(* Author: Florian Haftmann, TU Muenchen *)

header {* A simple cookbook example how to eliminate choice in programs. *}

theory Execute_Choice

imports Main "~~/src/HOL/Library/AList_Mapping"

begin

text {*

A trivial example:

*}

definition valuesum :: "('a, 'b :: ab_group_add) mapping => 'b" where

"valuesum m = (∑k ∈ Mapping.keys m. the (Mapping.lookup m k))"

text {*

Not that instead of defining @{term valuesum} with choice, we define it

directly and derive a description involving choice afterwards:

*}

lemma valuesum_rec:

assumes fin: "finite (dom (Mapping.lookup m))"

shows "valuesum m = (if Mapping.is_empty m then 0 else

let l = (SOME l. l ∈ Mapping.keys m) in the (Mapping.lookup m l) + valuesum (Mapping.delete l m))"

proof (cases "Mapping.is_empty m")

case True then show ?thesis by (simp add: is_empty_def keys_def valuesum_def)

next

case False

then have l: "∃l. l ∈ dom (Mapping.lookup m)" unfolding is_empty_def by transfer auto

then have "(let l = SOME l. l ∈ dom (Mapping.lookup m) in

the (Mapping.lookup m l) + (∑k ∈ dom (Mapping.lookup m) - {l}. the (Mapping.lookup m k))) =

(∑k ∈ dom (Mapping.lookup m). the (Mapping.lookup m k))"

proof (rule someI2_ex)

fix l

note fin

moreover assume "l ∈ dom (Mapping.lookup m)"

moreover obtain A where "A = dom (Mapping.lookup m) - {l}" by simp

ultimately have "dom (Mapping.lookup m) = insert l A" and "finite A" and "l ∉ A" by auto

then show "(let l = l

in the (Mapping.lookup m l) + (∑k ∈ dom (Mapping.lookup m) - {l}. the (Mapping.lookup m k))) =

(∑k ∈ dom (Mapping.lookup m). the (Mapping.lookup m k))"

by simp

qed

then show ?thesis unfolding is_empty_def valuesum_def by transfer simp

qed

text {*

In the context of the else-branch we can show that the exact choice is

irrelvant; in practice, finding this point where choice becomes irrelevant is the

most difficult thing!

*}

lemma valuesum_choice:

"finite (Mapping.keys M) ==> x ∈ Mapping.keys M ==> y ∈ Mapping.keys M ==>

the (Mapping.lookup M x) + valuesum (Mapping.delete x M) =

the (Mapping.lookup M y) + valuesum (Mapping.delete y M)"

unfolding valuesum_def by transfer (simp add: setsum_diff)

text {*

Given @{text valuesum_rec} as initial description, we stepwise refine it to something executable;

first, we formally insert the constructor @{term Mapping} and split the one equation into two,

where the second one provides the necessary context:

*}

lemma valuesum_rec_Mapping:

shows [code]: "valuesum (Mapping []) = 0"

and "valuesum (Mapping (x # xs)) = (let l = (SOME l. l ∈ Mapping.keys (Mapping (x # xs))) in

the (Mapping.lookup (Mapping (x # xs)) l) + valuesum (Mapping.delete l (Mapping (x # xs))))"

by (simp_all add: valuesum_rec finite_dom_map_of is_empty_Mapping null_def)

text {*

As a side effect the precondition disappears (but note this has nothing to do with choice!).

The first equation deals with the uncritical empty case and can already be used for code generation.

Using @{text valuesum_choice}, we are able to prove an executable version of @{term valuesum}:

*}

lemma valuesum_rec_exec [code]:

"valuesum (Mapping (x # xs)) = (let l = fst (hd (x # xs)) in

the (Mapping.lookup (Mapping (x # xs)) l) + valuesum (Mapping.delete l (Mapping (x # xs))))"

proof -

let ?M = "Mapping (x # xs)"

let ?l1 = "(SOME l. l ∈ Mapping.keys ?M)"

let ?l2 = "fst (hd (x # xs))"

have "finite (Mapping.keys ?M)" by (simp add: keys_Mapping)

moreover have "?l1 ∈ Mapping.keys ?M"

by (rule someI) (auto simp add: keys_Mapping)

moreover have "?l2 ∈ Mapping.keys ?M"

by (simp add: keys_Mapping)

ultimately have "the (Mapping.lookup ?M ?l1) + valuesum (Mapping.delete ?l1 ?M) =

the (Mapping.lookup ?M ?l2) + valuesum (Mapping.delete ?l2 ?M)"

by (rule valuesum_choice)

then show ?thesis by (simp add: valuesum_rec_Mapping)

qed

text {*

See how it works:

*}

value "valuesum (Mapping [(''abc'', (42::int)), (''def'', 1705)])"

end