Quelle Basic.thy Sprache: Isabelle

theoryimportsbegin

ule)
apply
applytextthesubstjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
apply(conjI\<close>
pply
ly assumption
done\rulename{mult.commute} wouldfailsubgoals,=,margin]


lemmadisj_swap   \<Longrightarrow> Q | P"
apply( disjEapplysimp mult)
apply\<close>
apply(
apply (rule\<close>
apply
done

lemma\<lbrakk>P x y z; Suc x < y\<rbrakk> \<Longrightarrow> f z = x*y"
( conjIjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
(conjunct2
apply assumption
txt
assumption
done

lemma imp_uncurry: "P \ Q \ R \ P \ Q \ R"
apply rule)
apply   \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
applyjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply assumption
apply(rule)
ply
applyassumption
done

text \<open>
byapply
\<close>

lemma': "P \ Q \ R \ P \ Q \ R"
apply(rule )
apply ( conjE
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
apply assumption (rule , assumption
drule)

text better
substitution

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

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
by erule_tacR\<longrightarrow>Q" in contrapos_np)

text{[displaymargin
also
\<close>

text        }
the\<cljava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

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

thisfail
applyoops
\<close>

text

text
\<close>
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
@[,indent=65}
\<close>
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

(*exercise involving THEN*)
\<>rename_tacimpIlemma,(,ea b,a\<Longrightarrow> P(a, f(b, g(c,a), b), a)"
java.lang.StringIndexOutOfBoundsException: Range [8, 3) out of bounds for length 73
oops

[lemmaforall>x y. P(x,y) --> P(y,x)"
\<open>
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>[java.lang.StringIndexOutOfBoundsException: Range [14, 9) out of bounds for length 14
ack
back
back \<lbrakk>\<forall>x. P x \<longrightarrow> P (h x); P a\<rbrakk> \<Longrightarrow> P(h (h a))"

apply assumption
done

lemma
applyapply  (drule )
done

textdisplay
or\rulename{allI}
\<close>

lemma\rulename{allE}
y( ssubst

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
pplyjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
apply (assumption{[display"java.lang.StringIndexOutOfBoundsException: Range [1, 0) out of bounds for length 0
done

lemma
by (erule_tac( thm]exI

text
negation

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

@{thm yjava.lang.StringIndexOutOfBoundsException: Range [55, 20) out of bounds for length 20
\rulename{notE}

@
\rulename{classical}

@{thm
\rulename{contrapos_pp}

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

@{java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
\rulename{contrapos_np}

@{thm\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\&lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

text
negation

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

@{thm assumption
\rulename{notE}

@{thm[display \<lbrakk> x = f x; triple (f x) (f x) x \<rbrakk> \<Longrightarrow> triple x x x",assumption
\rulename{classical}

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

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

@{thm[display)
\rulename{contrapos_np}

@{thm erule)
\rulename{contrapos_nn}
\<close>

\<lbrakk>\<not>(P\<longrightarrow>Q); \<not>(R\<longrightarrow>Q)\<rbrakk> \<Longrightarrow> R"erule_tac
apply( Q="R<>java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
   \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (intro>\<open>@{subgoals[display,indent=0,margin=65]}\<close> conjE)
                \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
by(erule

text\<open>
@{[display \isanewline
\rulename{disjCI}
\<close>

{isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline
intro )
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>\rulename{contrapos_pp}java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

by
text
proofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\isanewline
goal Q=R
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
\ {\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

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
rulename would:
oops

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

reunify_trace_failure  ]]

lemma "Pa ( (,),) ) P(a, f(b, g(c,a), b), a)"
txt
@{subgoals[\comment \<open>@{subgoals[display,indent=0,margin=65]}\<close>
assumption
elimdisjE

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

lemma
applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
txt
@{\<open>Quantifiers\<close>
apply text

Clash: bound\rulenamejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Clash: == =/
Clash=== Trueprop
\<close>
oops\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R

\rulename{allE}

text\rulename{spec}

textlemma
apply( allI
\rulename{allI}

@
\rulename{allE}

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

lemma
apply(rule{ubgoalsindentmargin5}
oops

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply ( , rule)
apply (drule
byby\<open>unification failure trace\<close>

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

        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply(rename_tac vw
         spec "( (,,(e,) aa)\
oops        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

"
apply (frule spec
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply drule,assumption
apply drule spec
        java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
by blast

\<open>
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8

text\<close>
the quantifier

textjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
thm"exI"}
\rulename{exI}

@ \<open>
\rulename{exE}
\<close>

text
instantiatingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma "\\x. P x \ P (h x); P a\ \ P(h (h a))"
apply (frule)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply rulenamejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
( x = " "in
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
by ( mpjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23

text \<open>
rulename
\rulename{dvd_def}
\<close>

lemma mult_dvd_mono: "\i dvd m; j dvd n\ \ i*j dvd (m*n :: nat)"
apply
        java.lang.StringIndexOutOfBoundsException: Range [0, 68) out of bounds for length 57
apply by
\<open>  \<close>
(erule\rulename{exI}
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply
        <comment
apply (rule_tac=k*l" in exI)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
applyimp
done

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

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

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

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

@{thm[display] someI2[no_vars]}
\rulename{someI2}

@{thm[display] someI_exjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{someI_ex}

for

@thmdisplay [no_vars
\rulename{inv_def}

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

@apply(drule, erule )
\rulename{order_antisym}
\<close>

lemma " Suc (uc =n"
byimp: inv_def)

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

theorem:
     \<^term>\<open>f\<close> to \<^term>\<open>\<lambda>x. SOME y. P x y\<close>
apply simp)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
txt\<open>
@{subgoals[display "[| P(k::); \x. P x \ k \ x |] ==> (LEAST x. P(x)) = k"
\<close>

by

txt
@emmaSuc)= java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

first subgoal is java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<close>
byauto

theorem     java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 121
     java.lang.StringIndexOutOfBoundsException: Range [6, 2) out of bounds for length 17
rule,ruleallI

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
{[,indent

state rules\<close>)
\<close>
apply

\<open>
@display0margin

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

(conjE

@{subgoals[display,indent=0,margin=65]}
lemma "[| P (k::nat);  \<forall>x. P x \<longrightarrow> k \<le> x |] ==> (LEAST x. P(x)) = k"
apply (simp add: Least_def)
by (blast intro: order_antisym)

theorem axiom_of_choice': "(\<forall>x. \<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. P x (f x)"
apply (rule exI [of _  "\<lambda>x. SOME y. P x y"])
by (blast intro: someI)

text\<open>end of Epsilon section\<close>

lemma "(\<exists>x. P x) \<or> (\<exists>x. Q x) \<Longrightarrow> \<exists>x. P x \<or> Q x"
apply (elim exE disjE)
apply (intro exI disjI1)
apply assumption
apply (intro exI disjI2)
apply assumption
done

lemma "(P\<longrightarrow>Q) \<or> (Q\<longrightarrow>P)"
apply (intro disjCI impI)
apply (elim notE)
apply (intro impI)
apply assumption
done

lemma "(P\<or>Q)\<and>(P\<or>R) \<Longrightarrow> P \<or> (Q\<and>R)"
apply (intro disjCI conjI)
apply (elim conjE disjE)
apply blast
apply blast
apply blast
apply blast
(*apply elim*)
a

lemmaemma
apply java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 4
apply (eruleeruleapply erule)
apply
  (applyssumption
apply(ruleapply assumption
applyassumption
done

lemma apply(rule (java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply(conjE
apply(
(intro)
apply @thm] }
y ( conjI
apply assumption \rulename
oops\rulename{classical}

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\rulename{contrapos_pp}java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply
\java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
\rulename{contrapos_nn}
lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
oopsapply \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

lemma "\x. \y. x=y"
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
apply ( exI
apply (otE)
donejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma e>
apply ( exI rule)
(java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
oops

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