src/HOL/IMP/Collecting_Examples.thy

generalized and simplified proofs of several theorems about convex sets

theory Collecting_Examples
imports Collecting Vars
begin

subsection "Pretty printing state sets"

text{* Tweak code generation to work with sets of non-equality types: *}
declare insert_code[code del] union_coset_filter[code del]
lemma insert_code [code]:  "insert x (set xs) = set (x#xs)"
by simp

text{* Compensate for the fact that sets may now have duplicates: *}
definition compact :: "'a set \<Rightarrow> 'a set" where
"compact X = X"

lemma [code]: "compact(set xs) = set(remdups xs)"
by(simp add: compact_def)

definition "vars_acom = compact o vars o strip"

text{* In order to display commands annotated with state sets, states must be
translated into a printable format as sets of variable-state pairs, for the
variables in the command: *}

definition show_acom :: "state set acom \<Rightarrow> (vname*val)set set acom" where
"show_acom C =
   annotate (\<lambda>p. (\<lambda>s. (\<lambda>x. (x, s x)) ` (vars_acom C)) ` anno C p) (strip C)"


subsection "Examples"

definition "c0 = WHILE Less (V ''x'') (N 3)
                DO ''x'' ::= Plus (V ''x'') (N 2)"
definition C0 :: "state set acom" where "C0 = annotate (%p. {}) c0"

text{* Collecting semantics: *}

value "show_acom (((step {<>}) ^^ 1) C0)"
value "show_acom (((step {<>}) ^^ 2) C0)"
value "show_acom (((step {<>}) ^^ 3) C0)"
value "show_acom (((step {<>}) ^^ 4) C0)"
value "show_acom (((step {<>}) ^^ 5) C0)"
value "show_acom (((step {<>}) ^^ 6) C0)"
value "show_acom (((step {<>}) ^^ 7) C0)"
value "show_acom (((step {<>}) ^^ 8) C0)"

text{* Small-step semantics: *}
value "show_acom (((step {}) ^^ 0) (step {<>} C0))"
value "show_acom (((step {}) ^^ 1) (step {<>} C0))"
value "show_acom (((step {}) ^^ 2) (step {<>} C0))"
value "show_acom (((step {}) ^^ 3) (step {<>} C0))"
value "show_acom (((step {}) ^^ 4) (step {<>} C0))"
value "show_acom (((step {}) ^^ 5) (step {<>} C0))"
value "show_acom (((step {}) ^^ 6) (step {<>} C0))"
value "show_acom (((step {}) ^^ 7) (step {<>} C0))"
value "show_acom (((step {}) ^^ 8) (step {<>} C0))"

end