(* Title: HOL/Tools/SMT/z3_proof_tools.ML
Author: Sascha Boehme, TU Muenchen
Helper functions required for Z3 proof reconstruction.
*)
signature Z3_PROOF_TOOLS =
sig
(*modifying terms*)
val as_meta_eq: cterm -> cterm
(*theorem nets*)
val thm_net_of: ('a -> thm) -> 'a list -> 'a Net.net
val net_instances: (int * thm) Net.net -> cterm -> (int * thm) list
val net_instance: thm Net.net -> cterm -> thm option
(*proof combinators*)
val under_assumption: (thm -> thm) -> cterm -> thm
val with_conv: conv -> (cterm -> thm) -> cterm -> thm
val discharge: thm -> thm -> thm
val varify: string list -> thm -> thm
val unfold_eqs: Proof.context -> thm list -> conv
val match_instantiate: (cterm -> cterm) -> cterm -> thm -> thm
val by_tac: (int -> tactic) -> cterm -> thm
val make_hyp_def: thm -> Proof.context -> thm * Proof.context
val by_abstraction: int -> bool * bool -> Proof.context -> thm list ->
(Proof.context -> cterm -> thm) -> cterm -> thm
(*a faster COMP*)
type compose_data
val precompose: (cterm -> cterm list) -> thm -> compose_data
val precompose2: (cterm -> cterm * cterm) -> thm -> compose_data
val compose: compose_data -> thm -> thm
(*unfolding of 'distinct'*)
val unfold_distinct_conv: conv
(*simpset*)
val add_simproc: Simplifier.simproc -> Context.generic -> Context.generic
val make_simpset: Proof.context -> thm list -> simpset
end
structure Z3_Proof_Tools: Z3_PROOF_TOOLS =
struct
(* modifying terms *)
fun as_meta_eq ct =
uncurry SMT_Utils.mk_cequals (Thm.dest_binop (SMT_Utils.dest_cprop ct))
(* theorem nets *)
fun thm_net_of f xthms =
let fun insert xthm = Net.insert_term (K false) (Thm.prop_of (f xthm), xthm)
in fold insert xthms Net.empty end
fun maybe_instantiate ct thm =
try Thm.first_order_match (Thm.cprop_of thm, ct)
|> Option.map (fn inst => Thm.instantiate inst thm)
local
fun instances_from_net match f net ct =
let
val lookup = if match then Net.match_term else Net.unify_term
val xthms = lookup net (Thm.term_of ct)
fun select ct = map_filter (f (maybe_instantiate ct)) xthms
fun select' ct =
let val thm = Thm.trivial ct
in map_filter (f (try (fn rule => rule COMP thm))) xthms end
in (case select ct of [] => select' ct | xthms' => xthms') end
in
fun net_instances net =
instances_from_net false (fn f => fn (i, thm) => Option.map (pair i) (f thm))
net
fun net_instance net = try hd o instances_from_net true I net
end
(* proof combinators *)
fun under_assumption f ct =
let val ct' = SMT_Utils.mk_cprop ct
in Thm.implies_intr ct' (f (Thm.assume ct')) end
fun with_conv conv prove ct =
let val eq = Thm.symmetric (conv ct)
in Thm.equal_elim eq (prove (Thm.lhs_of eq)) end
fun discharge p pq = Thm.implies_elim pq p
fun varify vars = Drule.generalize ([], vars)
fun unfold_eqs _ [] = Conv.all_conv
| unfold_eqs ctxt eqs =
Conv.top_sweep_conv (K (Conv.rewrs_conv eqs)) ctxt
fun match_instantiate f ct thm =
Thm.instantiate (Thm.match (f (Thm.cprop_of thm), ct)) thm
fun by_tac tac ct = Goal.norm_result (Goal.prove_internal [] ct (K (tac 1)))
(*
|- c x == t x ==> P (c x)
---------------------------
c == t |- P (c x)
*)
fun make_hyp_def thm ctxt =
let
val (lhs, rhs) = Thm.dest_binop (Thm.cprem_of thm 1)
val (cf, cvs) = Drule.strip_comb lhs
val eq = SMT_Utils.mk_cequals cf (fold_rev Thm.lambda cvs rhs)
fun apply cv th =
Thm.combination th (Thm.reflexive cv)
|> Conv.fconv_rule (Conv.arg_conv (Thm.beta_conversion false))
in
yield_singleton Assumption.add_assumes eq ctxt
|>> Thm.implies_elim thm o fold apply cvs
end
(* abstraction *)
local
fun abs_context ctxt = (ctxt, Termtab.empty, 1, false)
fun context_of (ctxt, _, _, _) = ctxt
fun replace (_, (cv, ct)) = Thm.forall_elim ct o Thm.forall_intr cv
fun abs_instantiate (_, tab, _, beta_norm) =
fold replace (Termtab.dest tab) #>
beta_norm ? Conv.fconv_rule (Thm.beta_conversion true)
fun lambda_abstract cvs t =
let
val frees = map Free (Term.add_frees t [])
val cvs' = filter (fn cv => member (op aconv) frees (Thm.term_of cv)) cvs
val vs = map (Term.dest_Free o Thm.term_of) cvs'
in (fold_rev absfree vs t, cvs') end
fun fresh_abstraction (_, cvs) ct (cx as (ctxt, tab, idx, beta_norm)) =
let val (t, cvs') = lambda_abstract cvs (Thm.term_of ct)
in
(case Termtab.lookup tab t of
SOME (cv, _) => (Drule.list_comb (cv, cvs'), cx)
| NONE =>
let
val (n, ctxt') = yield_singleton Variable.variant_fixes "x" ctxt
val cv = SMT_Utils.certify ctxt'
(Free (n, map SMT_Utils.typ_of cvs' ---> SMT_Utils.typ_of ct))
val cu = Drule.list_comb (cv, cvs')
val e = (t, (cv, fold_rev Thm.lambda cvs' ct))
val beta_norm' = beta_norm orelse not (null cvs')
in (cu, (ctxt', Termtab.update e tab, idx + 1, beta_norm')) end)
end
fun abs_comb f g dcvs ct =
let val (cf, cu) = Thm.dest_comb ct
in f dcvs cf ##>> g dcvs cu #>> uncurry Thm.apply end
fun abs_arg f = abs_comb (K pair) f
fun abs_args f dcvs ct =
(case Thm.term_of ct of
_ $ _ => abs_comb (abs_args f) f dcvs ct
| _ => pair ct)
fun abs_list f g dcvs ct =
(case Thm.term_of ct of
Const (@{const_name Nil}, _) => pair ct
| Const (@{const_name Cons}, _) $ _ $ _ =>
abs_comb (abs_arg f) (abs_list f g) dcvs ct
| _ => g dcvs ct)
fun abs_abs f (depth, cvs) ct =
let val (cv, cu) = Thm.dest_abs NONE ct
in f (depth, cv :: cvs) cu #>> Thm.lambda cv end
val is_atomic =
(fn Free _ => true | Var _ => true | Bound _ => true | _ => false)
fun abstract depth (ext_logic, with_theories) =
let
fun abstr1 cvs ct = abs_arg abstr cvs ct
and abstr2 cvs ct = abs_comb abstr1 abstr cvs ct
and abstr3 cvs ct = abs_comb abstr2 abstr cvs ct
and abstr_abs cvs ct = abs_arg (abs_abs abstr) cvs ct
and abstr (dcvs as (d, cvs)) ct =
(case Thm.term_of ct of
@{const Trueprop} $ _ => abstr1 dcvs ct
| @{const "==>"} $ _ $ _ => abstr2 dcvs ct
| @{const True} => pair ct
| @{const False} => pair ct
| @{const Not} $ _ => abstr1 dcvs ct
| @{const HOL.conj} $ _ $ _ => abstr2 dcvs ct
| @{const HOL.disj} $ _ $ _ => abstr2 dcvs ct
| @{const HOL.implies} $ _ $ _ => abstr2 dcvs ct
| Const (@{const_name HOL.eq}, _) $ _ $ _ => abstr2 dcvs ct
| Const (@{const_name distinct}, _) $ _ =>
if ext_logic then abs_arg (abs_list abstr fresh_abstraction) dcvs ct
else fresh_abstraction dcvs ct
| Const (@{const_name If}, _) $ _ $ _ $ _ =>
if ext_logic then abstr3 dcvs ct else fresh_abstraction dcvs ct
| Const (@{const_name All}, _) $ _ =>
if ext_logic then abstr_abs dcvs ct else fresh_abstraction dcvs ct
| Const (@{const_name Ex}, _) $ _ =>
if ext_logic then abstr_abs dcvs ct else fresh_abstraction dcvs ct
| t => (fn cx =>
if is_atomic t orelse can HOLogic.dest_number t then (ct, cx)
else if with_theories andalso
Z3_Interface.is_builtin_theory_term (context_of cx) t
then abs_args abstr dcvs ct cx
else if d = 0 then fresh_abstraction dcvs ct cx
else
(case Term.strip_comb t of
(Const _, _) => abs_args abstr (d-1, cvs) ct cx
| (Free _, _) => abs_args abstr (d-1, cvs) ct cx
| _ => fresh_abstraction dcvs ct cx)))
in abstr (depth, []) end
val cimp = Thm.cterm_of @{theory} @{const "==>"}
fun deepen depth f x =
if depth = 0 then f depth x
else (case try (f depth) x of SOME y => y | NONE => deepen (depth - 1) f x)
fun with_prems depth thms f ct =
fold_rev (Thm.mk_binop cimp o Thm.cprop_of) thms ct
|> deepen depth f
|> fold (fn prem => fn th => Thm.implies_elim th prem) thms
in
fun by_abstraction depth mode ctxt thms prove =
with_prems depth thms (fn d => fn ct =>
let val (cu, cx) = abstract d mode ct (abs_context ctxt)
in abs_instantiate cx (prove (context_of cx) cu) end)
end
(* a faster COMP *)
type compose_data = cterm list * (cterm -> cterm list) * thm
fun list2 (x, y) = [x, y]
fun precompose f rule = (f (Thm.cprem_of rule 1), f, rule)
fun precompose2 f rule = precompose (list2 o f) rule
fun compose (cvs, f, rule) thm =
discharge thm (Thm.instantiate ([], cvs ~~ f (Thm.cprop_of thm)) rule)
(* unfolding of 'distinct' *)
local
val set1 = @{lemma "x ~: set [] == ~False" by simp}
val set2 = @{lemma "x ~: set [x] == False" by simp}
val set3 = @{lemma "x ~: set [y] == x ~= y" by simp}
val set4 = @{lemma "x ~: set (x # ys) == False" by simp}
val set5 = @{lemma "x ~: set (y # ys) == x ~= y & x ~: set ys" by simp}
fun set_conv ct =
(Conv.rewrs_conv [set1, set2, set3, set4] else_conv
(Conv.rewr_conv set5 then_conv Conv.arg_conv set_conv)) ct
val dist1 = @{lemma "distinct [] == ~False" by (simp add: distinct_def)}
val dist2 = @{lemma "distinct [x] == ~False" by (simp add: distinct_def)}
val dist3 = @{lemma "distinct (x # xs) == x ~: set xs & distinct xs"
by (simp add: distinct_def)}
fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
in
fun unfold_distinct_conv ct =
(Conv.rewrs_conv [dist1, dist2] else_conv
(Conv.rewr_conv dist3 then_conv binop_conv set_conv unfold_distinct_conv)) ct
end
(* simpset *)
local
val antisym_le1 = mk_meta_eq @{thm order_class.antisym_conv}
val antisym_le2 = mk_meta_eq @{thm linorder_class.antisym_conv2}
val antisym_less1 = mk_meta_eq @{thm linorder_class.antisym_conv1}
val antisym_less2 = mk_meta_eq @{thm linorder_class.antisym_conv3}
fun eq_prop t thm = HOLogic.mk_Trueprop t aconv Thm.prop_of thm
fun dest_binop ((c as Const _) $ t $ u) = (c, t, u)
| dest_binop t = raise TERM ("dest_binop", [t])
fun prove_antisym_le ss t =
let
val (le, r, s) = dest_binop t
val less = Const (@{const_name less}, Term.fastype_of le)
val prems = Simplifier.prems_of ss
in
(case find_first (eq_prop (le $ s $ r)) prems of
NONE =>
find_first (eq_prop (HOLogic.mk_not (less $ r $ s))) prems
|> Option.map (fn thm => thm RS antisym_less1)
| SOME thm => SOME (thm RS antisym_le1))
end
handle THM _ => NONE
fun prove_antisym_less ss t =
let
val (less, r, s) = dest_binop (HOLogic.dest_not t)
val le = Const (@{const_name less_eq}, Term.fastype_of less)
val prems = Simplifier.prems_of ss
in
(case find_first (eq_prop (le $ r $ s)) prems of
NONE =>
find_first (eq_prop (HOLogic.mk_not (less $ s $ r))) prems
|> Option.map (fn thm => thm RS antisym_less2)
| SOME thm => SOME (thm RS antisym_le2))
end
handle THM _ => NONE
val basic_simpset = HOL_ss addsimps @{thms field_simps}
addsimps [@{thm times_divide_eq_right}, @{thm times_divide_eq_left}]
addsimps @{thms arith_special} addsimps @{thms arith_simps}
addsimps @{thms rel_simps}
addsimps @{thms array_rules}
addsimps @{thms term_true_def} addsimps @{thms term_false_def}
addsimps @{thms z3div_def} addsimps @{thms z3mod_def}
addsimprocs [@{simproc binary_int_div}, @{simproc binary_int_mod}]
addsimprocs [
Simplifier.simproc_global @{theory} "fast_int_arith" [
"(m::int) < n", "(m::int) <= n", "(m::int) = n"] (K Lin_Arith.simproc),
Simplifier.simproc_global @{theory} "antisym_le" ["(x::'a::order) <= y"]
(K prove_antisym_le),
Simplifier.simproc_global @{theory} "antisym_less" ["~ (x::'a::linorder) < y"]
(K prove_antisym_less)]
structure Simpset = Generic_Data
(
type T = simpset
val empty = basic_simpset
val extend = I
val merge = Simplifier.merge_ss
)
in
fun add_simproc simproc = Simpset.map (fn ss => ss addsimprocs [simproc])
fun make_simpset ctxt rules =
Simplifier.context ctxt (Simpset.get (Context.Proof ctxt)) addsimps rules
end
end