7 chapter {* Isabelle/Isar quick reference \label{ap:refcard} *}
9 section {* Proof commands *}
11 subsection {* Primitives and basic syntax *}
15 @{command "fix"}~@{text x} & augment context by @{text "\<And>x. \<box>"} \\
16 @{command "assume"}~@{text "a: \<phi>"} & augment context by @{text "\<phi> \<Longrightarrow> \<box>"} \\
17 @{command "then"} & indicate forward chaining of facts \\
18 @{command "have"}~@{text "a: \<phi>"} & prove local result \\
19 @{command "show"}~@{text "a: \<phi>"} & prove local result, refining some goal \\
20 @{command "using"}~@{text a} & indicate use of additional facts \\
21 @{command "unfolding"}~@{text a} & unfold definitional equations \\
22 @{command "proof"}~@{text "m\<^sub>1"}~\dots~@{command "qed"}~@{text "m\<^sub>2"} & indicate proof structure and refinements \\
23 @{command "{"}~\dots~@{command "}"} & indicate explicit blocks \\
24 @{command "next"} & switch blocks \\
25 @{command "note"}~@{text "a = b"} & reconsider facts \\
26 @{command "let"}~@{text "p = t"} & abbreviate terms by higher-order matching \\
29 \begin{matharray}{rcl}
30 @{text "theory\<dash>stmt"} & = & @{command "theorem"}~@{text "name: prop proof"} \Or @{command "definition"}~\dots \Or \dots \\[1ex]
31 @{text "proof"} & = & @{text "prfx\<^sup>*"}~@{command "proof"}~@{text "method stmt\<^sup>*"}~@{command "qed"}~@{text method} \\
32 & \Or & @{text "prfx\<^sup>*"}~@{command "done"} \\[1ex]
33 @{text prfx} & = & @{command "apply"}~@{text method} \\
34 & \Or & @{command "using"}~@{text "name\<^sup>+"} \\
35 & \Or & @{command "unfolding"}~@{text "name\<^sup>+"} \\
36 @{text stmt} & = & @{command "{"}~@{text "stmt\<^sup>*"}~@{command "}"} \\
37 & \Or & @{command "next"} \\
38 & \Or & @{command "note"}~@{text "name = name\<^sup>+"} \\
39 & \Or & @{command "let"}~@{text "term = term"} \\
40 & \Or & @{command "fix"}~@{text "var\<^sup>+"} \\
41 & \Or & @{command "assume"}~@{text "name: prop\<^sup>+"} \\
42 & \Or & @{command "then"}@{text "\<^sup>?"}~@{text goal} \\
43 @{text goal} & = & @{command "have"}~@{text "name: prop\<^sup>+ proof"} \\
44 & \Or & @{command "show"}~@{text "name: prop\<^sup>+ proof"} \\
49 subsection {* Abbreviations and synonyms *}
52 \begin{matharray}{rcl}
53 @{command "by"}~@{text "m\<^sub>1 m\<^sub>2"} & \equiv & @{command "proof"}~@{text "m\<^sub>1"}~@{command "qed"}~@{text "m\<^sub>2"} \\
54 @{command ".."} & \equiv & @{command "by"}~@{text rule} \\
55 @{command "."} & \equiv & @{command "by"}~@{text this} \\
56 @{command "hence"} & \equiv & @{command "then"}~@{command "have"} \\
57 @{command "thus"} & \equiv & @{command "then"}~@{command "show"} \\
58 @{command "from"}~@{text a} & \equiv & @{command "note"}~@{text a}~@{command "then"} \\
59 @{command "with"}~@{text a} & \equiv & @{command "from"}~@{text "a \<AND> this"} \\[1ex]
60 @{command "from"}~@{text this} & \equiv & @{command "then"} \\
61 @{command "from"}~@{text this}~@{command "have"} & \equiv & @{command "hence"} \\
62 @{command "from"}~@{text this}~@{command "show"} & \equiv & @{command "thus"} \\
67 subsection {* Derived elements *}
70 \begin{matharray}{rcl}
71 @{command "also"}@{text "\<^sub>0"} & \approx & @{command "note"}~@{text "calculation = this"} \\
72 @{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \approx & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\
73 @{command "finally"} & \approx & @{command "also"}~@{command "from"}~@{text calculation} \\[0.5ex]
74 @{command "moreover"} & \approx & @{command "note"}~@{text "calculation = calculation this"} \\
75 @{command "ultimately"} & \approx & @{command "moreover"}~@{command "from"}~@{text calculation} \\[0.5ex]
76 @{command "presume"}~@{text "a: \<phi>"} & \approx & @{command "assume"}~@{text "a: \<phi>"} \\
77 @{command "def"}~@{text "a: x \<equiv> t"} & \approx & @{command "fix"}~@{text x}~@{command "assume"}~@{text "a: x \<equiv> t"} \\
78 @{command "obtain"}~@{text "x \<WHERE> a: \<phi>"} & \approx & \dots~@{command "fix"}~@{text x}~@{command "assume"}~@{text "a: \<phi>"} \\
79 @{command "case"}~@{text c} & \approx & @{command "fix"}~@{text x}~@{command "assume"}~@{text "c: \<phi>"} \\
80 @{command "sorry"} & \approx & @{command "by"}~@{text cheating} \\
85 subsection {* Diagnostic commands *}
89 @{command "pr"} & print current state \\
90 @{command "thm"}~@{text a} & print fact \\
91 @{command "term"}~@{text t} & print term \\
92 @{command "prop"}~@{text \<phi>} & print meta-level proposition \\
93 @{command "typ"}~@{text \<tau>} & print meta-level type \\
98 section {* Proof methods *}
102 \multicolumn{2}{l}{\textbf{Single steps (forward-chaining facts)}} \\[0.5ex]
103 @{method assumption} & apply some assumption \\
104 @{method this} & apply current facts \\
105 @{method rule}~@{text a} & apply some rule \\
106 @{method rule} & apply standard rule (default for @{command "proof"}) \\
107 @{method contradiction} & apply @{text "\<not>"} elimination rule (any order) \\
108 @{method cases}~@{text t} & case analysis (provides cases) \\
109 @{method induct}~@{text x} & proof by induction (provides cases) \\[2ex]
111 \multicolumn{2}{l}{\textbf{Repeated steps (inserting facts)}} \\[0.5ex]
112 @{method "-"} & no rules \\
113 @{method intro}~@{text a} & introduction rules \\
114 @{method intro_classes} & class introduction rules \\
115 @{method elim}~@{text a} & elimination rules \\
116 @{method unfold}~@{text a} & definitional rewrite rules \\[2ex]
118 \multicolumn{2}{l}{\textbf{Automated proof tools (inserting facts)}} \\[0.5ex]
119 @{method iprover} & intuitionistic proof search \\
120 @{method blast}, @{method fast} & Classical Reasoner \\
121 @{method simp}, @{method simp_all} & Simplifier (+ Splitter) \\
122 @{method auto}, @{method force} & Simplifier + Classical Reasoner \\
123 @{method arith} & Arithmetic procedures \\
128 section {* Attributes *}
132 \multicolumn{2}{l}{\textbf{Operations}} \\[0.5ex]
133 @{attribute OF}~@{text a} & rule resolved with facts (skipping ``@{text _}'') \\
134 @{attribute of}~@{text t} & rule instantiated with terms (skipping ``@{text _}'') \\
135 @{attribute "where"}~@{text "x = t"} & rule instantiated with terms, by variable name \\
136 @{attribute symmetric} & resolution with symmetry rule \\
137 @{attribute THEN}~@{text b} & resolution with another rule \\
138 @{attribute rule_format} & result put into standard rule format \\
139 @{attribute elim_format} & destruct rule turned into elimination rule format \\[1ex]
141 \multicolumn{2}{l}{\textbf{Declarations}} \\[0.5ex]
142 @{attribute simp} & Simplifier rule \\
143 @{attribute intro}, @{attribute elim}, @{attribute dest} & Pure or Classical Reasoner rule \\
144 @{attribute iff} & Simplifier + Classical Reasoner rule \\
145 @{attribute split} & case split rule \\
146 @{attribute trans} & transitivity rule \\
147 @{attribute sym} & symmetry rule \\
152 section {* Rule declarations and methods *}
155 \begin{tabular}{l|lllll}
156 & @{method rule} & @{method iprover} & @{method blast} & @{method simp} & @{method auto} \\
157 & & & @{method fast} & @{method simp_all} & @{method force} \\
159 @{attribute Pure.elim}@{text "!"} @{attribute Pure.intro}@{text "!"}
160 & @{text "\<times>"} & @{text "\<times>"} \\
161 @{attribute Pure.elim} @{attribute Pure.intro}
162 & @{text "\<times>"} & @{text "\<times>"} \\
163 @{attribute elim}@{text "!"} @{attribute intro}@{text "!"}
164 & @{text "\<times>"} & & @{text "\<times>"} & & @{text "\<times>"} \\
165 @{attribute elim} @{attribute intro}
166 & @{text "\<times>"} & & @{text "\<times>"} & & @{text "\<times>"} \\
168 & @{text "\<times>"} & & @{text "\<times>"} & @{text "\<times>"} & @{text "\<times>"} \\
169 @{attribute iff}@{text "?"}
170 & @{text "\<times>"} \\
171 @{attribute elim}@{text "?"} @{attribute intro}@{text "?"}
172 & @{text "\<times>"} \\
174 & & & & @{text "\<times>"} & @{text "\<times>"} \\
176 & & & & @{text "\<times>"} & @{text "\<times>"} \\
178 & & & & @{text "\<times>"} & @{text "\<times>"} \\
183 section {* Emulating tactic scripts *}
185 subsection {* Commands *}
189 @{command "apply"}~@{text m} & apply proof method at initial position \\
190 @{command "apply_end"}~@{text m} & apply proof method near terminal position \\
191 @{command "done"} & complete proof \\
192 @{command "defer"}~@{text n} & move subgoal to end \\
193 @{command "prefer"}~@{text n} & move subgoal to beginning \\
194 @{command "back"} & backtrack last command \\
199 subsection {* Methods *}
203 @{method rule_tac}~@{text insts} & resolution (with instantiation) \\
204 @{method erule_tac}~@{text insts} & elim-resolution (with instantiation) \\
205 @{method drule_tac}~@{text insts} & destruct-resolution (with instantiation) \\
206 @{method frule_tac}~@{text insts} & forward-resolution (with instantiation) \\
207 @{method cut_tac}~@{text insts} & insert facts (with instantiation) \\
208 @{method thin_tac}~@{text \<phi>} & delete assumptions \\
209 @{method subgoal_tac}~@{text \<phi>} & new claims \\
210 @{method rename_tac}~@{text x} & rename innermost goal parameters \\
211 @{method rotate_tac}~@{text n} & rotate assumptions of goal \\
212 @{method tactic}~@{text "text"} & arbitrary ML tactic \\
213 @{method case_tac}~@{text t} & exhaustion (datatypes) \\
214 @{method induct_tac}~@{text x} & induction (datatypes) \\
215 @{method ind_cases}~@{text t} & exhaustion + simplification (inductive predicates) \\