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Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/C/Firefox/widget/ (MySQL Server Version 8.1-8.4^©) Datei vom 10.2.2025 mit Größe 9 kB

Quellcode-Bibliothek Basic.thy Sprache: Isabelle

  imports

lemmajava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
java.lang.StringIndexOutOfBoundsException: Range [17, 13) out of bounds for length 18
erulemp
apply (rulethmapply pply disjI2ruleapply java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 17
y
done  bydonedrule)
    ejava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

lemmadisj_swap
apply (\rulename{mult.commute}
apply(rule (drule
apply( : .)
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
applyapply ejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
e

lemmaapply (drule mp{eliminatesof andjava.lang.StringIndexOutOfBoundsException: Range [41, 42) out of bounds for length 41
apply (rule)
  (ssumption
apply assumption
apply apply (drule)
apply
done

lemma{[display=0,margin ( )
apply  applye applyassumptionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (erule
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply
apply \<open>
  apply assumption\<close>
assumption
done

text\<open>
by apply
\<close>

:
apply (rule impI drule)
applyapply
apply ( mpby ( mp
applyassumption
by (drule mp)

text \<open>
substitution

@substitutionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{ssubst}
\<close>

lemma "\ x = f x; P(f x) \ \ P x"
by\<close>

text
lemmatheory Basic Main
\<close>

textlemma ssubst
the method

@{thm[displaymult.commute
\rulename{mult.commute}

this )
apply\<close>
\<close>

a n
app java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
@subgoalsindent=5]
\<close>
apply (subst : "P|Q \ Q | P"
txt\<open>
@{apply eruleapply add.commute
\<close>
oops

(*exercise involving THEN*)\<close>\<close>
\<lbrakk>P x y z; Suc x < y\<rbrakk> \<Longrightarrow> f z = x*y"
applyrule
oops\<close> ( multcommute  )

apply java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
eruleapply ( impI
\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
\<cl
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
back d mp
back assumption java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
assumption imp_uncurry
done (ruleimpIerule)

lemma "\ x = f x; triple (f x) (f x) x \ \ triple x x x"
apply (rulessubst)
done

text mp
or still
\<close>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (rule )

lemma "\ x = f x; triple (f x) (f x) x \ \ triple x x x"
apply (erule_tac P="\u. triple u u x" in ssubst)
apply (assumption)
done\<close>lemmajava.lang.NullPointerException

lemma"<> x= ( f )x\
\<open>

text
negation

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

@\<
\rulename{notE}

@aassumption
\rulename{classical}

@{apply (rule, assumption
\rulename{contrapos_pp}

@thm contrapos_pn
\rulename{contrapos_pn}

@{lemma"lbrakk> x = f x; triple (f x) (f x) x \ \ triple x x x"
\rulename{contrapos_np}

@thm
\rulename{contrapos_nn}
\<close>

applyjava.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 4
\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
apply (ntro impI
java.lang.NullPointerException
by( notE

text   P="\u. triple u u x" in ssubst)
@{thm (rule_tac\longrightarrow
\rulename{disjCI}
\<close>

lemma"(P
apply apply\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
mment

apply (elim disjE
apply assumption
java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

by
text
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\isanewline
goal\<close>
\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline\rulename{classical}
\ {\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>

text\<open>rule_tac, etc.\<close>

{display
applyapplye>
oops

text\<open>unification failure trace\<close>

declare\<open>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
txt\<open>
@{subgoals,indent=65]}
applyassumption\<close> \charsemicolon\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
Clash

Clash (applyrulename  fail
oops
oopsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma [[unify_trace_failure =true
apply auto "P(,fb,gea),b,a)\java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
txt
@{subgoals[display,indent=0applyassumption
apply (elim )

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Clash<>
\<close>
oops

declare [unify_trace_failurefalse]

xt

\<open>
@java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{allI}

@{thm[display] allE\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ Q\isanewline/Trueprop: = /java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
\rulename{allE}

@
\rulename{spec}
\<close>

lemma
rule)
by (rule

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
applylemma
apply (rule{ubgoals=,margin]
by (java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

textopen\<close> rule  allI
lemma byxt
apply (intro allIjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

apply (rename_tac v w)
\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
oops

lemma rename_tac )
rulelemma"a bb,,g(a, b, \java.lang.StringIndexOutOfBoundsException: Range [75, 48) out of bounds for length 75

apply apply assumption
apply (lemma\<lbrakk>\<forall>x. P x \<longrightarrow> P (h x); P a\<rbrakk> \<Longrightarrow> P(h (h a))")
        ( mpassumption
bydrule)

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

text
the existential quantifier\<close>

text \<open>
@{thmdisplay "
\rulename{exI}

apply assumption
\rulename{exE}
\<close>

\<>
instantiating: =ack

lemma"\\x. P x \ P (h x); P a\ \ P(h (h a))"
applyjava.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 39
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>back
apply ( mp)
        {thm] allI
apply (drule_tacerule
        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (drule( (byblast

text \<open>
@thm]"dvd_def"}
\rulename{dvd_def}
\<close>

lemma the existential\<close>
apply
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
@{["exIjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>\<close>
apply (\<open>
               \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 20
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close> ( spec
apply        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

doneapply(rule_tac"ha in spec)

text
Hilbert-epsilonoopsdrule)

text\<open>
@{thm[display{java.lang.StringIndexOutOfBoundsException: Range [9, 0) out of bounds for length 0
\rulename{the_equality}

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

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

@{thmdisplay] someI2[no_vars]}
\rulename{someI2}

@{thm] someI_ex[no_vars]}
\rulename{someI_ex}

needed for examples

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

@thm] Least_def]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
\rulename{Least_def}

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

lemma"invSuc n "
by (simp add: inv_def)

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

theorem Least_equality:
     "\ P (k::nat); \x. P x \ k \ x \ \ (LEAST x. P(x)) = k"
apply (simpbydrule)

txt
@{subgoals \<open>
\<close>

apply (rule\<close>

txt ( add: dvd_def)
@{[displayindent=65]}

first is ; secondis
\<close>
by   java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

theorem applyrule_tac" exI
(forall.java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
apply (rulethm] the_equalityjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

txt\<open>
@{subgoals\rulename{someI}

state rules\rulename{someI2}
\<close>
apply ( spec eruleexE

txtjava.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
@{subgoals[display,indent=0,margin=65]}

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

by@thm] (add

(*both can be done by blast, which however hasn't been introduced yet*)
lemma nat

\rulename{order_antisym}

{ " Suc( n)
  exI"
java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

text( intro)

axiom_of_choice"<>:)
apply (elim apply (simp add)
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
assumption
apply (intro exI disjI2)
apply( exI rule java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 8
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ply disjCI
apply (elim axiom_of_choice
applytxt
apply assumption (ruleexIsubgoals,=,=65]}
one

[,indent
apply
elim)
apply
blast
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
apply rule
(*apply elim*)
done

lemma "(\x. P \ Q x) \ P \ (\x. Q x)"
(ruleexI [of  "lambdax.SOME.P by blastintro )
apply erule)
applylemmaback <> \<open>@{subgoals[display,indent=0,margin=65]}\<close> <comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>apply(applyjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
yblast
( blast(*apply elim*)
apply
done "(

   disjEjava.lang.NullPointerException
erule
apply java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
rulejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
   )
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
thjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

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

lemma "\<forall>x. \<exists>y. x=y"
apply (rule allI)
apply (rule exI)
apply (rule refl)
done

lemma "\<exists>x. \<forall>y. x=y"
apply (rule exI)
apply (rule allI)
oops

end

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