author | nipkow |
Mon, 02 Sep 2002 11:07:26 +0200 | |
changeset 13553 | 855f6bae851e |
parent 13550 | 5a176b8dda84 |
child 13596 | ee5f79b210c1 |
permissions | -rw-r--r-- |
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(* Title: HOL/HOL.thy |
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ID: $Id$ |
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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header {* The basis of Higher-Order Logic *} |
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theory HOL = CPure |
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files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"): |
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subsection {* Primitive logic *} |
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subsubsection {* Core syntax *} |
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classes type < logic |
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defaultsort type |
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global |
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typedecl bool |
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arities |
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bool :: type |
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fun :: (type, type) type |
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judgment |
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Trueprop :: "bool => prop" ("(_)" 5) |
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consts |
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Not :: "bool => bool" ("~ _" [40] 40) |
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True :: bool |
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False :: bool |
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If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) |
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arbitrary :: 'a |
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The :: "('a => bool) => 'a" |
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All :: "('a => bool) => bool" (binder "ALL " 10) |
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Ex :: "('a => bool) => bool" (binder "EX " 10) |
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Ex1 :: "('a => bool) => bool" (binder "EX! " 10) |
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Let :: "['a, 'a => 'b] => 'b" |
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"=" :: "['a, 'a] => bool" (infixl 50) |
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& :: "[bool, bool] => bool" (infixr 35) |
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"|" :: "[bool, bool] => bool" (infixr 30) |
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--> :: "[bool, bool] => bool" (infixr 25) |
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local |
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subsubsection {* Additional concrete syntax *} |
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nonterminals |
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letbinds letbind |
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case_syn cases_syn |
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syntax |
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"_not_equal" :: "['a, 'a] => bool" (infixl "~=" 50) |
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"_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10) |
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"_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10) |
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"" :: "letbind => letbinds" ("_") |
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"_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _") |
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"_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10) |
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"_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10) |
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"_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10) |
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"" :: "case_syn => cases_syn" ("_") |
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"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _") |
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translations |
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"x ~= y" == "~ (x = y)" |
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"THE x. P" == "The (%x. P)" |
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"_Let (_binds b bs) e" == "_Let b (_Let bs e)" |
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"let x = a in e" == "Let a (%x. e)" |
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syntax (output) |
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"=" :: "['a, 'a] => bool" (infix 50) |
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"_not_equal" :: "['a, 'a] => bool" (infix "~=" 50) |
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syntax (xsymbols) |
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Not :: "bool => bool" ("\<not> _" [40] 40) |
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"op &" :: "[bool, bool] => bool" (infixr "\<and>" 35) |
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"op |" :: "[bool, bool] => bool" (infixr "\<or>" 30) |
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"op -->" :: "[bool, bool] => bool" (infixr "\<longrightarrow>" 25) |
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"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) |
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"ALL " :: "[idts, bool] => bool" ("(3\<forall>_./ _)" [0, 10] 10) |
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"EX " :: "[idts, bool] => bool" ("(3\<exists>_./ _)" [0, 10] 10) |
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"EX! " :: "[idts, bool] => bool" ("(3\<exists>!_./ _)" [0, 10] 10) |
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"_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10) |
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(*"_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ \\<orelse> _")*) |
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syntax (xsymbols output) |
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"_not_equal" :: "['a, 'a] => bool" (infix "\<noteq>" 50) |
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syntax (HTML output) |
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Not :: "bool => bool" ("\<not> _" [40] 40) |
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syntax (HOL) |
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"ALL " :: "[idts, bool] => bool" ("(3! _./ _)" [0, 10] 10) |
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"EX " :: "[idts, bool] => bool" ("(3? _./ _)" [0, 10] 10) |
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"EX! " :: "[idts, bool] => bool" ("(3?! _./ _)" [0, 10] 10) |
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subsubsection {* Axioms and basic definitions *} |
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axioms |
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eq_reflection: "(x=y) ==> (x==y)" |
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refl: "t = (t::'a)" |
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subst: "[| s = t; P(s) |] ==> P(t::'a)" |
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ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)" |
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-- {* Extensionality is built into the meta-logic, and this rule expresses *} |
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-- {* a related property. It is an eta-expanded version of the traditional *} |
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-- {* rule, and similar to the ABS rule of HOL *} |
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the_eq_trivial: "(THE x. x = a) = (a::'a)" |
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impI: "(P ==> Q) ==> P-->Q" |
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mp: "[| P-->Q; P |] ==> Q" |
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defs |
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True_def: "True == ((%x::bool. x) = (%x. x))" |
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All_def: "All(P) == (P = (%x. True))" |
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Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q" |
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False_def: "False == (!P. P)" |
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not_def: "~ P == P-->False" |
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and_def: "P & Q == !R. (P-->Q-->R) --> R" |
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or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R" |
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Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)" |
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axioms |
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iff: "(P-->Q) --> (Q-->P) --> (P=Q)" |
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True_or_False: "(P=True) | (P=False)" |
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defs |
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Let_def: "Let s f == f(s)" |
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if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)" |
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arbitrary_def: "False ==> arbitrary == (THE x. False)" |
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-- {* @{term arbitrary} is completely unspecified, but is made to appear as a |
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definition syntactically *} |
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subsubsection {* Generic algebraic operations *} |
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axclass zero < type |
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axclass one < type |
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axclass plus < type |
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axclass minus < type |
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axclass times < type |
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axclass inverse < type |
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global |
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consts |
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"0" :: "'a::zero" ("0") |
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"1" :: "'a::one" ("1") |
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"+" :: "['a::plus, 'a] => 'a" (infixl 65) |
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- :: "['a::minus, 'a] => 'a" (infixl 65) |
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uminus :: "['a::minus] => 'a" ("- _" [81] 80) |
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* :: "['a::times, 'a] => 'a" (infixl 70) |
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syntax |
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"_index1" :: index ("\<^sub>1") |
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translations |
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(index) "\<^sub>1" == "_index 1" |
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local |
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typed_print_translation {* |
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let |
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fun tr' c = (c, fn show_sorts => fn T => fn ts => |
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if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match |
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else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T); |
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in [tr' "0", tr' "1"] end; |
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*} -- {* show types that are presumably too general *} |
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consts |
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abs :: "'a::minus => 'a" |
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inverse :: "'a::inverse => 'a" |
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divide :: "['a::inverse, 'a] => 'a" (infixl "'/" 70) |
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syntax (xsymbols) |
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abs :: "'a::minus => 'a" ("\<bar>_\<bar>") |
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syntax (HTML output) |
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abs :: "'a::minus => 'a" ("\<bar>_\<bar>") |
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axclass plus_ac0 < plus, zero |
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commute: "x + y = y + x" |
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assoc: "(x + y) + z = x + (y + z)" |
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zero: "0 + x = x" |
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subsection {* Theory and package setup *} |
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subsubsection {* Basic lemmas *} |
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use "HOL_lemmas.ML" |
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theorems case_split = case_split_thm [case_names True False] |
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subsubsection {* Intuitionistic Reasoning *} |
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lemma impE': |
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assumes 1: "P --> Q" |
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and 2: "Q ==> R" |
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and 3: "P --> Q ==> P" |
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shows R |
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proof - |
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from 3 and 1 have P . |
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with 1 have Q by (rule impE) |
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with 2 show R . |
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qed |
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lemma allE': |
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assumes 1: "ALL x. P x" |
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and 2: "P x ==> ALL x. P x ==> Q" |
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shows Q |
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proof - |
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from 1 have "P x" by (rule spec) |
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from this and 1 show Q by (rule 2) |
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qed |
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lemma notE': |
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assumes 1: "~ P" |
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and 2: "~ P ==> P" |
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shows R |
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proof - |
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from 2 and 1 have P . |
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with 1 show R by (rule notE) |
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qed |
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lemmas [CPure.elim!] = disjE iffE FalseE conjE exE |
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and [CPure.intro!] = iffI conjI impI TrueI notI allI refl |
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and [CPure.elim 2] = allE notE' impE' |
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and [CPure.intro] = exI disjI2 disjI1 |
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lemmas [trans] = trans |
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and [sym] = sym not_sym |
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and [CPure.elim?] = iffD1 iffD2 impE |
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subsubsection {* Atomizing meta-level connectives *} |
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lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)" |
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proof |
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assume "!!x. P x" |
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show "ALL x. P x" by (rule allI) |
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next |
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assume "ALL x. P x" |
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thus "!!x. P x" by (rule allE) |
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qed |
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lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" |
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proof |
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assume r: "A ==> B" |
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show "A --> B" by (rule impI) (rule r) |
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next |
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assume "A --> B" and A |
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thus B by (rule mp) |
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qed |
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lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" |
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proof |
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assume "x == y" |
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show "x = y" by (unfold prems) (rule refl) |
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next |
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assume "x = y" |
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thus "x == y" by (rule eq_reflection) |
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|
274 |
qed |
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275 |
|
12023 | 276 |
lemma atomize_conj [atomize]: |
277 |
"(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)" |
|
12003 | 278 |
proof |
11953 | 279 |
assume "!!C. (A ==> B ==> PROP C) ==> PROP C" |
280 |
show "A & B" by (rule conjI) |
|
281 |
next |
|
282 |
fix C |
|
283 |
assume "A & B" |
|
284 |
assume "A ==> B ==> PROP C" |
|
285 |
thus "PROP C" |
|
286 |
proof this |
|
287 |
show A by (rule conjunct1) |
|
288 |
show B by (rule conjunct2) |
|
289 |
qed |
|
290 |
qed |
|
291 |
||
12386 | 292 |
lemmas [symmetric, rulify] = atomize_all atomize_imp |
293 |
||
11750 | 294 |
|
295 |
subsubsection {* Classical Reasoner setup *} |
|
9529 | 296 |
|
10383 | 297 |
use "cladata.ML" |
298 |
setup hypsubst_setup |
|
11977 | 299 |
|
12386 | 300 |
ML_setup {* |
301 |
Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)); |
|
302 |
*} |
|
11977 | 303 |
|
10383 | 304 |
setup Classical.setup |
305 |
setup clasetup |
|
306 |
||
12386 | 307 |
lemmas [intro?] = ext |
308 |
and [elim?] = ex1_implies_ex |
|
11977 | 309 |
|
9869 | 310 |
use "blastdata.ML" |
311 |
setup Blast.setup |
|
4868 | 312 |
|
11750 | 313 |
|
314 |
subsubsection {* Simplifier setup *} |
|
315 |
||
12281 | 316 |
lemma meta_eq_to_obj_eq: "x == y ==> x = y" |
317 |
proof - |
|
318 |
assume r: "x == y" |
|
319 |
show "x = y" by (unfold r) (rule refl) |
|
320 |
qed |
|
321 |
||
322 |
lemma eta_contract_eq: "(%s. f s) = f" .. |
|
323 |
||
324 |
lemma simp_thms: |
|
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325 |
shows not_not: "(~ ~ P) = P" |
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326 |
and |
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|
327 |
"(P ~= Q) = (P = (~Q))" |
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|
328 |
"(P | ~P) = True" "(~P | P) = True" |
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|
329 |
"((~P) = (~Q)) = (P=Q)" |
12281 | 330 |
"(x = x) = True" |
331 |
"(~True) = False" "(~False) = True" |
|
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|
332 |
"(~P) ~= P" "P ~= (~P)" |
12281 | 333 |
"(True=P) = P" "(P=True) = P" "(False=P) = (~P)" "(P=False) = (~P)" |
334 |
"(True --> P) = P" "(False --> P) = True" |
|
335 |
"(P --> True) = True" "(P --> P) = True" |
|
336 |
"(P --> False) = (~P)" "(P --> ~P) = (~P)" |
|
337 |
"(P & True) = P" "(True & P) = P" |
|
338 |
"(P & False) = False" "(False & P) = False" |
|
339 |
"(P & P) = P" "(P & (P & Q)) = (P & Q)" |
|
340 |
"(P & ~P) = False" "(~P & P) = False" |
|
341 |
"(P | True) = True" "(True | P) = True" |
|
342 |
"(P | False) = P" "(False | P) = P" |
|
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|
343 |
"(P | P) = P" "(P | (P | Q)) = (P | Q)" and |
12281 | 344 |
"(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x" |
345 |
-- {* needed for the one-point-rule quantifier simplification procs *} |
|
346 |
-- {* essential for termination!! *} and |
|
347 |
"!!P. (EX x. x=t & P(x)) = P(t)" |
|
348 |
"!!P. (EX x. t=x & P(x)) = P(t)" |
|
349 |
"!!P. (ALL x. x=t --> P(x)) = P(t)" |
|
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350 |
"!!P. (ALL x. t=x --> P(x)) = P(t)" |
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|
351 |
by (blast, blast, blast, blast, blast, rules+) |
13421 | 352 |
|
12281 | 353 |
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))" |
12354 | 354 |
by rules |
12281 | 355 |
|
356 |
lemma ex_simps: |
|
357 |
"!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)" |
|
358 |
"!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))" |
|
359 |
"!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)" |
|
360 |
"!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))" |
|
361 |
"!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)" |
|
362 |
"!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))" |
|
363 |
-- {* Miniscoping: pushing in existential quantifiers. *} |
|
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|
364 |
by (rules | blast)+ |
12281 | 365 |
|
366 |
lemma all_simps: |
|
367 |
"!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)" |
|
368 |
"!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))" |
|
369 |
"!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)" |
|
370 |
"!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))" |
|
371 |
"!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)" |
|
372 |
"!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))" |
|
373 |
-- {* Miniscoping: pushing in universal quantifiers. *} |
|
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|
374 |
by (rules | blast)+ |
12281 | 375 |
|
376 |
lemma eq_ac: |
|
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|
377 |
shows eq_commute: "(a=b) = (b=a)" |
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|
378 |
and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" |
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|
379 |
and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+) |
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|
380 |
lemma neq_commute: "(a~=b) = (b~=a)" by rules |
12281 | 381 |
|
382 |
lemma conj_comms: |
|
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|
383 |
shows conj_commute: "(P&Q) = (Q&P)" |
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|
384 |
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+ |
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|
385 |
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules |
12281 | 386 |
|
387 |
lemma disj_comms: |
|
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|
388 |
shows disj_commute: "(P|Q) = (Q|P)" |
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|
389 |
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+ |
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|
390 |
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules |
12281 | 391 |
|
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|
392 |
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules |
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|
393 |
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules |
12281 | 394 |
|
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|
395 |
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules |
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|
396 |
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules |
12281 | 397 |
|
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|
398 |
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules |
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|
399 |
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by rules |
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|
400 |
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules |
12281 | 401 |
|
402 |
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *} |
|
403 |
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast |
|
404 |
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast |
|
405 |
||
406 |
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast |
|
407 |
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast |
|
408 |
||
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berghofe
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diff
changeset
|
409 |
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules |
12281 | 410 |
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast |
411 |
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast |
|
412 |
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast |
|
413 |
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast |
|
414 |
lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *} |
|
415 |
by blast |
|
416 |
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast |
|
417 |
||
12436
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changeset
|
418 |
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules |
12281 | 419 |
|
420 |
||
421 |
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q" |
|
422 |
-- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *} |
|
423 |
-- {* cases boil down to the same thing. *} |
|
424 |
by blast |
|
425 |
||
426 |
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast |
|
427 |
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast |
|
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diff
changeset
|
428 |
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules |
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
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diff
changeset
|
429 |
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules |
12281 | 430 |
|
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berghofe
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diff
changeset
|
431 |
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules |
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
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diff
changeset
|
432 |
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules |
12281 | 433 |
|
434 |
text {* |
|
435 |
\medskip The @{text "&"} congruence rule: not included by default! |
|
436 |
May slow rewrite proofs down by as much as 50\% *} |
|
437 |
||
438 |
lemma conj_cong: |
|
439 |
"(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))" |
|
12354 | 440 |
by rules |
12281 | 441 |
|
442 |
lemma rev_conj_cong: |
|
443 |
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))" |
|
12354 | 444 |
by rules |
12281 | 445 |
|
446 |
text {* The @{text "|"} congruence rule: not included by default! *} |
|
447 |
||
448 |
lemma disj_cong: |
|
449 |
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))" |
|
450 |
by blast |
|
451 |
||
452 |
lemma eq_sym_conv: "(x = y) = (y = x)" |
|
12354 | 453 |
by rules |
12281 | 454 |
|
455 |
||
456 |
text {* \medskip if-then-else rules *} |
|
457 |
||
458 |
lemma if_True: "(if True then x else y) = x" |
|
459 |
by (unfold if_def) blast |
|
460 |
||
461 |
lemma if_False: "(if False then x else y) = y" |
|
462 |
by (unfold if_def) blast |
|
463 |
||
464 |
lemma if_P: "P ==> (if P then x else y) = x" |
|
465 |
by (unfold if_def) blast |
|
466 |
||
467 |
lemma if_not_P: "~P ==> (if P then x else y) = y" |
|
468 |
by (unfold if_def) blast |
|
469 |
||
470 |
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" |
|
471 |
apply (rule case_split [of Q]) |
|
472 |
apply (subst if_P) |
|
473 |
prefer 3 apply (subst if_not_P) |
|
474 |
apply blast+ |
|
475 |
done |
|
476 |
||
477 |
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))" |
|
478 |
apply (subst split_if) |
|
479 |
apply blast |
|
480 |
done |
|
481 |
||
482 |
lemmas if_splits = split_if split_if_asm |
|
483 |
||
484 |
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))" |
|
485 |
by (rule split_if) |
|
486 |
||
487 |
lemma if_cancel: "(if c then x else x) = x" |
|
488 |
apply (subst split_if) |
|
489 |
apply blast |
|
490 |
done |
|
491 |
||
492 |
lemma if_eq_cancel: "(if x = y then y else x) = x" |
|
493 |
apply (subst split_if) |
|
494 |
apply blast |
|
495 |
done |
|
496 |
||
497 |
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))" |
|
498 |
-- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *} |
|
499 |
by (rule split_if) |
|
500 |
||
501 |
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))" |
|
502 |
-- {* And this form is useful for expanding @{text if}s on the LEFT. *} |
|
503 |
apply (subst split_if) |
|
504 |
apply blast |
|
505 |
done |
|
506 |
||
12436
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
berghofe
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12386
diff
changeset
|
507 |
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules |
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
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12386
diff
changeset
|
508 |
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules |
12281 | 509 |
|
9869 | 510 |
use "simpdata.ML" |
511 |
setup Simplifier.setup |
|
512 |
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup |
|
513 |
setup Splitter.setup setup Clasimp.setup |
|
514 |
||
13438
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
515 |
text{*Needs only HOL-lemmas:*} |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
516 |
lemma mk_left_commute: |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
517 |
assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
518 |
c: "\<And>x y. f x y = f y x" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
519 |
shows "f x (f y z) = f y (f x z)" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
520 |
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]]) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
521 |
|
11750 | 522 |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
523 |
subsubsection {* Generic cases and induction *} |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
524 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
525 |
constdefs |
11989 | 526 |
induct_forall :: "('a => bool) => bool" |
527 |
"induct_forall P == \<forall>x. P x" |
|
528 |
induct_implies :: "bool => bool => bool" |
|
529 |
"induct_implies A B == A --> B" |
|
530 |
induct_equal :: "'a => 'a => bool" |
|
531 |
"induct_equal x y == x = y" |
|
532 |
induct_conj :: "bool => bool => bool" |
|
533 |
"induct_conj A B == A & B" |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
534 |
|
11989 | 535 |
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))" |
536 |
by (simp only: atomize_all induct_forall_def) |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
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diff
changeset
|
537 |
|
11989 | 538 |
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)" |
539 |
by (simp only: atomize_imp induct_implies_def) |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
540 |
|
11989 | 541 |
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)" |
542 |
by (simp only: atomize_eq induct_equal_def) |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
543 |
|
11989 | 544 |
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) = |
545 |
induct_conj (induct_forall A) (induct_forall B)" |
|
12354 | 546 |
by (unfold induct_forall_def induct_conj_def) rules |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
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diff
changeset
|
547 |
|
11989 | 548 |
lemma induct_implies_conj: "induct_implies C (induct_conj A B) = |
549 |
induct_conj (induct_implies C A) (induct_implies C B)" |
|
12354 | 550 |
by (unfold induct_implies_def induct_conj_def) rules |
11989 | 551 |
|
552 |
lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)" |
|
12354 | 553 |
by (simp only: atomize_imp atomize_eq induct_conj_def) (rules intro: equal_intr_rule) |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
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diff
changeset
|
554 |
|
11989 | 555 |
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B" |
556 |
by (simp add: induct_implies_def) |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
557 |
|
12161 | 558 |
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq |
559 |
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq |
|
560 |
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def |
|
11989 | 561 |
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
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diff
changeset
|
562 |
|
11989 | 563 |
hide const induct_forall induct_implies induct_equal induct_conj |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
564 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
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diff
changeset
|
565 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
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diff
changeset
|
566 |
text {* Method setup. *} |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
567 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
568 |
ML {* |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
11770
diff
changeset
|
569 |
structure InductMethod = InductMethodFun |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
570 |
(struct |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
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diff
changeset
|
571 |
val dest_concls = HOLogic.dest_concls; |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
572 |
val cases_default = thm "case_split"; |
11989 | 573 |
val local_impI = thm "induct_impliesI"; |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
574 |
val conjI = thm "conjI"; |
11989 | 575 |
val atomize = thms "induct_atomize"; |
576 |
val rulify1 = thms "induct_rulify1"; |
|
577 |
val rulify2 = thms "induct_rulify2"; |
|
12240 | 578 |
val localize = [Thm.symmetric (thm "induct_implies_def")]; |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
579 |
end); |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
580 |
*} |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
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diff
changeset
|
581 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
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parents:
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diff
changeset
|
582 |
setup InductMethod.setup |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
583 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
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diff
changeset
|
584 |
|
11750 | 585 |
subsection {* Order signatures and orders *} |
586 |
||
587 |
axclass |
|
12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
12281
diff
changeset
|
588 |
ord < type |
11750 | 589 |
|
590 |
syntax |
|
591 |
"op <" :: "['a::ord, 'a] => bool" ("op <") |
|
592 |
"op <=" :: "['a::ord, 'a] => bool" ("op <=") |
|
593 |
||
594 |
global |
|
595 |
||
596 |
consts |
|
597 |
"op <" :: "['a::ord, 'a] => bool" ("(_/ < _)" [50, 51] 50) |
|
598 |
"op <=" :: "['a::ord, 'a] => bool" ("(_/ <= _)" [50, 51] 50) |
|
599 |
||
600 |
local |
|
601 |
||
12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
12023
diff
changeset
|
602 |
syntax (xsymbols) |
11750 | 603 |
"op <=" :: "['a::ord, 'a] => bool" ("op \<le>") |
604 |
"op <=" :: "['a::ord, 'a] => bool" ("(_/ \<le> _)" [50, 51] 50) |
|
605 |
||
606 |
||
607 |
subsubsection {* Monotonicity *} |
|
608 |
||
13412 | 609 |
locale mono = |
610 |
fixes f |
|
611 |
assumes mono: "A <= B ==> f A <= f B" |
|
11750 | 612 |
|
13421 | 613 |
lemmas monoI [intro?] = mono.intro |
13412 | 614 |
and monoD [dest?] = mono.mono |
11750 | 615 |
|
616 |
constdefs |
|
617 |
min :: "['a::ord, 'a] => 'a" |
|
618 |
"min a b == (if a <= b then a else b)" |
|
619 |
max :: "['a::ord, 'a] => 'a" |
|
620 |
"max a b == (if a <= b then b else a)" |
|
621 |
||
622 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
|
623 |
by (simp add: min_def) |
|
624 |
||
625 |
lemma min_of_mono: |
|
626 |
"ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)" |
|
627 |
by (simp add: min_def) |
|
628 |
||
629 |
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" |
|
630 |
by (simp add: max_def) |
|
631 |
||
632 |
lemma max_of_mono: |
|
633 |
"ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)" |
|
634 |
by (simp add: max_def) |
|
635 |
||
636 |
||
637 |
subsubsection "Orders" |
|
638 |
||
639 |
axclass order < ord |
|
640 |
order_refl [iff]: "x <= x" |
|
641 |
order_trans: "x <= y ==> y <= z ==> x <= z" |
|
642 |
order_antisym: "x <= y ==> y <= x ==> x = y" |
|
643 |
order_less_le: "(x < y) = (x <= y & x ~= y)" |
|
644 |
||
645 |
||
646 |
text {* Reflexivity. *} |
|
647 |
||
648 |
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y" |
|
649 |
-- {* This form is useful with the classical reasoner. *} |
|
650 |
apply (erule ssubst) |
|
651 |
apply (rule order_refl) |
|
652 |
done |
|
653 |
||
13553 | 654 |
lemma order_less_irrefl [iff]: "~ x < (x::'a::order)" |
11750 | 655 |
by (simp add: order_less_le) |
656 |
||
657 |
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)" |
|
658 |
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *} |
|
659 |
apply (simp add: order_less_le) |
|
12256 | 660 |
apply blast |
11750 | 661 |
done |
662 |
||
663 |
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard] |
|
664 |
||
665 |
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y" |
|
666 |
by (simp add: order_less_le) |
|
667 |
||
668 |
||
669 |
text {* Asymmetry. *} |
|
670 |
||
671 |
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)" |
|
672 |
by (simp add: order_less_le order_antisym) |
|
673 |
||
674 |
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P" |
|
675 |
apply (drule order_less_not_sym) |
|
676 |
apply (erule contrapos_np) |
|
677 |
apply simp |
|
678 |
done |
|
679 |
||
680 |
||
681 |
text {* Transitivity. *} |
|
682 |
||
683 |
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z" |
|
684 |
apply (simp add: order_less_le) |
|
685 |
apply (blast intro: order_trans order_antisym) |
|
686 |
done |
|
687 |
||
688 |
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z" |
|
689 |
apply (simp add: order_less_le) |
|
690 |
apply (blast intro: order_trans order_antisym) |
|
691 |
done |
|
692 |
||
693 |
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z" |
|
694 |
apply (simp add: order_less_le) |
|
695 |
apply (blast intro: order_trans order_antisym) |
|
696 |
done |
|
697 |
||
698 |
||
699 |
text {* Useful for simplification, but too risky to include by default. *} |
|
700 |
||
701 |
lemma order_less_imp_not_less: "(x::'a::order) < y ==> (~ y < x) = True" |
|
702 |
by (blast elim: order_less_asym) |
|
703 |
||
704 |
lemma order_less_imp_triv: "(x::'a::order) < y ==> (y < x --> P) = True" |
|
705 |
by (blast elim: order_less_asym) |
|
706 |
||
707 |
lemma order_less_imp_not_eq: "(x::'a::order) < y ==> (x = y) = False" |
|
708 |
by auto |
|
709 |
||
710 |
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==> (y = x) = False" |
|
711 |
by auto |
|
712 |
||
713 |
||
714 |
text {* Other operators. *} |
|
715 |
||
716 |
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least" |
|
717 |
apply (simp add: min_def) |
|
718 |
apply (blast intro: order_antisym) |
|
719 |
done |
|
720 |
||
721 |
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x" |
|
722 |
apply (simp add: max_def) |
|
723 |
apply (blast intro: order_antisym) |
|
724 |
done |
|
725 |
||
726 |
||
727 |
subsubsection {* Least value operator *} |
|
728 |
||
729 |
constdefs |
|
730 |
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) |
|
731 |
"Least P == THE x. P x & (ALL y. P y --> x <= y)" |
|
732 |
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *} |
|
733 |
||
734 |
lemma LeastI2: |
|
735 |
"[| P (x::'a::order); |
|
736 |
!!y. P y ==> x <= y; |
|
737 |
!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |] |
|
12281 | 738 |
==> Q (Least P)" |
11750 | 739 |
apply (unfold Least_def) |
740 |
apply (rule theI2) |
|
741 |
apply (blast intro: order_antisym)+ |
|
742 |
done |
|
743 |
||
744 |
lemma Least_equality: |
|
12281 | 745 |
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" |
11750 | 746 |
apply (simp add: Least_def) |
747 |
apply (rule the_equality) |
|
748 |
apply (auto intro!: order_antisym) |
|
749 |
done |
|
750 |
||
751 |
||
752 |
subsubsection "Linear / total orders" |
|
753 |
||
754 |
axclass linorder < order |
|
755 |
linorder_linear: "x <= y | y <= x" |
|
756 |
||
757 |
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x" |
|
758 |
apply (simp add: order_less_le) |
|
759 |
apply (insert linorder_linear) |
|
760 |
apply blast |
|
761 |
done |
|
762 |
||
763 |
lemma linorder_cases [case_names less equal greater]: |
|
764 |
"((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P" |
|
765 |
apply (insert linorder_less_linear) |
|
766 |
apply blast |
|
767 |
done |
|
768 |
||
769 |
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)" |
|
770 |
apply (simp add: order_less_le) |
|
771 |
apply (insert linorder_linear) |
|
772 |
apply (blast intro: order_antisym) |
|
773 |
done |
|
774 |
||
775 |
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)" |
|
776 |
apply (simp add: order_less_le) |
|
777 |
apply (insert linorder_linear) |
|
778 |
apply (blast intro: order_antisym) |
|
779 |
done |
|
780 |
||
781 |
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)" |
|
782 |
apply (cut_tac x = x and y = y in linorder_less_linear) |
|
783 |
apply auto |
|
784 |
done |
|
785 |
||
786 |
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R" |
|
787 |
apply (simp add: linorder_neq_iff) |
|
788 |
apply blast |
|
789 |
done |
|
790 |
||
791 |
||
792 |
subsubsection "Min and max on (linear) orders" |
|
793 |
||
794 |
lemma min_same [simp]: "min (x::'a::order) x = x" |
|
795 |
by (simp add: min_def) |
|
796 |
||
797 |
lemma max_same [simp]: "max (x::'a::order) x = x" |
|
798 |
by (simp add: max_def) |
|
799 |
||
800 |
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)" |
|
801 |
apply (simp add: max_def) |
|
802 |
apply (insert linorder_linear) |
|
803 |
apply (blast intro: order_trans) |
|
804 |
done |
|
805 |
||
806 |
lemma le_maxI1: "(x::'a::linorder) <= max x y" |
|
807 |
by (simp add: le_max_iff_disj) |
|
808 |
||
809 |
lemma le_maxI2: "(y::'a::linorder) <= max x y" |
|
810 |
-- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *} |
|
811 |
by (simp add: le_max_iff_disj) |
|
812 |
||
813 |
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)" |
|
814 |
apply (simp add: max_def order_le_less) |
|
815 |
apply (insert linorder_less_linear) |
|
816 |
apply (blast intro: order_less_trans) |
|
817 |
done |
|
818 |
||
819 |
lemma max_le_iff_conj [simp]: |
|
820 |
"!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)" |
|
821 |
apply (simp add: max_def) |
|
822 |
apply (insert linorder_linear) |
|
823 |
apply (blast intro: order_trans) |
|
824 |
done |
|
825 |
||
826 |
lemma max_less_iff_conj [simp]: |
|
827 |
"!!z::'a::linorder. (max x y < z) = (x < z & y < z)" |
|
828 |
apply (simp add: order_le_less max_def) |
|
829 |
apply (insert linorder_less_linear) |
|
830 |
apply (blast intro: order_less_trans) |
|
831 |
done |
|
832 |
||
833 |
lemma le_min_iff_conj [simp]: |
|
834 |
"!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)" |
|
12892 | 835 |
-- {* @{text "[iff]"} screws up a @{text blast} in MiniML *} |
11750 | 836 |
apply (simp add: min_def) |
837 |
apply (insert linorder_linear) |
|
838 |
apply (blast intro: order_trans) |
|
839 |
done |
|
840 |
||
841 |
lemma min_less_iff_conj [simp]: |
|
842 |
"!!z::'a::linorder. (z < min x y) = (z < x & z < y)" |
|
843 |
apply (simp add: order_le_less min_def) |
|
844 |
apply (insert linorder_less_linear) |
|
845 |
apply (blast intro: order_less_trans) |
|
846 |
done |
|
847 |
||
848 |
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)" |
|
849 |
apply (simp add: min_def) |
|
850 |
apply (insert linorder_linear) |
|
851 |
apply (blast intro: order_trans) |
|
852 |
done |
|
853 |
||
854 |
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)" |
|
855 |
apply (simp add: min_def order_le_less) |
|
856 |
apply (insert linorder_less_linear) |
|
857 |
apply (blast intro: order_less_trans) |
|
858 |
done |
|
859 |
||
13438
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
860 |
declare order_less_irrefl [iff] |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
861 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
862 |
lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
863 |
apply(simp add:max_def) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
864 |
apply(rule conjI) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
865 |
apply(blast intro:order_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
866 |
apply(simp add:linorder_not_le) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
867 |
apply(blast dest: order_less_trans order_le_less_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
868 |
done |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
869 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
870 |
lemma max_commute: "!!x::'a::linorder. max x y = max y x" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
871 |
apply(simp add:max_def) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
872 |
apply(rule conjI) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
873 |
apply(blast intro:order_antisym) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
874 |
apply(simp add:linorder_not_le) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
875 |
apply(blast dest: order_less_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
876 |
done |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
877 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
878 |
lemmas max_ac = max_assoc max_commute |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
879 |
mk_left_commute[of max,OF max_assoc max_commute] |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
880 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
881 |
lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
882 |
apply(simp add:min_def) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
883 |
apply(rule conjI) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
884 |
apply(blast intro:order_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
885 |
apply(simp add:linorder_not_le) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
886 |
apply(blast dest: order_less_trans order_le_less_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
887 |
done |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
888 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
889 |
lemma min_commute: "!!x::'a::linorder. min x y = min y x" |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
890 |
apply(simp add:min_def) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
891 |
apply(rule conjI) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
892 |
apply(blast intro:order_antisym) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
893 |
apply(simp add:linorder_not_le) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
894 |
apply(blast dest: order_less_trans) |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
895 |
done |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
896 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
897 |
lemmas min_ac = min_assoc min_commute |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
898 |
mk_left_commute[of min,OF min_assoc min_commute] |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
899 |
|
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
900 |
declare order_less_irrefl [iff del] |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
901 |
declare order_less_irrefl [simp] |
527811f00c56
added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents:
13421
diff
changeset
|
902 |
|
11750 | 903 |
lemma split_min: |
904 |
"P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))" |
|
905 |
by (simp add: min_def) |
|
906 |
||
907 |
lemma split_max: |
|
908 |
"P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))" |
|
909 |
by (simp add: max_def) |
|
910 |
||
911 |
||
912 |
subsubsection "Bounded quantifiers" |
|
913 |
||
914 |
syntax |
|
915 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) |
|
916 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) |
|
917 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) |
|
918 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
|
919 |
||
12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
12023
diff
changeset
|
920 |
syntax (xsymbols) |
11750 | 921 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) |
922 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) |
|
923 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) |
|
924 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
|
925 |
||
926 |
syntax (HOL) |
|
927 |
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) |
|
928 |
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) |
|
929 |
"_leAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) |
|
930 |
"_leEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
|
931 |
||
932 |
translations |
|
933 |
"ALL x<y. P" => "ALL x. x < y --> P" |
|
934 |
"EX x<y. P" => "EX x. x < y & P" |
|
935 |
"ALL x<=y. P" => "ALL x. x <= y --> P" |
|
936 |
"EX x<=y. P" => "EX x. x <= y & P" |
|
937 |
||
923 | 938 |
end |