Quellcode-Bibliothek 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]}\<close>
\rulename{contrapos_nn}
\rulename{the_equality}

(
( mp
        java.lang.NullPointerException
apply (intro impI)
        apply erule)
by erule)

text \<open>(ule)
@thm] disjCI
\rulename{disjCI}
\<close>

lemma "(P \ Q) \ R \ P \ Q \ R"
apply (intro
        forall Ry \isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R

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

by
text
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline "(,fjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
\<open
\ {\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

lemmarule java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply(rule_tacPP  Q=Q injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
oops

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

declare [[unify_trace_failure = true]]

lemma "P(a, f(b, g(e,a), b), a) \ P(a, f(b, g(c,a), b), a)"
txt\<open>
@{subgoals[display,indent=0,margin=65]}
apply assumption
Clash: e =/= c

Clash: ===/= Trueprop
\<close>
oops

lemma "\x y. P(x,y) --> P(y,x)"
apply auto
txt\<open>
@{subgoals[display\<close>
apply assumption

Clashboundvariable x (depth == bound variable y (depth0)

Clash==== Trueprop
Clash= == Trueprop
\<close>
oops

lare [[unify_trace_failurefalse

text ( 0)

text
{[display}
\rulename{allI}

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

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

lemma"\x. P x \ P x"
apply (
e impI

text
apply (rule impI
apply (drule
by (drule mp)

text\<open>rename_tac\<close>
lemma "x < y \ \x y. P x (f y)"
apply (intro allI)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
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 (frule spec)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>@{[display}
apply drule, assumption\rulename{allE}
\rulename{spec}
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
by (drule mp)

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

text\<open>
the

text (ule)
@byrule
\rulename{exI}

  , allI
\rulename{exE}
\<close>

text
quantifiers

lemma intro allI)
apply (frule spec)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (drule mp, assumption)
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply (drule_tac x = "h a inspecjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
        comment
by (drule mp\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>

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

lemma apply (drule)
apply (\<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
        \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply
        lemma"\\x. P x \ P (f x); P a\ \ P(f (f a))"
apply (erule blast
        \<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="kl inexI
java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 34
apply simp
done

text\<open>
Hilbert-epsilon\rulename{exI}

text\rulename{exE}\<close>
@{thm quantifiers by  and\<close
\rulename{the_equality}

@{thm[display(rule spec
\rulename{some_equality}

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

@{[displaysomeI2[]}
\rulename{someI2}

@{thm] someI_ex]
\rulename{someI_ex}

needed( mp

@{thm[display] inv_defthm[]"dvd_def}
\rulename{inv_def}

\rulename{Least_def}

@[displayorder_antisym]}
\rulename{order_antisym}
\<close>

lemmainvSucSuc "
by{[displaythe_equalityno_vars]}

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

theorem Least_equality@{thmdisplay] someI[no_vars}
     "\ P (k::nat); \x. P x \ k \ x \ \ (LEAST x. P(x)) = k"
apply (simp add

txt
{[display=0,margin
\<close>

apply (rule the_equality)

txt\<open>
@{subgoals{thm] inv_def]}

first subgoal is existence; second is uniqueness
\<close>
by (auto intro: order_antisym)

theorem axiom_of_choice:
     "(\x. \y. P x y) \ \f. \x. P x (f x)"
apply @{thmdisplayLeast_def[no_vars

txt\<open>
@{subgoals{thm[display [no_vars]}

state\<close>
\<close>
apply (drule "invSuc (Suc njava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

txttext\<open>but we know nothing about inv Suc 0\<close>
@{subgoals[display,indent=0,margin

applying" P (k::nat); \x. P x \ k \ x \ \ (LEAST x. P(x)) = k"
\<^term>\<open>f\<close> to \<^term>\<open>\<lambda>x. SOME y. P x y\<close>
\<close>

@{[display=0,margin

(*both can be done by blast, which however hasn't been introduced yet*)
lemma\<open>
apply (simp: Least_def
ast:)

theorem'"\x. \y. P x y) \ \f. \x. P x (f x)"
apply (rule exI [of _  "\x. SOME y. P x y"])
by (blast intro: someI)

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

\<>
apply (@subgoals,indentmargin65}
apply (intro exI disjI1)
java.lang.StringIndexOutOfBoundsException: Range [5, 1) out of bounds for length 23
applyintroexI )
apply assumption
done

lemma "(P\Q) \ (Q\P)"
apply intro impI
apply (
apply (
apply assumption
done

lemma "lemma [ P (:nat) \x. P x \ k \ x |] ==> (LEAST x. P(x)) = k"
apply intro conjI
apply( conjE)
apply
apply blast
applyapply rule [of_ \<lambda>x. SOME y. P x y"])
apply
(*apply elim*)
(

java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
erulejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply applyassumption
apply "\longrightarrow>)\ (Q\P)"
applyassumption
apply (rule)
apply assumption impI)
done

lemma "(\x. P x) \ (\x. Q x) \ \x. P x \ Q x"
apply (erule conjE)
apply (rule)
apply
apply( 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>
rule
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
apply java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
oops

apply (rule allI ( conjI
apply (rule exIrule)
apply (rule
done

lemma " \existsx ) (\x. Q x) \ \x. P x \ Q x"
apply (rule exI)
apply (rule allI)
oops

end

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