(*:maxLineLen=78:*)
theory Quick_Reference
imports Base Main
begin
chapter \<open>Isabelle/Isar quick reference \label{ap:refcard}\<close>
section \<open>Proof commands\<close>
subsection \<open>Primitives and basic syntax\<close>
text \<open>
\begin{tabular}{ll}
@{command "fix"}~\<open>x\<close> & augment context by \<open>\<And>x. \<box>\<close> \\
@{command "assume"}~\<open>a: \<phi>\<close> & augment context by \<open>\<phi> \<Longrightarrow> \<box>\<close> \\
@{command "then"} & indicate forward chaining of facts \\
@{command "have"}~\<open>a: \<phi>\<close> & prove local result \\
@{command "show"}~\<open>a: \<phi>\<close> & prove local result, refining some goal \\
@{command "using"}~\<open>a\<close> & indicate use of additional facts \\
@{command "unfolding"}~\<open>a\<close> & unfold definitional equations \\
@{command "proof"}~\<open>m\<^sub>1\<close>~\dots~@{command "qed"}~\<open>m\<^sub>2\<close> & indicate proof structure and refinements \\
@{command "{"}~\<open>\<dots>\<close>~@{command "}"} & indicate explicit blocks \\
@{command "next"} & switch blocks \\
@{command "note"}~\<open>a = b\<close> & reconsider facts \\
@{command "let"}~\<open>p = t\<close> & abbreviate terms by higher-order matching \\
@{command "write"}~\<open>c (mx)\<close> & declare local mixfix syntax \\
\end{tabular}
\<^medskip>
\begin{tabular}{rcl}
\<open>proof\<close> & = & \<open>prfx\<^sup>*\<close>~@{command "proof"}~\<open>method\<^sup>? stmt\<^sup>*\<close>~@{command "qed"}~\<open>method\<^sup>?\<close> \\
& \<open>|\<close> & \<open>prfx\<^sup>*\<close>~@{command "done"} \\
\<open>prfx\<close> & = & @{command "apply"}~\<open>method\<close> \\
& \<open>|\<close> & @{command "using"}~\<open>facts\<close> \\
& \<open>|\<close> & @{command "unfolding"}~\<open>facts\<close> \\
\<open>stmt\<close> & = & @{command "{"}~\<open>stmt\<^sup>*\<close>~@{command "}"} \\
& \<open>|\<close> & @{command "next"} \\
& \<open>|\<close> & @{command "note"}~\<open>name = facts\<close> \\
& \<open>|\<close> & @{command "let"}~\<open>term = term\<close> \\
& \<open>|\<close> & @{command "write"}~\<open>name (mixfix)\<close> \\
& \<open>|\<close> & @{command "fix"}~\<open>var\<^sup>+\<close> \\
& \<open>|\<close> & @{command "assume"}~\<open>name: props\<close> \\
& \<open>|\<close> & @{command "then"}\<open>\<^sup>?\<close>~\<open>goal\<close> \\
\<open>goal\<close> & = & @{command "have"}~\<open>name: props proof\<close> \\
& \<open>|\<close> & @{command "show"}~\<open>name: props proof\<close> \\
\end{tabular}
\<close>
subsection \<open>Abbreviations and synonyms\<close>
text \<open>
\begin{tabular}{rcl}
@{command "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> & \<open>\<equiv>\<close> &
@{command "proof"}~\<open>m\<^sub>1\<close>~@{command "qed"}~\<open>m\<^sub>2\<close> \\
@{command ".."} & \<open>\<equiv>\<close> & @{command "by"}~\<open>standard\<close> \\
@{command "."} & \<open>\<equiv>\<close> & @{command "by"}~\<open>this\<close> \\
@{command "hence"} & \<open>\<equiv>\<close> & @{command "then"}~@{command "have"} \\
@{command "thus"} & \<open>\<equiv>\<close> & @{command "then"}~@{command "show"} \\
@{command "from"}~\<open>a\<close> & \<open>\<equiv>\<close> & @{command "note"}~\<open>a\<close>~@{command "then"} \\
@{command "with"}~\<open>a\<close> & \<open>\<equiv>\<close> & @{command "from"}~\<open>a \<AND> this\<close> \\
@{command "from"}~\<open>this\<close> & \<open>\<equiv>\<close> & @{command "then"} \\
@{command "from"}~\<open>this\<close>~@{command "have"} & \<open>\<equiv>\<close> & @{command "hence"} \\
@{command "from"}~\<open>this\<close>~@{command "show"} & \<open>\<equiv>\<close> & @{command "thus"} \\
\end{tabular}
\<close>
subsection \<open>Derived elements\<close>
text \<open>
\begin{tabular}{rcl}
@{command "also"}\<open>\<^sub>0\<close> & \<open>\<approx>\<close> &
@{command "note"}~\<open>calculation = this\<close> \\
@{command "also"}\<open>\<^sub>n\<^sub>+\<^sub>1\<close> & \<open>\<approx>\<close> &
@{command "note"}~\<open>calculation = trans [OF calculation this]\<close> \\
@{command "finally"} & \<open>\<approx>\<close> &
@{command "also"}~@{command "from"}~\<open>calculation\<close> \\[0.5ex]
@{command "moreover"} & \<open>\<approx>\<close> &
@{command "note"}~\<open>calculation = calculation this\<close> \\
@{command "ultimately"} & \<open>\<approx>\<close> &
@{command "moreover"}~@{command "from"}~\<open>calculation\<close> \\[0.5ex]
@{command "presume"}~\<open>a: \<phi>\<close> & \<open>\<approx>\<close> &
@{command "assume"}~\<open>a: \<phi>\<close> \\
@{command "def"}~\<open>a: x \<equiv> t\<close> & \<open>\<approx>\<close> &
@{command "fix"}~\<open>x\<close>~@{command "assume"}~\<open>a: x \<equiv> t\<close> \\
@{command "obtain"}~\<open>x \<WHERE> a: \<phi>\<close> & \<open>\<approx>\<close> &
\<open>\<dots>\<close>~@{command "fix"}~\<open>x\<close>~@{command "assume"}~\<open>a: \<phi>\<close> \\
@{command "case"}~\<open>c\<close> & \<open>\<approx>\<close> &
@{command "fix"}~\<open>x\<close>~@{command "assume"}~\<open>c: \<phi>\<close> \\
@{command "sorry"} & \<open>\<approx>\<close> &
@{command "by"}~\<open>cheating\<close> \\
\end{tabular}
\<close>
subsection \<open>Diagnostic commands\<close>
text \<open>
\begin{tabular}{ll}
@{command "print_state"} & print proof state \\
@{command "print_statement"} & print fact in long statement form \\
@{command "thm"}~\<open>a\<close> & print fact \\
@{command "prop"}~\<open>\<phi>\<close> & print proposition \\
@{command "term"}~\<open>t\<close> & print term \\
@{command "typ"}~\<open>\<tau>\<close> & print type \\
\end{tabular}
\<close>
section \<open>Proof methods\<close>
text \<open>
\begin{tabular}{ll}
\multicolumn{2}{l}{\<^bold>\<open>Single steps (forward-chaining facts)\<close>} \\[0.5ex]
@{method assumption} & apply some assumption \\
@{method this} & apply current facts \\
@{method rule}~\<open>a\<close> & apply some rule \\
@{method standard} & apply standard rule (default for @{command "proof"}) \\
@{method contradiction} & apply \<open>\<not>\<close> elimination rule (any order) \\
@{method cases}~\<open>t\<close> & case analysis (provides cases) \\
@{method induct}~\<open>x\<close> & proof by induction (provides cases) \\[2ex]
\multicolumn{2}{l}{\<^bold>\<open>Repeated steps (inserting facts)\<close>} \\[0.5ex]
@{method "-"} & no rules \\
@{method intro}~\<open>a\<close> & introduction rules \\
@{method intro_classes} & class introduction rules \\
@{method elim}~\<open>a\<close> & elimination rules \\
@{method unfold}~\<open>a\<close> & definitional rewrite rules \\[2ex]
\multicolumn{2}{l}{\<^bold>\<open>Automated proof tools (inserting facts)\<close>} \\[0.5ex]
@{method iprover} & intuitionistic proof search \\
@{method blast}, @{method fast} & Classical Reasoner \\
@{method simp}, @{method simp_all} & Simplifier (+ Splitter) \\
@{method auto}, @{method force} & Simplifier + Classical Reasoner \\
@{method arith} & Arithmetic procedures \\
\end{tabular}
\<close>
section \<open>Attributes\<close>
text \<open>
\begin{tabular}{ll}
\multicolumn{2}{l}{\<^bold>\<open>Rules\<close>} \\[0.5ex]
@{attribute OF}~\<open>a\<close> & rule resolved with facts (skipping ``\<open>_\<close>'') \\
@{attribute of}~\<open>t\<close> & rule instantiated with terms (skipping ``\<open>_\<close>'') \\
@{attribute "where"}~\<open>x = t\<close> & rule instantiated with terms, by variable name \\
@{attribute symmetric} & resolution with symmetry rule \\
@{attribute THEN}~\<open>b\<close> & resolution with another rule \\
@{attribute rule_format} & result put into standard rule format \\
@{attribute elim_format} & destruct rule turned into elimination rule format \\[1ex]
\multicolumn{2}{l}{\<^bold>\<open>Declarations\<close>} \\[0.5ex]
@{attribute simp} & Simplifier rule \\
@{attribute intro}, @{attribute elim}, @{attribute dest} & Pure or Classical Reasoner rule \\
@{attribute iff} & Simplifier + Classical Reasoner rule \\
@{attribute split} & case split rule \\
@{attribute trans} & transitivity rule \\
@{attribute sym} & symmetry rule \\
\end{tabular}
\<close>
section \<open>Rule declarations and methods\<close>
text \<open>
\begin{tabular}{l|lllll}
& @{method rule} & @{method iprover} & @{method blast} & @{method simp} & @{method auto} \\
& & & @{method fast} & @{method simp_all} & @{method force} \\
\hline
@{attribute Pure.elim}\<open>!\<close> @{attribute Pure.intro}\<open>!\<close>
& \<open>\<times>\<close> & \<open>\<times>\<close> \\
@{attribute Pure.elim} @{attribute Pure.intro}
& \<open>\<times>\<close> & \<open>\<times>\<close> \\
@{attribute elim}\<open>!\<close> @{attribute intro}\<open>!\<close>
& \<open>\<times>\<close> & & \<open>\<times>\<close> & & \<open>\<times>\<close> \\
@{attribute elim} @{attribute intro}
& \<open>\<times>\<close> & & \<open>\<times>\<close> & & \<open>\<times>\<close> \\
@{attribute iff}
& \<open>\<times>\<close> & & \<open>\<times>\<close> & \<open>\<times>\<close> & \<open>\<times>\<close> \\
@{attribute iff}\<open>?\<close>
& \<open>\<times>\<close> \\
@{attribute elim}\<open>?\<close> @{attribute intro}\<open>?\<close>
& \<open>\<times>\<close> \\
@{attribute simp}
& & & & \<open>\<times>\<close> & \<open>\<times>\<close> \\
@{attribute cong}
& & & & \<open>\<times>\<close> & \<open>\<times>\<close> \\
@{attribute split}
& & & & \<open>\<times>\<close> & \<open>\<times>\<close> \\
\end{tabular}
\<close>
section \<open>Emulating tactic scripts\<close>
subsection \<open>Commands\<close>
text \<open>
\begin{tabular}{ll}
@{command "apply"}~\<open>m\<close> & apply proof method at initial position \\
@{command "apply_end"}~\<open>m\<close> & apply proof method near terminal position \\
@{command "done"} & complete proof \\
@{command "defer"}~\<open>n\<close> & move subgoal to end \\
@{command "prefer"}~\<open>n\<close> & move subgoal to beginning \\
@{command "back"} & backtrack last command \\
\end{tabular}
\<close>
subsection \<open>Methods\<close>
text \<open>
\begin{tabular}{ll}
@{method rule_tac}~\<open>insts\<close> & resolution (with instantiation) \\
@{method erule_tac}~\<open>insts\<close> & elim-resolution (with instantiation) \\
@{method drule_tac}~\<open>insts\<close> & destruct-resolution (with instantiation) \\
@{method frule_tac}~\<open>insts\<close> & forward-resolution (with instantiation) \\
@{method cut_tac}~\<open>insts\<close> & insert facts (with instantiation) \\
@{method thin_tac}~\<open>\<phi>\<close> & delete assumptions \\
@{method subgoal_tac}~\<open>\<phi>\<close> & new claims \\
@{method rename_tac}~\<open>x\<close> & rename innermost goal parameters \\
@{method rotate_tac}~\<open>n\<close> & rotate assumptions of goal \\
@{method tactic}~\<open>text\<close> & arbitrary ML tactic \\
@{method case_tac}~\<open>t\<close> & exhaustion (datatypes) \\
@{method induct_tac}~\<open>x\<close> & induction (datatypes) \\
@{method ind_cases}~\<open>t\<close> & exhaustion + simplification (inductive predicates) \\
\end{tabular}
\<close>
end