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Quellcode-Bibliothek Basic.thy   Sprache: Isabelle

 
   Main )

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 rule  rule)
 pply apply
apply assumption( conjI assumption
 apply assumption
apply assumption
done
    

lemma disj_swap: "Pjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 16
apply (erule erule (drule
 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 applya (rule)
apply ( )
apply assumption
done

\\<close> " x = f x; P(f x) \ \ P x"
applyrule)
 apply(drule)
 apply assumptionassumption
applyconj_swapPjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 17
apply assumption conjunct2text
donedrule

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Range [22, 17) out of bounds for length 17
 
applydrule usesjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
ssumption
( mp java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
  java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
donejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

text
by eliminates
\<close>

apply(  java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply 
lemma' " Q \ R \ P \ Q \ R"
apply( mp
 applyassumptiondrule)
by(drule)


text  java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17


@{thm[display] \rulename{ssubst}
\rulename{ssubst}
\<close>

   begin
 

   method] mult}
also rule
\<close>

text
assumptio 

@
\rulename{mult.commute}

this {[display=065java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 ( :mult)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
\<close>


lemmaassumption"
txt\<open>
@{subgoals[display,indent)
\<close>
apply(subst.)applydrule)
txtjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
@apply( ssubstjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
\<close>
oops

  java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
lemma "\P x y z; Suc x < y\ \ f z = x*y"
apply (rule mult.commute [THEN ssubst]) 
oops


lemma "\x = f x; triple (f x) (f x) x\ \ triple x x x"
apply[someI2
  \rulename{someI2}
back[display  x"texteruleconjE)
\> 
back\<
back java.lang.StringIndexOutOfBoundsException: Range [70, 22) out of bounds for length 22
 applyapplyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 4

lemma
apply (
done

text
orapplyelim)
\<close>

lemma " x = f x; triple (f x) (f x) x \ \ triple x x x"
by ( ssubst( exE


lemma " (rule conjI)
pplyajava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply(java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 18
done


lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by(java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4


texterulejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
negation impI

@display}
\rulename{notI}

@yrulejava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
\rulename{notE}

@{thm[display
\rulename{classical}

@{thm apply (elim java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{contrapos_pp}

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

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

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


 "java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 0
apply 
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (intro impI
         \<forall>x. \<exists>y. x=y"rule)
java.lang.StringIndexOutOfBoundsException: Range [11, 2) out of bounds for length 15

text
@java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{disjCI}
\<close>

(ruleapply( exI
apply java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
        

apply (elim conjE disjE)
 apply assumption
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

by (erule contrapos_np, rule
text
proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{6}}\isanewline
\isanewline
goal
{\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline
\ {\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>


apply( conjE  < 


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


text\<open>

declare [[unify_trace_failure =\isanewline

lemma"Pa java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
txt<java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4
@{subgoals=0
apply assumption
Clash\<open>rule_tac, etc.\<close>

Clash: == =/= Clash: == =/= Trueprop rule_tac = and=java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
\<close>
oops:  variable 1)/bound )

:  /
apply: =/Trueprop
dec = ]]
@
apply assumption

Clash: bound variable x (depth 1) =/= bound variabley (depth

Clash: == =/= @thm] allI
Clashdisplay
\<close>
\<close "\x. P x \ P x"

declare)


text\<open>Quantifiers\<close>

text\<open>
@{thm] allI
\rulename{allI}

@{thm( mpassumption)
\rulename{allE}

@{thm[display] spec}
\rulename{spec}
\<close>

lemma
apply( allI
 ( impI)

lemma \rulename{exI}
apply (ruleimpI ruleallI)
apply (drule spec\rulename{exE}
by

text\<open>
lemma "instantiating quantifiers explicitly by rule_tac and erule_tac\
apply(introallIjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>" in )
apply (rename_tac v w)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
oops


lemma "\\x. P x \ P (h x); P a\ \ P(h (h a))"
apply (\<> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
        
apply (drulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ( spec
 
by (drule mp

 java.lang.NullPointerException
byjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8


text(rule_tac"*" )
the existential quantifier\<close>

text \<open>
@{thm[display]"exI"} java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
\rulename{exI}

@{
\rulename{exE}
\<close>


text\<open>
ntiating explicitlyrule_tac erule_tac>

lemma "\rulename{the_equality}
apply (rulespec)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>\rulename{some_equality}
apply (\rulename{so
        thm] someI2no_vars
apply (drule_tac[display[no_vars}
        \rulename{someI_ex}
by drule)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
@{thmdisplay"}
\rulename{dvd_def}
\<close>

lemma mult_dvd_mono
apply (simp add: dvd_def)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (erule exE) 
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (erule exE) 
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (rename_tac l)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (rule_tac x="k*l" in exI) 
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
applysimp
done

text\<open>@thmdisplay] order_antisym[no_vars
Hilbert-epsilon " Suc (Suc n) = n"

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

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

@[display]java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
\rulename{someI}

@{thm[display
\rulename{someI2}

@{@{subgoals,indent=65]}
\rulename{someI_ex}

needed

@{[display[no_vars
\rulename{inv_def}

{[] Least_def]}
\rulename{Least_def}

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


lemma n) = n"
by (simp add: inv_def)

java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

theorem
     java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
apply (simp add: Least_def)
 
txt
{subgoals,indent=65]}
\<close>
   
apply (rule the_equality)

txt
@{subgoals add)

firstby (bl intro: order_antisym
\<close>
by (auto axiom_of_choice: (


theoremjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (rule exI, rule

txt<open
{[display=0,margin=6]java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

state after intro rules
\<close>
apply (drule spec, erule exE)

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

applying @text{someI (intro exI disjI2
\<^term>\<open>f\<close> to \<^term>\<open>\<lambda>x. SOME y. P x y\<close>
\<close>

by ( ( disjCI)

(*both can be done by blast, which however hasn't been introduced yet*)
"|k:;\x. P x \ k \ x |] ==> (LEAST x. P(x)) = k"
apply (simp( disjCI)
by  elim disjE

theoremblast
apply ( exIof  "\x. SOME y. P x y"])
by (blast intro: someI)

text


lemma"(\x. P x) \ (\x. Q x) \ \x. P x \ Q x"
applylemma"(x. P \ Q x) \ P \ (\x. Q x)"
 apply (intro exI disjI1apply(erule exE)
 apply assumption
apply (intro exI disjI2)
apply assumption
done

lemma"P
applyapply 
apply (elim exI
apply (intro
applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done( exE

 rule)
applyjava.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 18
applyapply( allI) 
apply blast
apply blast
apply blast
apply blast
(*apply elim*)
done

lemma "(\x. P \ Q x) \ P \ (\x. Q x)"
apply (erule  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply lemma"\x. \y. x=y"
apply (rule)
 apply assumption
apply ( exI
 apply assumption
done

lemma"(<>.Px)\java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
apply (erule conjE
apply (erule exE)
apply (erule exE)
apply (rule exI)
apply (rule conjI)
 apply assumption
oops

lemma "\y. R y y \ \x. \y. R x y"
apply (rule exI) 
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (rule allI) 
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (drule spec) 
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
oops

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

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

end

99%


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