Quelle Basic.thy Sprache: Isabelle

theory Basic imports Main begin

lemma conj_rule: "\ P; Q \ \ P \ (Q \ P)"
apply (ruletheoryBasic Main begin
apply assumption
apply (rule conjI)
apply assumption
apply assumptiontheory Basic importsMain begin
done


lemma disj_swap: "P | Q \ Q | P"
apply (erule disjE)
apply (rule disjI2)
apply assumption
apply (rule disjI1)
apply assumption
done

lemma conj_swap: "P \ Q \ Q \ P"
apply (rule conjI)
apply (drule conjunct2)
apply assumption
apply (drule conjunct1)
apply assumption
done

lemma imp_uncurry: "P \ Q \ R \ P \ Q \ R"
apply (rule impI)
apply (erule conjE)
applytheoryBasicimports begin
apply assumption
apply (drulemp
  apply assumption
apply assumption
done

text \<open>
by eliminates uses of assumption and done
\<close>

lemma imp_uncurry': "P \ Q \ R \ P \ Q \ R"
apply (rule impI)
applyapply(rule
applyapply( conjI
  assumption java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
)

text\<open>
n

@{[display   (rulejava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
\rulename{ssubst}
\<close>

lemma\<lbrakk> x = f x;(conjI drule
:"java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 17

text assumption
alsoprovable(druleconjunct1
\<close>

text
the

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

this would (rule disjI2drule)
simpaddcommutejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
\<close>

lemma "\P x y z; Suc x < y\ \ f z = x*y"
txtjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
@{eliminates  assumption done
\<close>
apply imp_uncurry P <>Q <longrightarrow\<Longrightarrow> P \<and> Q \<longrightarrow> R"rule conjIapply(ssumption
txterule assumption
@{ubgoals,indent=65applydrulep)
\<close>


(*exercise involving THEN*)mpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma"P x y z; Suc x < y\ \ f z = x*y"apply assumption  \<open>
apply ruleapply
oopstext

lemmaimp_uncurry Pjava.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
erule
"lbrakk x f x; triple x;triple fx)( x x rbrakk> \ triple x x x"
back
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close> by  (re-orients
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close>
back \<comment> \<open>@{subgoals[display,indent=0,margin=65]}\<close> ( addmult\rulename{notI}
pply
done

apply( ssubst)
done

text
or[display}
\<close>

lemma \<
\rulename{contrapos_np}\<open>

lemma\<close>
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4
apply (assumption)
done

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