| 1 | (**************************************************************************) |
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| 2 | (* ___ *) |
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| 3 | (* ||M|| *) |
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| 4 | (* ||A|| A project by Andrea Asperti *) |
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| 5 | (* ||T|| *) |
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| 6 | (* ||I|| Developers: *) |
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| 7 | (* ||T|| The HELM team. *) |
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| 8 | (* ||A|| http://helm.cs.unibo.it *) |
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| 9 | (* \ / *) |
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| 10 | (* \ / This file is distributed under the terms of the *) |
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| 11 | (* v GNU General Public License Version 2 *) |
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| 12 | (* *) |
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| 13 | (**************************************************************************) |
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| 14 | |
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| 15 | include "Traces.ma". |
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| 16 | |
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| 17 | record monoid: Type[1] ≝ |
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| 18 | { carrier :> Type[0] |
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| 19 | ; op: carrier → carrier → carrier |
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| 20 | ; e: carrier |
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| 21 | ; neutral_r : ∀x. op … x e = x |
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| 22 | ; neutral_l : ∀x. op … e x = x |
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| 23 | ; is_associative: ∀x,y,z. op … (op … x y) z = op … x (op … y z) |
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| 24 | }. |
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| 25 | |
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| 26 | record monoid_action (I : monoid) (M : Type[0]) : Type[0] ≝ |
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| 27 | { act :2> I → M → M |
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| 28 | ; act_neutral : ∀x.act (e …) x = x |
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| 29 | ; act_op : ∀i,j,x.act (op … i j) x = act j (act i x) |
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| 30 | }. |
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| 31 | |
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| 32 | record concrete_transition_sys : Type[2] ≝ |
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| 33 | { trans_sys :> abstract_status |
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| 34 | }. |
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| 35 | |
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| 36 | record abstract_transition_sys : Type[1] ≝ |
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| 37 | { abs_state_t :> Type[0] |
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| 38 | ; abs_instr : monoid |
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| 39 | ; act_abs : monoid_action abs_instr abs_state_t |
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| 40 | }. |
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| 41 | |
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| 42 | notation "〚 term 19 C 〛 term 90 A" with precedence 10 for @{ 'sem $C $A }. |
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| 43 | interpretation "act_abs" 'sem f x = (act ?? (act_abs ?) f x). |
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| 44 | |
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| 45 | record galois_rel (C : concrete_transition_sys) |
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| 46 | (A : abstract_transition_sys) : Type[0] ≝ |
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| 47 | { rel_galois :2> C → A → Prop |
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| 48 | }. |
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| 49 | |
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| 50 | record galois_connection : Type[2] ≝ |
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| 51 | { concr : concrete_transition_sys |
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| 52 | ; abs_d : abstract_transition_sys |
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| 53 | ; galois_rel:> galois_rel concr abs_d |
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| 54 | }. |
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| 55 | |
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| 56 | record abstract_interpretation : Type[2] ≝ |
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| 57 | { galois_conn:> galois_connection |
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| 58 | ; instr_t : Type[0] |
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| 59 | ; fetch : concr galois_conn → option instr_t |
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| 60 | ; instr_map : instr_t → abs_instr … (abs_d galois_conn) |
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| 61 | ; instr_map_ok : |
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| 62 | ∀s,s': concr … galois_conn. ∀a: abs_d … galois_conn.∀l,i. |
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| 63 | as_execute … l s s' → |
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| 64 | fetch … s = Some ? i → |
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| 65 | ∀I. instr_map … i = I → |
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| 66 | galois_conn s a → galois_conn s' (〚I〛 a) |
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| 67 | }. |
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