author | wenzelm |
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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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*) |
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header {* The basis of Higher-Order Logic *} |
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theory HOL |
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imports CPure |
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uses ("simpdata.ML") "Tools/res_atpset.ML" |
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begin |
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subsection {* Primitive logic *} |
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subsubsection {* Core syntax *} |
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classes type |
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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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arbitrary :: 'a |
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undefined :: '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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consts |
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If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) |
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subsubsection {* Additional concrete syntax *} |
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notation (output) |
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"op =" (infix "=" 50) |
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abbreviation |
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not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where |
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"x ~= y == ~ (x = y)" |
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notation (output) |
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not_equal (infix "~=" 50) |
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notation (xsymbols) |
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Not ("\<not> _" [40] 40) and |
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"op &" (infixr "\<and>" 35) and |
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"op |" (infixr "\<or>" 30) and |
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"op -->" (infixr "\<longrightarrow>" 25) and |
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not_equal (infix "\<noteq>" 50) |
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notation (HTML output) |
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Not ("\<not> _" [40] 40) and |
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"op &" (infixr "\<and>" 35) and |
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"op |" (infixr "\<or>" 30) and |
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not_equal (infix "\<noteq>" 50) |
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abbreviation (iff) |
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iff :: "[bool, bool] => bool" (infixr "<->" 25) where |
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"A <-> B == A = B" |
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notation (xsymbols) |
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iff (infixr "\<longleftrightarrow>" 25) |
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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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"_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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"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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print_translation {* |
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(* To avoid eta-contraction of body: *) |
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[("The", fn [Abs abs] => |
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let val (x,t) = atomic_abs_tr' abs |
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in Syntax.const "_The" $ x $ t end)] |
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*} |
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syntax (xsymbols) |
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"_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10) |
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notation (xsymbols) |
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All (binder "\<forall>" 10) and |
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Ex (binder "\<exists>" 10) and |
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Ex1 (binder "\<exists>!" 10) |
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notation (HTML output) |
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All (binder "\<forall>" 10) and |
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Ex (binder "\<exists>" 10) and |
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Ex1 (binder "\<exists>!" 10) |
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notation (HOL) |
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All (binder "! " 10) and |
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Ex (binder "? " 10) and |
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Ex1 (binder "?! " 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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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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finalconsts |
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"op =" |
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"op -->" |
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The |
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arbitrary |
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undefined |
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subsubsection {* Generic algebraic operations *} |
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class zero = |
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fixes zero :: "'a" ("\<^loc>0") |
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class one = |
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fixes one :: "'a" ("\<^loc>1") |
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hide (open) const zero one |
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class plus = |
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class minus = |
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fixes uminus :: "'a \<Rightarrow> 'a" |
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and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>-" 65) |
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and abs :: "'a \<Rightarrow> 'a" |
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class times = |
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fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>*" 70) |
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class inverse = |
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fixes inverse :: "'a \<Rightarrow> 'a" |
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and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>'/" 70) |
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notation |
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uminus ("- _" [81] 80) |
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notation (xsymbols) |
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abs ("\<bar>_\<bar>") |
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notation (HTML output) |
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abs ("\<bar>_\<bar>") |
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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) "\<^bsub>\<struct>\<^esub>" |
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typed_print_translation {* |
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let |
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val thy = the_context (); |
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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); |
21410 | 224 |
in map (tr' o Sign.const_syntax_name thy) ["HOL.one", "HOL.zero"] end; |
11750 | 225 |
*} -- {* show types that are presumably too general *} |
226 |
||
227 |
||
20944 | 228 |
subsection {* Fundamental rules *} |
229 |
||
20973 | 230 |
subsubsection {* Equality *} |
20944 | 231 |
|
232 |
text {* Thanks to Stephan Merz *} |
|
233 |
lemma subst: |
|
234 |
assumes eq: "s = t" and p: "P s" |
|
235 |
shows "P t" |
|
236 |
proof - |
|
237 |
from eq have meta: "s \<equiv> t" |
|
238 |
by (rule eq_reflection) |
|
239 |
from p show ?thesis |
|
240 |
by (unfold meta) |
|
241 |
qed |
|
15411 | 242 |
|
18457 | 243 |
lemma sym: "s = t ==> t = s" |
244 |
by (erule subst) (rule refl) |
|
15411 | 245 |
|
18457 | 246 |
lemma ssubst: "t = s ==> P s ==> P t" |
247 |
by (drule sym) (erule subst) |
|
15411 | 248 |
|
249 |
lemma trans: "[| r=s; s=t |] ==> r=t" |
|
18457 | 250 |
by (erule subst) |
15411 | 251 |
|
20944 | 252 |
lemma def_imp_eq: |
253 |
assumes meq: "A == B" |
|
254 |
shows "A = B" |
|
18457 | 255 |
by (unfold meq) (rule refl) |
256 |
||
20944 | 257 |
(*a mere copy*) |
258 |
lemma meta_eq_to_obj_eq: |
|
259 |
assumes meq: "A == B" |
|
260 |
shows "A = B" |
|
261 |
by (unfold meq) (rule refl) |
|
15411 | 262 |
|
21502 | 263 |
text {* Useful with @{text erule} for proving equalities from known equalities. *} |
20944 | 264 |
(* a = b |
15411 | 265 |
| | |
266 |
c = d *) |
|
267 |
lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d" |
|
268 |
apply (rule trans) |
|
269 |
apply (rule trans) |
|
270 |
apply (rule sym) |
|
271 |
apply assumption+ |
|
272 |
done |
|
273 |
||
15524
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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|
274 |
text {* For calculational reasoning: *} |
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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parents:
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changeset
|
275 |
|
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
nipkow
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|
276 |
lemma forw_subst: "a = b ==> P b ==> P a" |
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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|
277 |
by (rule ssubst) |
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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|
278 |
|
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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|
279 |
lemma back_subst: "P a ==> a = b ==> P b" |
2ef571f80a55
Moved oderings from HOL into the new Orderings.thy
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|
280 |
by (rule subst) |
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Moved oderings from HOL into the new Orderings.thy
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changeset
|
281 |
|
15411 | 282 |
|
20944 | 283 |
subsubsection {*Congruence rules for application*} |
15411 | 284 |
|
285 |
(*similar to AP_THM in Gordon's HOL*) |
|
286 |
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)" |
|
287 |
apply (erule subst) |
|
288 |
apply (rule refl) |
|
289 |
done |
|
290 |
||
291 |
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*) |
|
292 |
lemma arg_cong: "x=y ==> f(x)=f(y)" |
|
293 |
apply (erule subst) |
|
294 |
apply (rule refl) |
|
295 |
done |
|
296 |
||
15655 | 297 |
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d" |
298 |
apply (erule ssubst)+ |
|
299 |
apply (rule refl) |
|
300 |
done |
|
301 |
||
15411 | 302 |
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)" |
303 |
apply (erule subst)+ |
|
304 |
apply (rule refl) |
|
305 |
done |
|
306 |
||
307 |
||
20944 | 308 |
subsubsection {*Equality of booleans -- iff*} |
15411 | 309 |
|
21504 | 310 |
lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q" |
311 |
by (iprover intro: iff [THEN mp, THEN mp] impI assms) |
|
15411 | 312 |
|
313 |
lemma iffD2: "[| P=Q; Q |] ==> P" |
|
18457 | 314 |
by (erule ssubst) |
15411 | 315 |
|
316 |
lemma rev_iffD2: "[| Q; P=Q |] ==> P" |
|
18457 | 317 |
by (erule iffD2) |
15411 | 318 |
|
21504 | 319 |
lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P" |
320 |
by (drule sym) (rule iffD2) |
|
321 |
||
322 |
lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P" |
|
323 |
by (drule sym) (rule rev_iffD2) |
|
15411 | 324 |
|
325 |
lemma iffE: |
|
326 |
assumes major: "P=Q" |
|
21504 | 327 |
and minor: "[| P --> Q; Q --> P |] ==> R" |
18457 | 328 |
shows R |
329 |
by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1]) |
|
15411 | 330 |
|
331 |
||
20944 | 332 |
subsubsection {*True*} |
15411 | 333 |
|
334 |
lemma TrueI: "True" |
|
21504 | 335 |
unfolding True_def by (rule refl) |
15411 | 336 |
|
21504 | 337 |
lemma eqTrueI: "P ==> P = True" |
18457 | 338 |
by (iprover intro: iffI TrueI) |
15411 | 339 |
|
21504 | 340 |
lemma eqTrueE: "P = True ==> P" |
341 |
by (erule iffD2) (rule TrueI) |
|
15411 | 342 |
|
343 |
||
20944 | 344 |
subsubsection {*Universal quantifier*} |
15411 | 345 |
|
21504 | 346 |
lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)" |
347 |
unfolding All_def by (iprover intro: ext eqTrueI assms) |
|
15411 | 348 |
|
349 |
lemma spec: "ALL x::'a. P(x) ==> P(x)" |
|
350 |
apply (unfold All_def) |
|
351 |
apply (rule eqTrueE) |
|
352 |
apply (erule fun_cong) |
|
353 |
done |
|
354 |
||
355 |
lemma allE: |
|
356 |
assumes major: "ALL x. P(x)" |
|
21504 | 357 |
and minor: "P(x) ==> R" |
358 |
shows R |
|
359 |
by (iprover intro: minor major [THEN spec]) |
|
15411 | 360 |
|
361 |
lemma all_dupE: |
|
362 |
assumes major: "ALL x. P(x)" |
|
21504 | 363 |
and minor: "[| P(x); ALL x. P(x) |] ==> R" |
364 |
shows R |
|
365 |
by (iprover intro: minor major major [THEN spec]) |
|
15411 | 366 |
|
367 |
||
21504 | 368 |
subsubsection {* False *} |
369 |
||
370 |
text {* |
|
371 |
Depends upon @{text spec}; it is impossible to do propositional |
|
372 |
logic before quantifiers! |
|
373 |
*} |
|
15411 | 374 |
|
375 |
lemma FalseE: "False ==> P" |
|
21504 | 376 |
apply (unfold False_def) |
377 |
apply (erule spec) |
|
378 |
done |
|
15411 | 379 |
|
21504 | 380 |
lemma False_neq_True: "False = True ==> P" |
381 |
by (erule eqTrueE [THEN FalseE]) |
|
15411 | 382 |
|
383 |
||
21504 | 384 |
subsubsection {* Negation *} |
15411 | 385 |
|
386 |
lemma notI: |
|
21504 | 387 |
assumes "P ==> False" |
15411 | 388 |
shows "~P" |
21504 | 389 |
apply (unfold not_def) |
390 |
apply (iprover intro: impI assms) |
|
391 |
done |
|
15411 | 392 |
|
393 |
lemma False_not_True: "False ~= True" |
|
21504 | 394 |
apply (rule notI) |
395 |
apply (erule False_neq_True) |
|
396 |
done |
|
15411 | 397 |
|
398 |
lemma True_not_False: "True ~= False" |
|
21504 | 399 |
apply (rule notI) |
400 |
apply (drule sym) |
|
401 |
apply (erule False_neq_True) |
|
402 |
done |
|
15411 | 403 |
|
404 |
lemma notE: "[| ~P; P |] ==> R" |
|
21504 | 405 |
apply (unfold not_def) |
406 |
apply (erule mp [THEN FalseE]) |
|
407 |
apply assumption |
|
408 |
done |
|
15411 | 409 |
|
21504 | 410 |
lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P" |
411 |
by (erule notE [THEN notI]) (erule meta_mp) |
|
15411 | 412 |
|
413 |
||
20944 | 414 |
subsubsection {*Implication*} |
15411 | 415 |
|
416 |
lemma impE: |
|
417 |
assumes "P-->Q" "P" "Q ==> R" |
|
418 |
shows "R" |
|
17589 | 419 |
by (iprover intro: prems mp) |
15411 | 420 |
|
421 |
(* Reduces Q to P-->Q, allowing substitution in P. *) |
|
422 |
lemma rev_mp: "[| P; P --> Q |] ==> Q" |
|
17589 | 423 |
by (iprover intro: mp) |
15411 | 424 |
|
425 |
lemma contrapos_nn: |
|
426 |
assumes major: "~Q" |
|
427 |
and minor: "P==>Q" |
|
428 |
shows "~P" |
|
17589 | 429 |
by (iprover intro: notI minor major [THEN notE]) |
15411 | 430 |
|
431 |
(*not used at all, but we already have the other 3 combinations *) |
|
432 |
lemma contrapos_pn: |
|
433 |
assumes major: "Q" |
|
434 |
and minor: "P ==> ~Q" |
|
435 |
shows "~P" |
|
17589 | 436 |
by (iprover intro: notI minor major notE) |
15411 | 437 |
|
438 |
lemma not_sym: "t ~= s ==> s ~= t" |
|
21250 | 439 |
by (erule contrapos_nn) (erule sym) |
440 |
||
441 |
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y" |
|
442 |
by (erule subst, erule ssubst, assumption) |
|
15411 | 443 |
|
444 |
(*still used in HOLCF*) |
|
445 |
lemma rev_contrapos: |
|
446 |
assumes pq: "P ==> Q" |
|
447 |
and nq: "~Q" |
|
448 |
shows "~P" |
|
449 |
apply (rule nq [THEN contrapos_nn]) |
|
450 |
apply (erule pq) |
|
451 |
done |
|
452 |
||
20944 | 453 |
subsubsection {*Existential quantifier*} |
15411 | 454 |
|
455 |
lemma exI: "P x ==> EX x::'a. P x" |
|
456 |
apply (unfold Ex_def) |
|
17589 | 457 |
apply (iprover intro: allI allE impI mp) |
15411 | 458 |
done |
459 |
||
460 |
lemma exE: |
|
461 |
assumes major: "EX x::'a. P(x)" |
|
462 |
and minor: "!!x. P(x) ==> Q" |
|
463 |
shows "Q" |
|
464 |
apply (rule major [unfolded Ex_def, THEN spec, THEN mp]) |
|
17589 | 465 |
apply (iprover intro: impI [THEN allI] minor) |
15411 | 466 |
done |
467 |
||
468 |
||
20944 | 469 |
subsubsection {*Conjunction*} |
15411 | 470 |
|
471 |
lemma conjI: "[| P; Q |] ==> P&Q" |
|
472 |
apply (unfold and_def) |
|
17589 | 473 |
apply (iprover intro: impI [THEN allI] mp) |
15411 | 474 |
done |
475 |
||
476 |
lemma conjunct1: "[| P & Q |] ==> P" |
|
477 |
apply (unfold and_def) |
|
17589 | 478 |
apply (iprover intro: impI dest: spec mp) |
15411 | 479 |
done |
480 |
||
481 |
lemma conjunct2: "[| P & Q |] ==> Q" |
|
482 |
apply (unfold and_def) |
|
17589 | 483 |
apply (iprover intro: impI dest: spec mp) |
15411 | 484 |
done |
485 |
||
486 |
lemma conjE: |
|
487 |
assumes major: "P&Q" |
|
488 |
and minor: "[| P; Q |] ==> R" |
|
489 |
shows "R" |
|
490 |
apply (rule minor) |
|
491 |
apply (rule major [THEN conjunct1]) |
|
492 |
apply (rule major [THEN conjunct2]) |
|
493 |
done |
|
494 |
||
495 |
lemma context_conjI: |
|
496 |
assumes prems: "P" "P ==> Q" shows "P & Q" |
|
17589 | 497 |
by (iprover intro: conjI prems) |
15411 | 498 |
|
499 |
||
20944 | 500 |
subsubsection {*Disjunction*} |
15411 | 501 |
|
502 |
lemma disjI1: "P ==> P|Q" |
|
503 |
apply (unfold or_def) |
|
17589 | 504 |
apply (iprover intro: allI impI mp) |
15411 | 505 |
done |
506 |
||
507 |
lemma disjI2: "Q ==> P|Q" |
|
508 |
apply (unfold or_def) |
|
17589 | 509 |
apply (iprover intro: allI impI mp) |
15411 | 510 |
done |
511 |
||
512 |
lemma disjE: |
|
513 |
assumes major: "P|Q" |
|
514 |
and minorP: "P ==> R" |
|
515 |
and minorQ: "Q ==> R" |
|
516 |
shows "R" |
|
17589 | 517 |
by (iprover intro: minorP minorQ impI |
15411 | 518 |
major [unfolded or_def, THEN spec, THEN mp, THEN mp]) |
519 |
||
520 |
||
20944 | 521 |
subsubsection {*Classical logic*} |
15411 | 522 |
|
523 |
lemma classical: |
|
524 |
assumes prem: "~P ==> P" |
|
525 |
shows "P" |
|
526 |
apply (rule True_or_False [THEN disjE, THEN eqTrueE]) |
|
527 |
apply assumption |
|
528 |
apply (rule notI [THEN prem, THEN eqTrueI]) |
|
529 |
apply (erule subst) |
|
530 |
apply assumption |
|
531 |
done |
|
532 |
||
533 |
lemmas ccontr = FalseE [THEN classical, standard] |
|
534 |
||
535 |
(*notE with premises exchanged; it discharges ~R so that it can be used to |
|
536 |
make elimination rules*) |
|
537 |
lemma rev_notE: |
|
538 |
assumes premp: "P" |
|
539 |
and premnot: "~R ==> ~P" |
|
540 |
shows "R" |
|
541 |
apply (rule ccontr) |
|
542 |
apply (erule notE [OF premnot premp]) |
|
543 |
done |
|
544 |
||
545 |
(*Double negation law*) |
|
546 |
lemma notnotD: "~~P ==> P" |
|
547 |
apply (rule classical) |
|
548 |
apply (erule notE) |
|
549 |
apply assumption |
|
550 |
done |
|
551 |
||
552 |
lemma contrapos_pp: |
|
553 |
assumes p1: "Q" |
|
554 |
and p2: "~P ==> ~Q" |
|
555 |
shows "P" |
|
17589 | 556 |
by (iprover intro: classical p1 p2 notE) |
15411 | 557 |
|
558 |
||
20944 | 559 |
subsubsection {*Unique existence*} |
15411 | 560 |
|
561 |
lemma ex1I: |
|
562 |
assumes prems: "P a" "!!x. P(x) ==> x=a" |
|
563 |
shows "EX! x. P(x)" |
|
17589 | 564 |
by (unfold Ex1_def, iprover intro: prems exI conjI allI impI) |
15411 | 565 |
|
566 |
text{*Sometimes easier to use: the premises have no shared variables. Safe!*} |
|
567 |
lemma ex_ex1I: |
|
568 |
assumes ex_prem: "EX x. P(x)" |
|
569 |
and eq: "!!x y. [| P(x); P(y) |] ==> x=y" |
|
570 |
shows "EX! x. P(x)" |
|
17589 | 571 |
by (iprover intro: ex_prem [THEN exE] ex1I eq) |
15411 | 572 |
|
573 |
lemma ex1E: |
|
574 |
assumes major: "EX! x. P(x)" |
|
575 |
and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R" |
|
576 |
shows "R" |
|
577 |
apply (rule major [unfolded Ex1_def, THEN exE]) |
|
578 |
apply (erule conjE) |
|
17589 | 579 |
apply (iprover intro: minor) |
15411 | 580 |
done |
581 |
||
582 |
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x" |
|
583 |
apply (erule ex1E) |
|
584 |
apply (rule exI) |
|
585 |
apply assumption |
|
586 |
done |
|
587 |
||
588 |
||
20944 | 589 |
subsubsection {*THE: definite description operator*} |
15411 | 590 |
|
591 |
lemma the_equality: |
|
592 |
assumes prema: "P a" |
|
593 |
and premx: "!!x. P x ==> x=a" |
|
594 |
shows "(THE x. P x) = a" |
|
595 |
apply (rule trans [OF _ the_eq_trivial]) |
|
596 |
apply (rule_tac f = "The" in arg_cong) |
|
597 |
apply (rule ext) |
|
598 |
apply (rule iffI) |
|
599 |
apply (erule premx) |
|
600 |
apply (erule ssubst, rule prema) |
|
601 |
done |
|
602 |
||
603 |
lemma theI: |
|
604 |
assumes "P a" and "!!x. P x ==> x=a" |
|
605 |
shows "P (THE x. P x)" |
|
17589 | 606 |
by (iprover intro: prems the_equality [THEN ssubst]) |
15411 | 607 |
|
608 |
lemma theI': "EX! x. P x ==> P (THE x. P x)" |
|
609 |
apply (erule ex1E) |
|
610 |
apply (erule theI) |
|
611 |
apply (erule allE) |
|
612 |
apply (erule mp) |
|
613 |
apply assumption |
|
614 |
done |
|
615 |
||
616 |
(*Easier to apply than theI: only one occurrence of P*) |
|
617 |
lemma theI2: |
|
618 |
assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x" |
|
619 |
shows "Q (THE x. P x)" |
|
17589 | 620 |
by (iprover intro: prems theI) |
15411 | 621 |
|
18697 | 622 |
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a" |
15411 | 623 |
apply (rule the_equality) |
624 |
apply assumption |
|
625 |
apply (erule ex1E) |
|
626 |
apply (erule all_dupE) |
|
627 |
apply (drule mp) |
|
628 |
apply assumption |
|
629 |
apply (erule ssubst) |
|
630 |
apply (erule allE) |
|
631 |
apply (erule mp) |
|
632 |
apply assumption |
|
633 |
done |
|
634 |
||
635 |
lemma the_sym_eq_trivial: "(THE y. x=y) = x" |
|
636 |
apply (rule the_equality) |
|
637 |
apply (rule refl) |
|
638 |
apply (erule sym) |
|
639 |
done |
|
640 |
||
641 |
||
20944 | 642 |
subsubsection {*Classical intro rules for disjunction and existential quantifiers*} |
15411 | 643 |
|
644 |
lemma disjCI: |
|
645 |
assumes "~Q ==> P" shows "P|Q" |
|
646 |
apply (rule classical) |
|
17589 | 647 |
apply (iprover intro: prems disjI1 disjI2 notI elim: notE) |
15411 | 648 |
done |
649 |
||
650 |
lemma excluded_middle: "~P | P" |
|
17589 | 651 |
by (iprover intro: disjCI) |
15411 | 652 |
|
20944 | 653 |
text {* |
654 |
case distinction as a natural deduction rule. |
|
655 |
Note that @{term "~P"} is the second case, not the first |
|
656 |
*} |
|
15411 | 657 |
lemma case_split_thm: |
658 |
assumes prem1: "P ==> Q" |
|
659 |
and prem2: "~P ==> Q" |
|
660 |
shows "Q" |
|
661 |
apply (rule excluded_middle [THEN disjE]) |
|
662 |
apply (erule prem2) |
|
663 |
apply (erule prem1) |
|
664 |
done |
|
20944 | 665 |
lemmas case_split = case_split_thm [case_names True False] |
15411 | 666 |
|
667 |
(*Classical implies (-->) elimination. *) |
|
668 |
lemma impCE: |
|
669 |
assumes major: "P-->Q" |
|
670 |
and minor: "~P ==> R" "Q ==> R" |
|
671 |
shows "R" |
|
672 |
apply (rule excluded_middle [of P, THEN disjE]) |
|
17589 | 673 |
apply (iprover intro: minor major [THEN mp])+ |
15411 | 674 |
done |
675 |
||
676 |
(*This version of --> elimination works on Q before P. It works best for |
|
677 |
those cases in which P holds "almost everywhere". Can't install as |
|
678 |
default: would break old proofs.*) |
|
679 |
lemma impCE': |
|
680 |
assumes major: "P-->Q" |
|
681 |
and minor: "Q ==> R" "~P ==> R" |
|
682 |
shows "R" |
|
683 |
apply (rule excluded_middle [of P, THEN disjE]) |
|
17589 | 684 |
apply (iprover intro: minor major [THEN mp])+ |
15411 | 685 |
done |
686 |
||
687 |
(*Classical <-> elimination. *) |
|
688 |
lemma iffCE: |
|
689 |
assumes major: "P=Q" |
|
690 |
and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R" |
|
691 |
shows "R" |
|
692 |
apply (rule major [THEN iffE]) |
|
17589 | 693 |
apply (iprover intro: minor elim: impCE notE) |
15411 | 694 |
done |
695 |
||
696 |
lemma exCI: |
|
697 |
assumes "ALL x. ~P(x) ==> P(a)" |
|
698 |
shows "EX x. P(x)" |
|
699 |
apply (rule ccontr) |
|
17589 | 700 |
apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"]) |
15411 | 701 |
done |
702 |
||
703 |
||
12386 | 704 |
subsubsection {* Intuitionistic Reasoning *} |
705 |
||
706 |
lemma impE': |
|
12937
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parents:
12892
diff
changeset
|
707 |
assumes 1: "P --> Q" |
0c4fd7529467
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diff
changeset
|
708 |
and 2: "Q ==> R" |
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parents:
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diff
changeset
|
709 |
and 3: "P --> Q ==> P" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
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diff
changeset
|
710 |
shows R |
12386 | 711 |
proof - |
712 |
from 3 and 1 have P . |
|
713 |
with 1 have Q by (rule impE) |
|
714 |
with 2 show R . |
|
715 |
qed |
|
716 |
||
717 |
lemma allE': |
|
12937
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wenzelm
parents:
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diff
changeset
|
718 |
assumes 1: "ALL x. P x" |
0c4fd7529467
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wenzelm
parents:
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diff
changeset
|
719 |
and 2: "P x ==> ALL x. P x ==> Q" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
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diff
changeset
|
720 |
shows Q |
12386 | 721 |
proof - |
722 |
from 1 have "P x" by (rule spec) |
|
723 |
from this and 1 show Q by (rule 2) |
|
724 |
qed |
|
725 |
||
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
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diff
changeset
|
726 |
lemma notE': |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
727 |
assumes 1: "~ P" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
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diff
changeset
|
728 |
and 2: "~ P ==> P" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
729 |
shows R |
12386 | 730 |
proof - |
731 |
from 2 and 1 have P . |
|
732 |
with 1 show R by (rule notE) |
|
733 |
qed |
|
734 |
||
15801 | 735 |
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE |
736 |
and [Pure.intro!] = iffI conjI impI TrueI notI allI refl |
|
737 |
and [Pure.elim 2] = allE notE' impE' |
|
738 |
and [Pure.intro] = exI disjI2 disjI1 |
|
12386 | 739 |
|
740 |
lemmas [trans] = trans |
|
741 |
and [sym] = sym not_sym |
|
15801 | 742 |
and [Pure.elim?] = iffD1 iffD2 impE |
11750 | 743 |
|
11438
3d9222b80989
declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents:
11432
diff
changeset
|
744 |
|
11750 | 745 |
subsubsection {* Atomizing meta-level connectives *} |
746 |
||
747 |
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)" |
|
12003 | 748 |
proof |
9488 | 749 |
assume "!!x. P x" |
10383 | 750 |
show "ALL x. P x" by (rule allI) |
9488 | 751 |
next |
752 |
assume "ALL x. P x" |
|
10383 | 753 |
thus "!!x. P x" by (rule allE) |
9488 | 754 |
qed |
755 |
||
11750 | 756 |
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" |
12003 | 757 |
proof |
9488 | 758 |
assume r: "A ==> B" |
10383 | 759 |
show "A --> B" by (rule impI) (rule r) |
9488 | 760 |
next |
761 |
assume "A --> B" and A |
|
10383 | 762 |
thus B by (rule mp) |
9488 | 763 |
qed |
764 |
||
14749 | 765 |
lemma atomize_not: "(A ==> False) == Trueprop (~A)" |
766 |
proof |
|
767 |
assume r: "A ==> False" |
|
768 |
show "~A" by (rule notI) (rule r) |
|
769 |
next |
|
770 |
assume "~A" and A |
|
771 |
thus False by (rule notE) |
|
772 |
qed |
|
773 |
||
11750 | 774 |
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" |
12003 | 775 |
proof |
10432
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
776 |
assume "x == y" |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
777 |
show "x = y" by (unfold prems) (rule refl) |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
778 |
next |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
779 |
assume "x = y" |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
780 |
thus "x == y" by (rule eq_reflection) |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
781 |
qed |
3dfbc913d184
added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents:
10383
diff
changeset
|
782 |
|
12023 | 783 |
lemma atomize_conj [atomize]: |
19121 | 784 |
includes meta_conjunction_syntax |
785 |
shows "(A && B) == Trueprop (A & B)" |
|
12003 | 786 |
proof |
19121 | 787 |
assume conj: "A && B" |
788 |
show "A & B" |
|
789 |
proof (rule conjI) |
|
790 |
from conj show A by (rule conjunctionD1) |
|
791 |
from conj show B by (rule conjunctionD2) |
|
792 |
qed |
|
11953 | 793 |
next |
19121 | 794 |
assume conj: "A & B" |
795 |
show "A && B" |
|
796 |
proof - |
|
797 |
from conj show A .. |
|
798 |
from conj show B .. |
|
11953 | 799 |
qed |
800 |
qed |
|
801 |
||
12386 | 802 |
lemmas [symmetric, rulify] = atomize_all atomize_imp |
18832 | 803 |
and [symmetric, defn] = atomize_all atomize_imp atomize_eq |
12386 | 804 |
|
11750 | 805 |
|
20944 | 806 |
subsection {* Package setup *} |
807 |
||
11750 | 808 |
subsubsection {* Classical Reasoner setup *} |
9529 | 809 |
|
20944 | 810 |
lemma thin_refl: |
811 |
"\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" . |
|
812 |
||
21151 | 813 |
ML {* |
814 |
structure Hypsubst = HypsubstFun( |
|
815 |
struct |
|
816 |
structure Simplifier = Simplifier |
|
21218 | 817 |
val dest_eq = HOLogic.dest_eq |
21151 | 818 |
val dest_Trueprop = HOLogic.dest_Trueprop |
819 |
val dest_imp = HOLogic.dest_imp |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
820 |
val eq_reflection = thm "HOL.eq_reflection" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
821 |
val rev_eq_reflection = thm "HOL.def_imp_eq" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
822 |
val imp_intr = thm "HOL.impI" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
823 |
val rev_mp = thm "HOL.rev_mp" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
824 |
val subst = thm "HOL.subst" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
825 |
val sym = thm "HOL.sym" |
21151 | 826 |
val thin_refl = thm "thin_refl"; |
827 |
end); |
|
21671 | 828 |
open Hypsubst; |
21151 | 829 |
|
830 |
structure Classical = ClassicalFun( |
|
831 |
struct |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
832 |
val mp = thm "HOL.mp" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
833 |
val not_elim = thm "HOL.notE" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
834 |
val classical = thm "HOL.classical" |
21151 | 835 |
val sizef = Drule.size_of_thm |
836 |
val hyp_subst_tacs = [Hypsubst.hyp_subst_tac] |
|
837 |
end); |
|
838 |
||
839 |
structure BasicClassical: BASIC_CLASSICAL = Classical; |
|
21671 | 840 |
open BasicClassical; |
21151 | 841 |
*} |
842 |
||
21009 | 843 |
setup {* |
844 |
let |
|
845 |
(*prevent substitution on bool*) |
|
846 |
fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso |
|
847 |
Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false) |
|
848 |
(nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm; |
|
849 |
in |
|
21151 | 850 |
Hypsubst.hypsubst_setup |
851 |
#> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac) |
|
852 |
#> Classical.setup |
|
853 |
#> ResAtpset.setup |
|
21009 | 854 |
end |
855 |
*} |
|
856 |
||
857 |
declare iffI [intro!] |
|
858 |
and notI [intro!] |
|
859 |
and impI [intro!] |
|
860 |
and disjCI [intro!] |
|
861 |
and conjI [intro!] |
|
862 |
and TrueI [intro!] |
|
863 |
and refl [intro!] |
|
864 |
||
865 |
declare iffCE [elim!] |
|
866 |
and FalseE [elim!] |
|
867 |
and impCE [elim!] |
|
868 |
and disjE [elim!] |
|
869 |
and conjE [elim!] |
|
870 |
and conjE [elim!] |
|
871 |
||
872 |
declare ex_ex1I [intro!] |
|
873 |
and allI [intro!] |
|
874 |
and the_equality [intro] |
|
875 |
and exI [intro] |
|
876 |
||
877 |
declare exE [elim!] |
|
878 |
allE [elim] |
|
879 |
||
880 |
ML {* |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
881 |
val HOL_cs = Classical.claset_of (the_context ()); |
21009 | 882 |
*} |
19162 | 883 |
|
20223 | 884 |
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P" |
885 |
apply (erule swap) |
|
886 |
apply (erule (1) meta_mp) |
|
887 |
done |
|
10383 | 888 |
|
18689
a50587cd8414
prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents:
18595
diff
changeset
|
889 |
declare ex_ex1I [rule del, intro! 2] |
a50587cd8414
prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents:
18595
diff
changeset
|
890 |
and ex1I [intro] |
a50587cd8414
prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents:
18595
diff
changeset
|
891 |
|
12386 | 892 |
lemmas [intro?] = ext |
893 |
and [elim?] = ex1_implies_ex |
|
11977 | 894 |
|
20944 | 895 |
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*) |
20973 | 896 |
lemma alt_ex1E [elim!]: |
20944 | 897 |
assumes major: "\<exists>!x. P x" |
898 |
and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R" |
|
899 |
shows R |
|
900 |
apply (rule ex1E [OF major]) |
|
901 |
apply (rule prem) |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
902 |
apply (tactic {* ares_tac [thm "allI"] 1 *})+ |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
903 |
apply (tactic {* etac (Classical.dup_elim (thm "allE")) 1 *}) |
20944 | 904 |
by iprover |
905 |
||
21151 | 906 |
ML {* |
907 |
structure Blast = BlastFun( |
|
908 |
struct |
|
909 |
type claset = Classical.claset |
|
910 |
val equality_name = "op =" |
|
911 |
val not_name = "Not" |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
912 |
val notE = thm "HOL.notE" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
913 |
val ccontr = thm "HOL.ccontr" |
21151 | 914 |
val contr_tac = Classical.contr_tac |
915 |
val dup_intr = Classical.dup_intr |
|
916 |
val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac |
|
21671 | 917 |
val claset = Classical.claset |
21151 | 918 |
val rep_cs = Classical.rep_cs |
919 |
val cla_modifiers = Classical.cla_modifiers |
|
920 |
val cla_meth' = Classical.cla_meth' |
|
921 |
end); |
|
21671 | 922 |
val Blast_tac = Blast.Blast_tac; |
923 |
val blast_tac = Blast.blast_tac; |
|
20944 | 924 |
*} |
925 |
||
21151 | 926 |
setup Blast.setup |
927 |
||
20944 | 928 |
|
929 |
subsubsection {* Simplifier *} |
|
12281 | 930 |
|
931 |
lemma eta_contract_eq: "(%s. f s) = f" .. |
|
932 |
||
933 |
lemma simp_thms: |
|
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
934 |
shows not_not: "(~ ~ P) = P" |
15354 | 935 |
and Not_eq_iff: "((~P) = (~Q)) = (P = Q)" |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
936 |
and |
12436
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
berghofe
parents:
12386
diff
changeset
|
937 |
"(P ~= Q) = (P = (~Q))" |
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
berghofe
parents:
12386
diff
changeset
|
938 |
"(P | ~P) = True" "(~P | P) = True" |
12281 | 939 |
"(x = x) = True" |
20944 | 940 |
and not_True_eq_False: "(\<not> True) = False" |
941 |
and not_False_eq_True: "(\<not> False) = True" |
|
942 |
and |
|
12436
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
berghofe
parents:
12386
diff
changeset
|
943 |
"(~P) ~= P" "P ~= (~P)" |
20944 | 944 |
"(True=P) = P" |
945 |
and eq_True: "(P = True) = P" |
|
946 |
and "(False=P) = (~P)" |
|
947 |
and eq_False: "(P = False) = (\<not> P)" |
|
948 |
and |
|
12281 | 949 |
"(True --> P) = P" "(False --> P) = True" |
950 |
"(P --> True) = True" "(P --> P) = True" |
|
951 |
"(P --> False) = (~P)" "(P --> ~P) = (~P)" |
|
952 |
"(P & True) = P" "(True & P) = P" |
|
953 |
"(P & False) = False" "(False & P) = False" |
|
954 |
"(P & P) = P" "(P & (P & Q)) = (P & Q)" |
|
955 |
"(P & ~P) = False" "(~P & P) = False" |
|
956 |
"(P | True) = True" "(True | P) = True" |
|
957 |
"(P | False) = P" "(False | P) = P" |
|
12436
a2df07fefed7
Replaced several occurrences of "blast" by "rules".
berghofe
parents:
12386
diff
changeset
|
958 |
"(P | P) = P" "(P | (P | Q)) = (P | Q)" and |
12281 | 959 |
"(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x" |
960 |
-- {* needed for the one-point-rule quantifier simplification procs *} |
|
961 |
-- {* essential for termination!! *} and |
|
962 |
"!!P. (EX x. x=t & P(x)) = P(t)" |
|
963 |
"!!P. (EX x. t=x & P(x)) = P(t)" |
|
964 |
"!!P. (ALL x. x=t --> P(x)) = P(t)" |
|
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
965 |
"!!P. (ALL x. t=x --> P(x)) = P(t)" |
17589 | 966 |
by (blast, blast, blast, blast, blast, iprover+) |
13421 | 967 |
|
14201 | 968 |
lemma disj_absorb: "(A | A) = A" |
969 |
by blast |
|
970 |
||
971 |
lemma disj_left_absorb: "(A | (A | B)) = (A | B)" |
|
972 |
by blast |
|
973 |
||
974 |
lemma conj_absorb: "(A & A) = A" |
|
975 |
by blast |
|
976 |
||
977 |
lemma conj_left_absorb: "(A & (A & B)) = (A & B)" |
|
978 |
by blast |
|
979 |
||
12281 | 980 |
lemma eq_ac: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
981 |
shows eq_commute: "(a=b) = (b=a)" |
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
982 |
and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" |
17589 | 983 |
and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+) |
984 |
lemma neq_commute: "(a~=b) = (b~=a)" by iprover |
|
12281 | 985 |
|
986 |
lemma conj_comms: |
|
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
987 |
shows conj_commute: "(P&Q) = (Q&P)" |
17589 | 988 |
and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+ |
989 |
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover |
|
12281 | 990 |
|
19174 | 991 |
lemmas conj_ac = conj_commute conj_left_commute conj_assoc |
992 |
||
12281 | 993 |
lemma disj_comms: |
12937
0c4fd7529467
clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents:
12892
diff
changeset
|
994 |
shows disj_commute: "(P|Q) = (Q|P)" |
17589 | 995 |
and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+ |
996 |
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover |
|
12281 | 997 |
|
19174 | 998 |
lemmas disj_ac = disj_commute disj_left_commute disj_assoc |
999 |
||
17589 | 1000 |
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover |
1001 |
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover |
|
12281 | 1002 |
|
17589 | 1003 |
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover |
1004 |
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover |
|
12281 | 1005 |
|
17589 | 1006 |
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover |
1007 |
lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover |
|
1008 |
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover |
|
12281 | 1009 |
|
1010 |
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *} |
|
1011 |
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast |
|
1012 |
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast |
|
1013 |
||
1014 |
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast |
|
1015 |
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast |
|
1016 |
||
21151 | 1017 |
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))" |
1018 |
by iprover |
|
1019 |
||
17589 | 1020 |
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover |
12281 | 1021 |
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast |
1022 |
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast |
|
1023 |
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast |
|
1024 |
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast |
|
1025 |
lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *} |
|
1026 |
by blast |
|
1027 |
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast |
|
1028 |
||
17589 | 1029 |
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover |
12281 | 1030 |
|
1031 |
||
1032 |
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q" |
|
1033 |
-- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *} |
|
1034 |
-- {* cases boil down to the same thing. *} |
|
1035 |
by blast |
|
1036 |
||
1037 |
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast |
|
1038 |
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast |
|
17589 | 1039 |
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover |
1040 |
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover |
|
12281 | 1041 |
|
17589 | 1042 |
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover |
1043 |
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover |
|
12281 | 1044 |
|
1045 |
text {* |
|
1046 |
\medskip The @{text "&"} congruence rule: not included by default! |
|
1047 |
May slow rewrite proofs down by as much as 50\% *} |
|
1048 |
||
1049 |
lemma conj_cong: |
|
1050 |
"(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))" |
|
17589 | 1051 |
by iprover |
12281 | 1052 |
|
1053 |
lemma rev_conj_cong: |
|
1054 |
"(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))" |
|
17589 | 1055 |
by iprover |
12281 | 1056 |
|
1057 |
text {* The @{text "|"} congruence rule: not included by default! *} |
|
1058 |
||
1059 |
lemma disj_cong: |
|
1060 |
"(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))" |
|
1061 |
by blast |
|
1062 |
||
1063 |
||
1064 |
text {* \medskip if-then-else rules *} |
|
1065 |
||
1066 |
lemma if_True: "(if True then x else y) = x" |
|
1067 |
by (unfold if_def) blast |
|
1068 |
||
1069 |
lemma if_False: "(if False then x else y) = y" |
|
1070 |
by (unfold if_def) blast |
|
1071 |
||
1072 |
lemma if_P: "P ==> (if P then x else y) = x" |
|
1073 |
by (unfold if_def) blast |
|
1074 |
||
1075 |
lemma if_not_P: "~P ==> (if P then x else y) = y" |
|
1076 |
by (unfold if_def) blast |
|
1077 |
||
1078 |
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" |
|
1079 |
apply (rule case_split [of Q]) |
|
15481 | 1080 |
apply (simplesubst if_P) |
1081 |
prefer 3 apply (simplesubst if_not_P, blast+) |
|
12281 | 1082 |
done |
1083 |
||
1084 |
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))" |
|
15481 | 1085 |
by (simplesubst split_if, blast) |
12281 | 1086 |
|
1087 |
lemmas if_splits = split_if split_if_asm |
|
1088 |
||
1089 |
lemma if_cancel: "(if c then x else x) = x" |
|
15481 | 1090 |
by (simplesubst split_if, blast) |
12281 | 1091 |
|
1092 |
lemma if_eq_cancel: "(if x = y then y else x) = x" |
|
15481 | 1093 |
by (simplesubst split_if, blast) |
12281 | 1094 |
|
1095 |
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))" |
|
19796 | 1096 |
-- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *} |
12281 | 1097 |
by (rule split_if) |
1098 |
||
1099 |
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))" |
|
19796 | 1100 |
-- {* And this form is useful for expanding @{text "if"}s on the LEFT. *} |
15481 | 1101 |
apply (simplesubst split_if, blast) |
12281 | 1102 |
done |
1103 |
||
17589 | 1104 |
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover |
1105 |
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover |
|
12281 | 1106 |
|
15423 | 1107 |
text {* \medskip let rules for simproc *} |
1108 |
||
1109 |
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g" |
|
1110 |
by (unfold Let_def) |
|
1111 |
||
1112 |
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g" |
|
1113 |
by (unfold Let_def) |
|
1114 |
||
16633
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1115 |
text {* |
16999 | 1116 |
The following copy of the implication operator is useful for |
1117 |
fine-tuning congruence rules. It instructs the simplifier to simplify |
|
1118 |
its premise. |
|
16633
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1119 |
*} |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1120 |
|
17197 | 1121 |
constdefs |
1122 |
simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1) |
|
1123 |
"simp_implies \<equiv> op ==>" |
|
16633
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1124 |
|
18457 | 1125 |
lemma simp_impliesI: |
16633
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1126 |
assumes PQ: "(PROP P \<Longrightarrow> PROP Q)" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1127 |
shows "PROP P =simp=> PROP Q" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1128 |
apply (unfold simp_implies_def) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1129 |
apply (rule PQ) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1130 |
apply assumption |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1131 |
done |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1132 |
|
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1133 |
lemma simp_impliesE: |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1134 |
assumes PQ:"PROP P =simp=> PROP Q" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1135 |
and P: "PROP P" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1136 |
and QR: "PROP Q \<Longrightarrow> PROP R" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1137 |
shows "PROP R" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1138 |
apply (rule QR) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1139 |
apply (rule PQ [unfolded simp_implies_def]) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1140 |
apply (rule P) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1141 |
done |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1142 |
|
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1143 |
lemma simp_implies_cong: |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1144 |
assumes PP' :"PROP P == PROP P'" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1145 |
and P'QQ': "PROP P' ==> (PROP Q == PROP Q')" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1146 |
shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1147 |
proof (unfold simp_implies_def, rule equal_intr_rule) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1148 |
assume PQ: "PROP P \<Longrightarrow> PROP Q" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1149 |
and P': "PROP P'" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1150 |
from PP' [symmetric] and P' have "PROP P" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1151 |
by (rule equal_elim_rule1) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1152 |
hence "PROP Q" by (rule PQ) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1153 |
with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1154 |
next |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1155 |
assume P'Q': "PROP P' \<Longrightarrow> PROP Q'" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1156 |
and P: "PROP P" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1157 |
from PP' and P have P': "PROP P'" by (rule equal_elim_rule1) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1158 |
hence "PROP Q'" by (rule P'Q') |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1159 |
with P'QQ' [OF P', symmetric] show "PROP Q" |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1160 |
by (rule equal_elim_rule1) |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1161 |
qed |
208ebc9311f2
Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents:
16587
diff
changeset
|
1162 |
|
20944 | 1163 |
lemma uncurry: |
1164 |
assumes "P \<longrightarrow> Q \<longrightarrow> R" |
|
1165 |
shows "P \<and> Q \<longrightarrow> R" |
|
1166 |
using prems by blast |
|
1167 |
||
1168 |
lemma iff_allI: |
|
1169 |
assumes "\<And>x. P x = Q x" |
|
1170 |
shows "(\<forall>x. P x) = (\<forall>x. Q x)" |
|
1171 |
using prems by blast |
|
1172 |
||
1173 |
lemma iff_exI: |
|
1174 |
assumes "\<And>x. P x = Q x" |
|
1175 |
shows "(\<exists>x. P x) = (\<exists>x. Q x)" |
|
1176 |
using prems by blast |
|
1177 |
||
1178 |
lemma all_comm: |
|
1179 |
"(\<forall>x y. P x y) = (\<forall>y x. P x y)" |
|
1180 |
by blast |
|
1181 |
||
1182 |
lemma ex_comm: |
|
1183 |
"(\<exists>x y. P x y) = (\<exists>y x. P x y)" |
|
1184 |
by blast |
|
1185 |
||
9869 | 1186 |
use "simpdata.ML" |
21671 | 1187 |
ML {* open Simpdata *} |
1188 |
||
21151 | 1189 |
setup {* |
1190 |
Simplifier.method_setup Splitter.split_modifiers |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1191 |
#> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy)) |
21151 | 1192 |
#> Splitter.setup |
1193 |
#> Clasimp.setup |
|
1194 |
#> EqSubst.setup |
|
1195 |
*} |
|
1196 |
||
1197 |
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P" |
|
1198 |
proof |
|
1199 |
assume prem: "True \<Longrightarrow> PROP P" |
|
1200 |
from prem [OF TrueI] show "PROP P" . |
|
1201 |
next |
|
1202 |
assume "PROP P" |
|
1203 |
show "PROP P" . |
|
1204 |
qed |
|
1205 |
||
1206 |
lemma ex_simps: |
|
1207 |
"!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)" |
|
1208 |
"!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))" |
|
1209 |
"!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)" |
|
1210 |
"!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))" |
|
1211 |
"!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)" |
|
1212 |
"!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))" |
|
1213 |
-- {* Miniscoping: pushing in existential quantifiers. *} |
|
1214 |
by (iprover | blast)+ |
|
1215 |
||
1216 |
lemma all_simps: |
|
1217 |
"!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)" |
|
1218 |
"!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))" |
|
1219 |
"!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)" |
|
1220 |
"!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))" |
|
1221 |
"!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)" |
|
1222 |
"!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))" |
|
1223 |
-- {* Miniscoping: pushing in universal quantifiers. *} |
|
1224 |
by (iprover | blast)+ |
|
15481 | 1225 |
|
21671 | 1226 |
lemmas [simp] = |
1227 |
triv_forall_equality (*prunes params*) |
|
1228 |
True_implies_equals (*prune asms `True'*) |
|
1229 |
if_True |
|
1230 |
if_False |
|
1231 |
if_cancel |
|
1232 |
if_eq_cancel |
|
1233 |
imp_disjL |
|
20973 | 1234 |
(*In general it seems wrong to add distributive laws by default: they |
1235 |
might cause exponential blow-up. But imp_disjL has been in for a while |
|
1236 |
and cannot be removed without affecting existing proofs. Moreover, |
|
1237 |
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the |
|
1238 |
grounds that it allows simplification of R in the two cases.*) |
|
21671 | 1239 |
conj_assoc |
1240 |
disj_assoc |
|
1241 |
de_Morgan_conj |
|
1242 |
de_Morgan_disj |
|
1243 |
imp_disj1 |
|
1244 |
imp_disj2 |
|
1245 |
not_imp |
|
1246 |
disj_not1 |
|
1247 |
not_all |
|
1248 |
not_ex |
|
1249 |
cases_simp |
|
1250 |
the_eq_trivial |
|
1251 |
the_sym_eq_trivial |
|
1252 |
ex_simps |
|
1253 |
all_simps |
|
1254 |
simp_thms |
|
1255 |
||
1256 |
lemmas [cong] = imp_cong simp_implies_cong |
|
1257 |
lemmas [split] = split_if |
|
20973 | 1258 |
|
1259 |
ML {* |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1260 |
val HOL_ss = Simplifier.simpset_of (the_context ()); |
20973 | 1261 |
*} |
1262 |
||
20944 | 1263 |
text {* Simplifies x assuming c and y assuming ~c *} |
1264 |
lemma if_cong: |
|
1265 |
assumes "b = c" |
|
1266 |
and "c \<Longrightarrow> x = u" |
|
1267 |
and "\<not> c \<Longrightarrow> y = v" |
|
1268 |
shows "(if b then x else y) = (if c then u else v)" |
|
1269 |
unfolding if_def using prems by simp |
|
1270 |
||
1271 |
text {* Prevents simplification of x and y: |
|
1272 |
faster and allows the execution of functional programs. *} |
|
1273 |
lemma if_weak_cong [cong]: |
|
1274 |
assumes "b = c" |
|
1275 |
shows "(if b then x else y) = (if c then x else y)" |
|
1276 |
using prems by (rule arg_cong) |
|
1277 |
||
1278 |
text {* Prevents simplification of t: much faster *} |
|
1279 |
lemma let_weak_cong: |
|
1280 |
assumes "a = b" |
|
1281 |
shows "(let x = a in t x) = (let x = b in t x)" |
|
1282 |
using prems by (rule arg_cong) |
|
1283 |
||
1284 |
text {* To tidy up the result of a simproc. Only the RHS will be simplified. *} |
|
1285 |
lemma eq_cong2: |
|
1286 |
assumes "u = u'" |
|
1287 |
shows "(t \<equiv> u) \<equiv> (t \<equiv> u')" |
|
1288 |
using prems by simp |
|
1289 |
||
1290 |
lemma if_distrib: |
|
1291 |
"f (if c then x else y) = (if c then f x else f y)" |
|
1292 |
by simp |
|
1293 |
||
1294 |
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand |
|
21502 | 1295 |
side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *} |
20944 | 1296 |
lemma restrict_to_left: |
1297 |
assumes "x = y" |
|
1298 |
shows "(x = z) = (y = z)" |
|
1299 |
using prems by simp |
|
1300 |
||
17459 | 1301 |
|
20944 | 1302 |
subsubsection {* Generic cases and induction *} |
17459 | 1303 |
|
20944 | 1304 |
text {* Rule projections: *} |
18887 | 1305 |
|
20944 | 1306 |
ML {* |
1307 |
structure ProjectRule = ProjectRuleFun |
|
1308 |
(struct |
|
1309 |
val conjunct1 = thm "conjunct1"; |
|
1310 |
val conjunct2 = thm "conjunct2"; |
|
1311 |
val mp = thm "mp"; |
|
1312 |
end) |
|
17459 | 1313 |
*} |
1314 |
||
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1315 |
constdefs |
18457 | 1316 |
induct_forall where "induct_forall P == \<forall>x. P x" |
1317 |
induct_implies where "induct_implies A B == A \<longrightarrow> B" |
|
1318 |
induct_equal where "induct_equal x y == x = y" |
|
1319 |
induct_conj where "induct_conj A B == A \<and> B" |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1320 |
|
11989 | 1321 |
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))" |
18457 | 1322 |
by (unfold atomize_all induct_forall_def) |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1323 |
|
11989 | 1324 |
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)" |
18457 | 1325 |
by (unfold atomize_imp induct_implies_def) |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1326 |
|
11989 | 1327 |
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)" |
18457 | 1328 |
by (unfold atomize_eq induct_equal_def) |
1329 |
||
1330 |
lemma induct_conj_eq: |
|
1331 |
includes meta_conjunction_syntax |
|
1332 |
shows "(A && B) == Trueprop (induct_conj A B)" |
|
1333 |
by (unfold atomize_conj induct_conj_def) |
|
1334 |
||
1335 |
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq |
|
1336 |
lemmas induct_rulify [symmetric, standard] = induct_atomize |
|
1337 |
lemmas induct_rulify_fallback = |
|
1338 |
induct_forall_def induct_implies_def induct_equal_def induct_conj_def |
|
1339 |
||
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1340 |
|
11989 | 1341 |
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) = |
1342 |
induct_conj (induct_forall A) (induct_forall B)" |
|
17589 | 1343 |
by (unfold induct_forall_def induct_conj_def) iprover |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1344 |
|
11989 | 1345 |
lemma induct_implies_conj: "induct_implies C (induct_conj A B) = |
1346 |
induct_conj (induct_implies C A) (induct_implies C B)" |
|
17589 | 1347 |
by (unfold induct_implies_def induct_conj_def) iprover |
11989 | 1348 |
|
13598
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1349 |
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)" |
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1350 |
proof |
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1351 |
assume r: "induct_conj A B ==> PROP C" and A B |
18457 | 1352 |
show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`) |
13598
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1353 |
next |
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1354 |
assume r: "A ==> B ==> PROP C" and "induct_conj A B" |
18457 | 1355 |
show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def]) |
13598
8bc77b17f59f
Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents:
13596
diff
changeset
|
1356 |
qed |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1357 |
|
11989 | 1358 |
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry |
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1359 |
|
11989 | 1360 |
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
|
1361 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1362 |
text {* Method setup. *} |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1363 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1364 |
ML {* |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1365 |
structure InductMethod = InductMethodFun |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1366 |
(struct |
15411 | 1367 |
val cases_default = thm "case_split" |
1368 |
val atomize = thms "induct_atomize" |
|
18457 | 1369 |
val rulify = thms "induct_rulify" |
1370 |
val rulify_fallback = thms "induct_rulify_fallback" |
|
11824
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1371 |
end); |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1372 |
*} |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1373 |
|
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1374 |
setup InductMethod.setup |
f4c1882dde2c
setup generic cases and induction (from Inductive.thy);
wenzelm
parents:
11770
diff
changeset
|
1375 |
|
18457 | 1376 |
|
20944 | 1377 |
|
1378 |
subsection {* Other simple lemmas and lemma duplicates *} |
|
1379 |
||
1380 |
lemmas eq_sym_conv = eq_commute |
|
1381 |
lemmas if_def2 = if_bool_eq_conj |
|
1382 |
||
1383 |
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x" |
|
1384 |
by blast+ |
|
1385 |
||
1386 |
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))" |
|
1387 |
apply (rule iffI) |
|
1388 |
apply (rule_tac a = "%x. THE y. P x y" in ex1I) |
|
1389 |
apply (fast dest!: theI') |
|
1390 |
apply (fast intro: ext the1_equality [symmetric]) |
|
1391 |
apply (erule ex1E) |
|
1392 |
apply (rule allI) |
|
1393 |
apply (rule ex1I) |
|
1394 |
apply (erule spec) |
|
1395 |
apply (erule_tac x = "%z. if z = x then y else f z" in allE) |
|
1396 |
apply (erule impE) |
|
1397 |
apply (rule allI) |
|
1398 |
apply (rule_tac P = "xa = x" in case_split_thm) |
|
1399 |
apply (drule_tac [3] x = x in fun_cong, simp_all) |
|
1400 |
done |
|
1401 |
||
1402 |
lemma mk_left_commute: |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1403 |
fixes f (infix "\<otimes>" 60) |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1404 |
assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1405 |
c: "\<And>x y. x \<otimes> y = y \<otimes> x" |
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21524
diff
changeset
|
1406 |
shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
20944 | 1407 |
by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]]) |
1408 |
||
21671 | 1409 |
|
1410 |
subsection {* Basic ML bindings *} |
|
1411 |
||
1412 |
ML {* |
|
1413 |
val FalseE = thm "FalseE" |
|
1414 |
val Let_def = thm "Let_def" |
|
1415 |
val TrueI = thm "TrueI"; |
|
1416 |
val allE = thm "allE"; |
|
1417 |
val allI = thm "allI"; |
|
1418 |
val all_dupE = thm "all_dupE" |
|
1419 |
val arg_cong = thm "arg_cong"; |
|
1420 |
val box_equals = thm "box_equals" |
|
1421 |
val ccontr = thm "ccontr"; |
|
1422 |
val classical = thm "classical"; |
|
1423 |
val conjE = thm "conjE"; |
|
1424 |
val conjI = thm "conjI"; |
|
1425 |
val conjunct1 = thm "conjunct1"; |
|
1426 |
val conjunct2 = thm "conjunct2"; |
|
1427 |
val disjCI = thm "disjCI"; |
|
1428 |
val disjE = thm "disjE"; |
|
1429 |
val disjI1 = thm "disjI1" |
|
1430 |
val disjI2 = thm "disjI2" |
|
1431 |
val eq_reflection = thm "eq_reflection"; |
|
1432 |
val ex1E = thm "ex1E" |
|
1433 |
val ex1I = thm "ex1I" |
|
1434 |
val ex1_implies_ex = thm "ex1_implies_ex" |
|
1435 |
val exE = thm "exE"; |
|
1436 |
val exI = thm "exI"; |
|
1437 |
val excluded_middle = thm "excluded_middle" |
|
1438 |
val ext = thm "ext" |
|
1439 |
val fun_cong = thm "fun_cong" |
|
1440 |
val iffD1 = thm "iffD1"; |
|
1441 |
val iffD2 = thm "iffD2"; |
|
1442 |
val iffI = thm "iffI"; |
|
1443 |
val impE = thm "impE" |
|
1444 |
val impI = thm "impI"; |
|
1445 |
val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq"; |
|
1446 |
val mp = thm "mp"; |
|
1447 |
val notE = thm "notE"; |
|
1448 |
val notI = thm "notI"; |
|
1449 |
val not_all = thm "not_all"; |
|
1450 |
val not_ex = thm "not_ex"; |
|
1451 |
val not_iff = thm "not_iff"; |
|
1452 |
val not_not = thm "not_not"; |
|
1453 |
val not_sym = thm "not_sym" |
|
1454 |
val refl = thm "refl"; |
|
1455 |
val rev_mp = thm "rev_mp" |
|
1456 |
val spec = thm "spec"; |
|
1457 |
val ssubst = thm "ssubst" |
|
1458 |
val subst = thm "subst"; |
|
1459 |
val sym = thm "sym"; |
|
1460 |
val trans = thm "trans"; |
|
1461 |
*} |
|
1462 |
||
1463 |
||
1464 |
subsection {* Legacy tactics *} |
|
1465 |
||
1466 |
ML {* |
|
1467 |
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i); |
|
1468 |
||
1469 |
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *) |
|
1470 |
local |
|
1471 |
fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t |
|
1472 |
| wrong_prem (Bound _) = true |
|
1473 |
| wrong_prem _ = false; |
|
1474 |
val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of); |
|
1475 |
val spec = thm "spec" |
|
1476 |
val mp = thm "mp" |
|
1477 |
in |
|
1478 |
fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]); |
|
1479 |
fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]; |
|
1480 |
end; |
|
1481 |
*} |
|
1482 |
||
14357 | 1483 |
end |