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products/sources/formale Sprachen/Isabelle/Doc/Tutorial/Rules/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 10 kB

Quellcode-Bibliothek Basic.thy Sprache: Isabelle

java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

)
(rule
y
apply (rule)
applyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply assumption
done


lemma disj_swap: "P | Q \ Q | P"
apply( bydrulejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
pplyrule)
apply disjI1assumptionjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  conjI( conjunct2
apply
done

  java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 17
apply (rule
apply (drule)
apply  apply assumptiony assumption
y ( conjunct1
apply assumption\<close>
e

lemma
apply(ule)
apply erule : "P |java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 17
apply mpsimpmult
apply assumption
apply (drule mp)
  apply assumption>
apply assumption
done

texteyjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
eliminates of and
\<close>

lemma':"\longrightarrow Q\ R \ P \ Q \ R"
apply ( (java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply (erulejava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
applym)
e  assumption
by (drule )

text \<open>
substitution

@{thm java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
done
\<close>

" x = f x; P(f x) \ \ P x"
by (apply(erule java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

text
also provable simpre-orients)
\<close>

text\<open>
the \<open>

@{gation
\rulename{mult.commute}

this
apply(imp: mult
\<close>

lemma@ java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
txt
@goals{[]java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
\<close>
apply (
txt
@{[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<close>

(*exercise involving THEN*)
lemma "\P x y z; Suc x < y\ \ f z = x*y"
apply
oops

lemma "\x = f x; triple (f x) (f x) x\ \ triple x x x"
apply erule)
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close> \<open>
back
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply assumption
done\rulename{notE}

lemma"
apply (erule ssubst )
done

text\<open>
or better still\rulename{classical}java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<close>

lemma "\ x = f x; triple (f x) (f x) x \ \ triple x x x"
by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma "\ x = f x; triple (f x) (f x) x \ \ triple x x x"
\apply (ntroimpI
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done\<close>

lemma"\\(P\Q); \(R\Q)\ \ R"
by(java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 53

\rulename{disjCI}\<close> ( \<or> Q) \<and> R \<Longrightarrow> P \<or> Q \<and> R"java.lang.StringIndexOutOfBoundsException: Range [73, 9) out of bounds for length 73
negation

erule
\rulename{notI}

@\<open>
\rulename{notE}

@{thmthm]java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
\rulename{classical}

@{thm[display]apply( disjCIconjI
\rulename{contrapos_pp}

@{thm[display] contrapos_pn}
\rulename{contrapos_pn}

@{thm\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
\rulename{contrapos_np}

@hm]java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
\rulename{contrapos_nn}
\<close>

lemma "\\(P\Q); \(R\Q)\ \ R"
apply(erule_tac="\

apply (intro impI \<open>{subgoals[display,indent=0,margin=65]}[display=0,margin
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
by

text
@
\rulename{disjCI}
\<close>

lemma"P\java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
apply (\<open>
        <>\java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 10

apply( conjE
apply


by (erule contrapos_np
text
\lemma
\isanewline
goal[[ = false]java.lang.StringIndexOutOfBoundsException: Range [38, 39) out of bounds for length 10
{java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ Q\isanewline
\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
\<close>

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 17
\\<
oops( apply (rule allIdisplay0margin]

java.lang.NullPointerException

declare"java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

(bga)
txt\<open>
@yjava.lang.NullPointerException
apply
Clashapply ,)

(drule )
\<close>
oops

lemmajava.lang.NullPointerException
apply auto
txt
@{subgoalsrulename{exI}
applyassumption

\<close>

textopen
Clash= \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
\<close>
oops

declare [[unify_trace_failure = false]]

text \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

text (drule,assumption
@{display}
\rulename{allI}

@{
\rulename{allE}

@ erule)
\rulename{spec}
\<close>

lemma "\x. P x \ P x"
apply pplyjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
byjava.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 11

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply rule[displayjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
apply
@{

text\java.lang.NullPointerException
lemma "x < y lemma "x < y \ \x y. P x (f y)"
applyfrule

({java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
@(java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 35
oops \rulename{contrapos_pn}

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply\
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply
apply drule
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
bydrule)

lemma "\\x. P x \ P (f x); P a\ \ P(f (f a))"
by blast

text
theexistentialquantifier\<close>

text \<open>
@{pplyerule exE)
\rulename{exI}

@{
\rulename{exE}
\<close>

text\<> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
instantiating

lemma "java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
apply
\java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
apply (\rulename{order_antisym}
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close> "inv Suc (Suc n)=njava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
apply (drule_tac x = "h a" in
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
( mp

text
@{thm
\rulename{dvd_def}
\<close>

lemma
apply simpdvd_def
        subgoals,=0,margin
apply (erule subgoalisexistence  uniqueness
        byneededexamples
apply {thm[display] inv_def]}
        \rulename{inv_def}
apply (rename_tac
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (rule_tac x="k*l"in)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
     "\x y. P x y) \ \f. \x. P x (f x)"
done

text\<open>
Hilbert-epsilon theorems\<close>

text\<open>
@{[display[no_vars]}
\rulename{the_equality}

@{thm[display] some_equality[no_vars]}
\rulename{some_equality}

@{thm[display
\rulename{someI}

@{thmstate after intro
\rulename{someI2}

@java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{someI_ex}

needed inv ( n) n"

@{by (s addinv_def
\rulename{inv_def}

{[displayLeast_def : Least_def
\rulename{Least_def}

@{thm[\<close>
\rulename{order_antisym}
\<close>

emmainv (uc ="
by (simpapply(rule [of _  \<lambda>x. SOME y. P x y"])

text\<open>but we know nothing about inv Suc 0\<close>

theorem\<close>text auto: order_antisym
"lbrakk P (k::nat; \x. P x \ k \ x \ \ (LEAST x. P(x)) = k"
: Least_def

txt\<open>
@{subgoals[display      apply java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
\<close>

apply (

txt\<open>
@{subgoals[display,indentdone@subgoalsdisplay=0,margin=6

firststate after intro
\<close>
by (intro  impI

theorem:
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
apply (rule exI{[displayindentmargin] @text}

txt\<open>
@{subgoalsdisplay=0,margin=65]}

state\<close>
\<close>
apply(  disjE

txt\<open>
@{apply

applying @textapply blast
\<^term>\<open>f\<close> to \<^term>\<open>\<lambda>x. SOME y. P x y\<close>
\<close>

by( someI

(*both can be done by blast, which however hasn't been introduced yet*)
apply exI _  "<>x y Pxy( :someI
apply( conjE
lastbackcommentjava.lang.NullPointerException

    assumptionapply (apply assumptiondone
applyapply (elimapply done
by (blast intro: someIapplyapplyyblastapplyblastapplyapply(applyblast(*apply elim*)

"

text\<open>
(exEjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assumption
rule exI)
apply assumption assumptionjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
done

lemma "(P\Q) \ (Q\P)"
apply( eruleonjE
(assumption
applyintro)
apply[] notI
done{()

lemma"rulename{notE}
applydisjCI
apply(apply
apply blast
apply blast
applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply blast  <comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
(*apply elim*)
done  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

lemma "(\x. P \ Q x) \ P \ (\x. Q x)"
apply  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply \lemma"\x. \y. x=y"
apply (ruleapply (rule
apply assumptionapply one{java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 3
done\<open>

\\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ Q\isanewline
apply( exE
apply (erule( notE
apply ( exI
apply (rule{[displaydisjCI}
apply assumptionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
oops

<>y. R \{\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\comment
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply spec
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
oops

lemma "\x. \y. x=y"
apply (rule allI)
apply (rule exI)
apply (rule refl)
done

lemma "\x. \y. x=y"
apply (rule[display,indent\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
apply (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [22, 4) out of bounds for length 4

end

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