proof
|
haftmann@31956
|
1877 |
assume "PROP ?ofclass"
|
haftmann@39086
|
1878 |
show "PROP ?equal"
|
haftmann@39086
|
1879 |
by (tactic {* ALLGOALS (rtac (Thm.unconstrainT @{thm eq_equal})) *})
|
haftmann@31956
|
1880 |
(fact `PROP ?ofclass`)
|
haftmann@31956
|
1881 |
next
|
haftmann@39086
|
1882 |
assume "PROP ?equal"
|
haftmann@31956
|
1883 |
show "PROP ?ofclass" proof
|
haftmann@39086
|
1884 |
qed (simp add: `PROP ?equal`)
|
haftmann@31956
|
1885 |
qed
|
haftmann@31956
|
1886 |
|
haftmann@31956
|
1887 |
setup {*
|
haftmann@39086
|
1888 |
Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a\<Colon>equal \<Rightarrow> 'a \<Rightarrow> bool"})
|
haftmann@31956
|
1889 |
*}
|
haftmann@31956
|
1890 |
|
haftmann@31956
|
1891 |
setup {*
|
haftmann@39086
|
1892 |
Nbe.add_const_alias @{thm equal_alias_cert}
|
haftmann@28513
|
1893 |
*}
|
haftmann@28513
|
1894 |
|
haftmann@30929
|
1895 |
text {* Cases *}
|
haftmann@30929
|
1896 |
|
haftmann@30929
|
1897 |
lemma Let_case_cert:
|
haftmann@30929
|
1898 |
assumes "CASE \<equiv> (\<lambda>x. Let x f)"
|
haftmann@30929
|
1899 |
shows "CASE x \<equiv> f x"
|
haftmann@30929
|
1900 |
using assms by simp_all
|
haftmann@30929
|
1901 |
|
haftmann@30929
|
1902 |
setup {*
|
haftmann@30929
|
1903 |
Code.add_case @{thm Let_case_cert}
|
haftmann@30929
|
1904 |
#> Code.add_undefined @{const_name undefined}
|
haftmann@30929
|
1905 |
*}
|
haftmann@30929
|
1906 |
|
haftmann@30929
|
1907 |
code_abort undefined
|
haftmann@30929
|
1908 |
|
haftmann@39218
|
1909 |
|
haftmann@30929
|
1910 |
subsubsection {* Generic code generator target languages *}
|
haftmann@30929
|
1911 |
|
haftmann@39218
|
1912 |
text {* type @{typ bool} *}
|
haftmann@30929
|
1913 |
|
haftmann@30929
|
1914 |
code_type bool
|
haftmann@30929
|
1915 |
(SML "bool")
|
haftmann@30929
|
1916 |
(OCaml "bool")
|
haftmann@30929
|
1917 |
(Haskell "Bool")
|
haftmann@34294
|
1918 |
(Scala "Boolean")
|
haftmann@30929
|
1919 |
|
bulwahn@43277
|
1920 |
code_const True and False and Not and HOL.conj and HOL.disj and HOL.implies and If
|
haftmann@30929
|
1921 |
(SML "true" and "false" and "not"
|
haftmann@30929
|
1922 |
and infixl 1 "andalso" and infixl 0 "orelse"
|
bulwahn@43277
|
1923 |
and "!(if (_)/ then (_)/ else true)"
|
haftmann@30929
|
1924 |
and "!(if (_)/ then (_)/ else (_))")
|
haftmann@30929
|
1925 |
(OCaml "true" and "false" and "not"
|
haftmann@39947
|
1926 |
and infixl 3 "&&" and infixl 2 "||"
|
bulwahn@43277
|
1927 |
and "!(if (_)/ then (_)/ else true)"
|
haftmann@30929
|
1928 |
and "!(if (_)/ then (_)/ else (_))")
|
haftmann@30929
|
1929 |
(Haskell "True" and "False" and "not"
|
haftmann@43049
|
1930 |
and infixr 3 "&&" and infixr 2 "||"
|
bulwahn@43277
|
1931 |
and "!(if (_)/ then (_)/ else True)"
|
haftmann@30929
|
1932 |
and "!(if (_)/ then (_)/ else (_))")
|
haftmann@39006
|
1933 |
(Scala "true" and "false" and "'! _"
|
haftmann@34301
|
1934 |
and infixl 3 "&&" and infixl 1 "||"
|
bulwahn@43277
|
1935 |
and "!(if ((_))/ (_)/ else true)"
|
haftmann@34301
|
1936 |
and "!(if ((_))/ (_)/ else (_))")
|
haftmann@34294
|
1937 |
|
haftmann@30929
|
1938 |
code_reserved SML
|
haftmann@30929
|
1939 |
bool true false not
|
haftmann@30929
|
1940 |
|
haftmann@30929
|
1941 |
code_reserved OCaml
|
haftmann@30929
|
1942 |
bool not
|
haftmann@30929
|
1943 |
|
haftmann@34294
|
1944 |
code_reserved Scala
|
haftmann@34294
|
1945 |
Boolean
|
haftmann@34294
|
1946 |
|
haftmann@39270
|
1947 |
code_modulename SML Pure HOL
|
haftmann@39270
|
1948 |
code_modulename OCaml Pure HOL
|
haftmann@39270
|
1949 |
code_modulename Haskell Pure HOL
|
haftmann@39270
|
1950 |
|
haftmann@30929
|
1951 |
text {* using built-in Haskell equality *}
|
haftmann@30929
|
1952 |
|
haftmann@39086
|
1953 |
code_class equal
|
haftmann@30929
|
1954 |
(Haskell "Eq")
|
haftmann@30929
|
1955 |
|
haftmann@39086
|
1956 |
code_const "HOL.equal"
|
haftmann@39499
|
1957 |
(Haskell infix 4 "==")
|
haftmann@30929
|
1958 |
|
haftmann@39093
|
1959 |
code_const HOL.eq
|
haftmann@39499
|
1960 |
(Haskell infix 4 "==")
|
haftmann@30929
|
1961 |
|
haftmann@30929
|
1962 |
text {* undefined *}
|
haftmann@30929
|
1963 |
|
haftmann@30929
|
1964 |
code_const undefined
|
haftmann@30929
|
1965 |
(SML "!(raise/ Fail/ \"undefined\")")
|
haftmann@30929
|
1966 |
(OCaml "failwith/ \"undefined\"")
|
haftmann@30929
|
1967 |
(Haskell "error/ \"undefined\"")
|
haftmann@34886
|
1968 |
(Scala "!error(\"undefined\")")
|
haftmann@30929
|
1969 |
|
haftmann@30929
|
1970 |
subsubsection {* Evaluation and normalization by evaluation *}
|
haftmann@30929
|
1971 |
|
haftmann@30929
|
1972 |
ML {*
|
haftmann@30929
|
1973 |
fun gen_eval_method conv ctxt = SIMPLE_METHOD'
|
wenzelm@43305
|
1974 |
(CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 (conv ctxt))) ctxt)
|
haftmann@30929
|
1975 |
THEN' rtac TrueI)
|
haftmann@30929
|
1976 |
*}
|
haftmann@30929
|
1977 |
|
wenzelm@43305
|
1978 |
method_setup eval = {*
|
wenzelm@43305
|
1979 |
Scan.succeed (gen_eval_method (Code_Runtime.dynamic_holds_conv o Proof_Context.theory_of))
|
wenzelm@43305
|
1980 |
*} "solve goal by evaluation"
|
haftmann@30929
|
1981 |
|
wenzelm@43305
|
1982 |
method_setup evaluation = {*
|
wenzelm@43305
|
1983 |
Scan.succeed (gen_eval_method Codegen.evaluation_conv)
|
wenzelm@43305
|
1984 |
*} "solve goal by evaluation"
|
haftmann@30929
|
1985 |
|
haftmann@30929
|
1986 |
method_setup normalization = {*
|
haftmann@41488
|
1987 |
Scan.succeed (fn ctxt => SIMPLE_METHOD'
|
wenzelm@43232
|
1988 |
(CHANGED_PROP o (CONVERSION (Nbe.dynamic_conv (Proof_Context.theory_of ctxt))
|
haftmann@41488
|
1989 |
THEN' (fn k => TRY (rtac TrueI k)))))
|
haftmann@30929
|
1990 |
*} "solve goal by normalization"
|
haftmann@30929
|
1991 |
|
wenzelm@31913
|
1992 |
|
haftmann@33084
|
1993 |
subsection {* Counterexample Search Units *}
|
haftmann@33084
|
1994 |
|
haftmann@30929
|
1995 |
subsubsection {* Quickcheck *}
|
haftmann@30929
|
1996 |
|
haftmann@33084
|
1997 |
quickcheck_params [size = 5, iterations = 50]
|
haftmann@33084
|
1998 |
|
haftmann@30929
|
1999 |
|
haftmann@33084
|
2000 |
subsubsection {* Nitpick setup *}
|
blanchet@30309
|
2001 |
|
blanchet@29800
|
2002 |
ML {*
|
blanchet@42663
|
2003 |
structure Nitpick_Unfolds = Named_Thms
|
blanchet@30254
|
2004 |
(
|
blanchet@42663
|
2005 |
val name = "nitpick_unfold"
|
blanchet@30254
|
2006 |
val description = "alternative definitions of constants as needed by Nitpick"
|
blanchet@30254
|
2007 |
)
|
blanchet@33056
|
2008 |
structure Nitpick_Simps = Named_Thms
|
blanchet@29800
|
2009 |
(
|
blanchet@33056
|
2010 |
val name = "nitpick_simp"
|
blanchet@29806
|
2011 |
val description = "equational specification of constants as needed by Nitpick"
|
blanchet@29800
|
2012 |
)
|
blanchet@33056
|
2013 |
structure Nitpick_Psimps = Named_Thms
|
blanchet@29800
|
2014 |
(
|
blanchet@33056
|
2015 |
val name = "nitpick_psimp"
|
blanchet@29806
|
2016 |
val description = "partial equational specification of constants as needed by Nitpick"
|
blanchet@29800
|
2017 |
)
|
blanchet@35807
|
2018 |
structure Nitpick_Choice_Specs = Named_Thms
|
blanchet@35807
|
2019 |
(
|
blanchet@35808
|
2020 |
val name = "nitpick_choice_spec"
|
blanchet@35807
|
2021 |
val description = "choice specification of constants as needed by Nitpick"
|
blanchet@35807
|
2022 |
)
|
blanchet@29800
|
2023 |
*}
|
wenzelm@30981
|
2024 |
|
wenzelm@30981
|
2025 |
setup {*
|
blanchet@42663
|
2026 |
Nitpick_Unfolds.setup
|
blanchet@33056
|
2027 |
#> Nitpick_Simps.setup
|
blanchet@33056
|
2028 |
#> Nitpick_Psimps.setup
|
blanchet@35807
|
2029 |
#> Nitpick_Choice_Specs.setup
|
wenzelm@30981
|
2030 |
*}
|
wenzelm@30981
|
2031 |
|
blanchet@42663
|
2032 |
declare if_bool_eq_conj [nitpick_unfold, no_atp]
|
blanchet@42663
|
2033 |
if_bool_eq_disj [no_atp]
|
blanchet@42663
|
2034 |
|
blanchet@29800
|
2035 |
|
haftmann@33084
|
2036 |
subsection {* Preprocessing for the predicate compiler *}
|
haftmann@33084
|
2037 |
|
haftmann@33084
|
2038 |
ML {*
|
haftmann@33084
|
2039 |
structure Predicate_Compile_Alternative_Defs = Named_Thms
|
haftmann@33084
|
2040 |
(
|
haftmann@33084
|
2041 |
val name = "code_pred_def"
|
haftmann@33084
|
2042 |
val description = "alternative definitions of constants for the Predicate Compiler"
|
haftmann@33084
|
2043 |
)
|
haftmann@33084
|
2044 |
structure Predicate_Compile_Inline_Defs = Named_Thms
|
haftmann@33084
|
2045 |
(
|
haftmann@33084
|
2046 |
val name = "code_pred_inline"
|
haftmann@33084
|
2047 |
val description = "inlining definitions for the Predicate Compiler"
|
haftmann@33084
|
2048 |
)
|
bulwahn@36246
|
2049 |
structure Predicate_Compile_Simps = Named_Thms
|
bulwahn@36246
|
2050 |
(
|
bulwahn@36246
|
2051 |
val name = "code_pred_simp"
|
bulwahn@36246
|
2052 |
val description = "simplification rules for the optimisations in the Predicate Compiler"
|
bulwahn@36246
|
2053 |
)
|
haftmann@33084
|
2054 |
*}
|
haftmann@33084
|
2055 |
|
haftmann@33084
|
2056 |
setup {*
|
haftmann@33084
|
2057 |
Predicate_Compile_Alternative_Defs.setup
|
haftmann@33084
|
2058 |
#> Predicate_Compile_Inline_Defs.setup
|
bulwahn@36246
|
2059 |
#> Predicate_Compile_Simps.setup
|
haftmann@33084
|
2060 |
*}
|
haftmann@33084
|
2061 |
|
haftmann@33084
|
2062 |
|
haftmann@22839
|
2063 |
subsection {* Legacy tactics and ML bindings *}
|
wenzelm@21671
|
2064 |
|
wenzelm@21671
|
2065 |
ML {*
|
wenzelm@21671
|
2066 |
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
|
wenzelm@21671
|
2067 |
|
wenzelm@21671
|
2068 |
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
|
wenzelm@21671
|
2069 |
local
|
wenzelm@35364
|
2070 |
fun wrong_prem (Const (@{const_name All}, _) $ Abs (_, _, t)) = wrong_prem t
|
wenzelm@21671
|
2071 |
| wrong_prem (Bound _) = true
|
wenzelm@21671
|
2072 |
| wrong_prem _ = false;
|
wenzelm@21671
|
2073 |
val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
|
wenzelm@21671
|
2074 |
in
|
wenzelm@21671
|
2075 |
fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
|
wenzelm@21671
|
2076 |
fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
|
wenzelm@21671
|
2077 |
end;
|
haftmann@22839
|
2078 |
|
wenzelm@39406
|
2079 |
val all_conj_distrib = @{thm all_conj_distrib};
|
wenzelm@39406
|
2080 |
val all_simps = @{thms all_simps};
|
wenzelm@39406
|
2081 |
val atomize_not = @{thm atomize_not};
|
wenzelm@39406
|
2082 |
val case_split = @{thm case_split};
|
wenzelm@39406
|
2083 |
val cases_simp = @{thm cases_simp};
|
wenzelm@39406
|
2084 |
val choice_eq = @{thm choice_eq};
|
wenzelm@39406
|
2085 |
val cong = @{thm cong};
|
wenzelm@39406
|
2086 |
val conj_comms = @{thms conj_comms};
|
wenzelm@39406
|
2087 |
val conj_cong = @{thm conj_cong};
|
wenzelm@39406
|
2088 |
val de_Morgan_conj = @{thm de_Morgan_conj};
|
wenzelm@39406
|
2089 |
val de_Morgan_disj = @{thm de_Morgan_disj};
|
wenzelm@39406
|
2090 |
val disj_assoc = @{thm disj_assoc};
|
wenzelm@39406
|
2091 |
val disj_comms = @{thms disj_comms};
|
wenzelm@39406
|
2092 |
val disj_cong = @{thm disj_cong};
|
wenzelm@39406
|
2093 |
val eq_ac = @{thms eq_ac};
|
wenzelm@39406
|
2094 |
val eq_cong2 = @{thm eq_cong2}
|
wenzelm@39406
|
2095 |
val Eq_FalseI = @{thm Eq_FalseI};
|
wenzelm@39406
|
2096 |
val Eq_TrueI = @{thm Eq_TrueI};
|
wenzelm@39406
|
2097 |
val Ex1_def = @{thm Ex1_def};
|
wenzelm@39406
|
2098 |
val ex_disj_distrib = @{thm ex_disj_distrib};
|
wenzelm@39406
|
2099 |
val ex_simps = @{thms ex_simps};
|
wenzelm@39406
|
2100 |
val if_cancel = @{thm if_cancel};
|
wenzelm@39406
|
2101 |
val if_eq_cancel = @{thm if_eq_cancel};
|
wenzelm@39406
|
2102 |
val if_False = @{thm if_False};
|
wenzelm@39406
|
2103 |
val iff_conv_conj_imp = @{thm iff_conv_conj_imp};
|
wenzelm@39406
|
2104 |
val iff = @{thm iff};
|
wenzelm@39406
|
2105 |
val if_splits = @{thms if_splits};
|
wenzelm@39406
|
2106 |
val if_True = @{thm if_True};
|
wenzelm@39406
|
2107 |
val if_weak_cong = @{thm if_weak_cong};
|
wenzelm@39406
|
2108 |
val imp_all = @{thm imp_all};
|
wenzelm@39406
|
2109 |
val imp_cong = @{thm imp_cong};
|
wenzelm@39406
|
2110 |
val imp_conjL = @{thm imp_conjL};
|
wenzelm@39406
|
2111 |
val imp_conjR = @{thm imp_conjR};
|
wenzelm@39406
|
2112 |
val imp_conv_disj = @{thm imp_conv_disj};
|
wenzelm@39406
|
2113 |
val simp_implies_def = @{thm simp_implies_def};
|
wenzelm@39406
|
2114 |
val simp_thms = @{thms simp_thms};
|
wenzelm@39406
|
2115 |
val split_if = @{thm split_if};
|
wenzelm@39406
|
2116 |
val the1_equality = @{thm the1_equality};
|
wenzelm@39406
|
2117 |
val theI = @{thm theI};
|
wenzelm@39406
|
2118 |
val theI' = @{thm theI'};
|
wenzelm@39406
|
2119 |
val True_implies_equals = @{thm True_implies_equals};
|
chaieb@23037
|
2120 |
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})
|
chaieb@23037
|
2121 |
|
wenzelm@21671
|
2122 |
*}
|
wenzelm@21671
|
2123 |
|
haftmann@39095
|
2124 |
hide_const (open) eq equal
|
haftmann@39095
|
2125 |
|
kleing@14357
|
2126 |
end
|