author | boehmes |
Wed, 12 May 2010 23:54:04 +0200 | |
changeset 36899 | bcd6fce5bf06 |
parent 36898 | 8e55aa1306c5 |
child 36936 | c52d1c130898 |
permissions | -rw-r--r-- |
36898 | 1 |
(* Title: HOL/Tools/SMT/smt_normalize.ML |
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Author: Sascha Boehme, TU Muenchen |
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Normalization steps on theorems required by SMT solvers: |
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* simplify trivial distincts (those with less than three elements), |
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* rewrite bool case expressions as if expressions, |
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* normalize numerals (e.g. replace negative numerals by negated positive |
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numerals), |
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* embed natural numbers into integers, |
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* add extra rules specifying types and constants which occur frequently, |
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* fully translate into object logic, add universal closure, |
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* lift lambda terms, |
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* make applications explicit for functions with varying number of arguments. |
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*) |
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signature SMT_NORMALIZE = |
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sig |
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type extra_norm = thm list -> Proof.context -> thm list * Proof.context |
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val normalize: extra_norm -> thm list -> Proof.context -> |
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thm list * Proof.context |
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36899
bcd6fce5bf06
layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents:
36898
diff
changeset
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val atomize_conv: Proof.context -> conv |
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val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv |
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end |
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structure SMT_Normalize: SMT_NORMALIZE = |
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struct |
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infix 2 ?? |
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fun (test ?? f) x = if test x then f x else x |
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fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct |
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fun if_true_conv c cv = if_conv c cv Conv.all_conv |
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(* simplification of trivial distincts (distinct should have at least |
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three elements in the argument list) *) |
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local |
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fun is_trivial_distinct (Const (@{const_name distinct}, _) $ t) = |
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length (HOLogic.dest_list t) <= 2 |
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| is_trivial_distinct _ = false |
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val thms = @{lemma |
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"distinct [] == True" |
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"distinct [x] == True" |
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"distinct [x, y] == (x ~= y)" |
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by simp_all} |
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fun distinct_conv _ = |
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if_true_conv is_trivial_distinct (More_Conv.rewrs_conv thms) |
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in |
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fun trivial_distinct ctxt = |
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map ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ?? |
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Conv.fconv_rule (More_Conv.top_conv distinct_conv ctxt)) |
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end |
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(* rewrite bool case expressions as if expressions *) |
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local |
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val is_bool_case = (fn |
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Const (@{const_name "bool.bool_case"}, _) $ _ $ _ $ _ => true |
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| _ => false) |
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val thms = @{lemma |
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"(case P of True => x | False => y) == (if P then x else y)" |
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"(case P of False => y | True => x) == (if P then x else y)" |
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by (rule eq_reflection, simp)+} |
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val unfold_conv = if_true_conv is_bool_case (More_Conv.rewrs_conv thms) |
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in |
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fun rewrite_bool_cases ctxt = |
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map ((Term.exists_subterm is_bool_case o Thm.prop_of) ?? |
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Conv.fconv_rule (More_Conv.top_conv (K unfold_conv) ctxt)) |
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end |
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(* normalization of numerals: rewriting of negative integer numerals into |
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positive numerals, Numeral0 into 0, Numeral1 into 1 *) |
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local |
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fun is_number_sort ctxt T = |
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Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring}) |
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fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) $ _) = |
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(case try HOLogic.dest_number t of |
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SOME (T, i) => is_number_sort ctxt T andalso i < 2 |
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| NONE => false) |
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| is_strange_number _ _ = false |
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val pos_numeral_ss = HOL_ss |
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addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}] |
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addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}] |
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addsimps @{thms Int.pred_bin_simps} |
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addsimps @{thms Int.normalize_bin_simps} |
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addsimps @{lemma |
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"Int.Min = - Int.Bit1 Int.Pls" |
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"Int.Bit0 (- Int.Pls) = - Int.Pls" |
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"Int.Bit0 (- k) = - Int.Bit0 k" |
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"Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)" |
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by simp_all (simp add: pred_def)} |
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fun pos_conv ctxt = if_conv (is_strange_number ctxt) |
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(Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss)) |
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Conv.no_conv |
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in |
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fun normalize_numerals ctxt = |
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map ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ?? |
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Conv.fconv_rule (More_Conv.top_sweep_conv pos_conv ctxt)) |
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end |
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(* embedding of standard natural number operations into integer operations *) |
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local |
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val nat_embedding = @{lemma |
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"nat (int n) = n" |
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"i >= 0 --> int (nat i) = i" |
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"i < 0 --> int (nat i) = 0" |
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by simp_all} |
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val nat_rewriting = @{lemma |
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"0 = nat 0" |
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"1 = nat 1" |
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"number_of i = nat (number_of i)" |
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"int (nat 0) = 0" |
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"int (nat 1) = 1" |
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"a < b = (int a < int b)" |
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"a <= b = (int a <= int b)" |
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"Suc a = nat (int a + 1)" |
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"a + b = nat (int a + int b)" |
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"a - b = nat (int a - int b)" |
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"a * b = nat (int a * int b)" |
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"a div b = nat (int a div int b)" |
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"a mod b = nat (int a mod int b)" |
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"min a b = nat (min (int a) (int b))" |
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"max a b = nat (max (int a) (int b))" |
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"int (nat (int a + int b)) = int a + int b" |
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"int (nat (int a * int b)) = int a * int b" |
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"int (nat (int a div int b)) = int a div int b" |
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"int (nat (int a mod int b)) = int a mod int b" |
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"int (nat (min (int a) (int b))) = min (int a) (int b)" |
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"int (nat (max (int a) (int b))) = max (int a) (int b)" |
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by (simp_all add: nat_mult_distrib nat_div_distrib nat_mod_distrib |
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int_mult[symmetric] zdiv_int[symmetric] zmod_int[symmetric])} |
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fun on_positive num f x = |
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(case try HOLogic.dest_number (Thm.term_of num) of |
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SOME (_, i) => if i >= 0 then SOME (f x) else NONE |
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| NONE => NONE) |
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val cancel_int_nat_ss = HOL_ss |
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addsimps [@{thm Nat_Numeral.nat_number_of}] |
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addsimps [@{thm Nat_Numeral.int_nat_number_of}] |
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addsimps @{thms neg_simps} |
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fun cancel_int_nat_simproc _ ss ct = |
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let |
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val num = Thm.dest_arg (Thm.dest_arg ct) |
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val goal = Thm.mk_binop @{cterm "op == :: int => _"} ct num |
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val simpset = Simplifier.inherit_context ss cancel_int_nat_ss |
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fun tac _ = Simplifier.simp_tac simpset 1 |
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in on_positive num (Goal.prove_internal [] goal) tac end |
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val nat_ss = HOL_ss |
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addsimps nat_rewriting |
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addsimprocs [Simplifier.make_simproc { |
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name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}], |
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proc = cancel_int_nat_simproc, identifier = [] }] |
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fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss) |
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val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat})) |
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val uses_nat_int = |
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Term.exists_subterm (member (op aconv) [@{term int}, @{term nat}]) |
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in |
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fun nat_as_int ctxt = |
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map ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt)) #> |
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exists (uses_nat_int o Thm.prop_of) ?? append nat_embedding |
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end |
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(* further normalizations: beta/eta, universal closure, atomize *) |
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val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)} |
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fun eta_expand_conv cv ctxt = |
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Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt |
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local |
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val eta_conv = eta_expand_conv |
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fun keep_conv ctxt = More_Conv.binder_conv norm_conv ctxt |
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and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt) |
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and keep_let_conv ctxt = Conv.combination_conv |
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(Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt) |
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and unfold_let_conv ctxt = Conv.combination_conv |
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(Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt) |
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and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt |
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and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt |
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and unfold_ball_conv ctxt = unfold_conv @{thm Ball_def} ctxt |
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and unfold_bex_conv ctxt = unfold_conv @{thm Bex_def} ctxt |
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and norm_conv ctxt ct = |
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(case Thm.term_of ct of |
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Const (@{const_name All}, _) $ Abs _ => keep_conv |
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| Const (@{const_name All}, _) $ _ => eta_binder_conv |
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| Const (@{const_name All}, _) => eta_conv eta_binder_conv |
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| Const (@{const_name Ex}, _) $ Abs _ => keep_conv |
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| Const (@{const_name Ex}, _) $ _ => eta_binder_conv |
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| Const (@{const_name Ex}, _) => eta_conv eta_binder_conv |
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| Const (@{const_name Let}, _) $ _ $ Abs _ => keep_let_conv |
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| Const (@{const_name Let}, _) $ _ $ _ => unfold_let_conv |
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| Const (@{const_name Let}, _) $ _ => eta_conv unfold_let_conv |
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| Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv) |
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| Const (@{const_name Ex1}, _) $ _ => unfold_ex1_conv |
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| Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv |
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| Const (@{const_name Ball}, _) $ _ $ _ => unfold_ball_conv |
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| Const (@{const_name Ball}, _) $ _ => eta_conv unfold_ball_conv |
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| Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv) |
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| Const (@{const_name Bex}, _) $ _ $ _ => unfold_bex_conv |
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| Const (@{const_name Bex}, _) $ _ => eta_conv unfold_bex_conv |
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| Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv) |
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| Abs _ => Conv.abs_conv (norm_conv o snd) |
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| _ $ _ => Conv.comb_conv o norm_conv |
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| _ => K Conv.all_conv) ctxt ct |
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fun is_normed t = |
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(case t of |
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Const (@{const_name All}, _) $ Abs (_, _, u) => is_normed u |
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| Const (@{const_name All}, _) $ _ => false |
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| Const (@{const_name All}, _) => false |
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| Const (@{const_name Ex}, _) $ Abs (_, _, u) => is_normed u |
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| Const (@{const_name Ex}, _) $ _ => false |
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| Const (@{const_name Ex}, _) => false |
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| Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) => |
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is_normed u1 andalso is_normed u2 |
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| Const (@{const_name Let}, _) $ _ $ _ => false |
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| Const (@{const_name Let}, _) $ _ => false |
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| Const (@{const_name Let}, _) => false |
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| Const (@{const_name Ex1}, _) => false |
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| Const (@{const_name Ball}, _) => false |
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| Const (@{const_name Bex}, _) => false |
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| Abs (_, _, u) => is_normed u |
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| u1 $ u2 => is_normed u1 andalso is_normed u2 |
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| _ => true) |
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in |
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fun norm_binder_conv ctxt = if_conv is_normed Conv.all_conv (norm_conv ctxt) |
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end |
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fun norm_def ctxt thm = |
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(case Thm.prop_of thm of |
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@{term Trueprop} $ (Const (@{const_name "op ="}, _) $ _ $ Abs _) => |
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norm_def ctxt (thm RS @{thm fun_cong}) |
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| Const (@{const_name "=="}, _) $ _ $ Abs _ => |
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norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq}) |
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| _ => thm) |
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fun atomize_conv ctxt ct = |
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(case Thm.term_of ct of |
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@{term "op ==>"} $ _ $ _ => |
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Conv.binop_conv (atomize_conv ctxt) then_conv |
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Conv.rewr_conv @{thm atomize_imp} |
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| Const (@{const_name "=="}, _) $ _ $ _ => |
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Conv.binop_conv (atomize_conv ctxt) then_conv |
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Conv.rewr_conv @{thm atomize_eq} |
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| Const (@{const_name all}, _) $ Abs _ => |
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More_Conv.binder_conv atomize_conv ctxt then_conv |
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Conv.rewr_conv @{thm atomize_all} |
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| _ => Conv.all_conv) ct |
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fun normalize_rule ctxt = |
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Conv.fconv_rule ( |
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(* reduce lambda abstractions, except at known binders: *) |
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Thm.beta_conversion true then_conv |
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Thm.eta_conversion then_conv |
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norm_binder_conv ctxt) #> |
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norm_def ctxt #> |
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Drule.forall_intr_vars #> |
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Conv.fconv_rule (atomize_conv ctxt) |
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(* lift lambda terms into additional rules *) |
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local |
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val meta_eq = @{cpat "op =="} |
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val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq)) |
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fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq |
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fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu |
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fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt) |
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fun used_vars cvs ct = |
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let |
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val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs) |
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val add = (fn SOME ct => insert (op aconvc) ct | _ => I) |
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in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end |
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fun apply cv thm = |
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let val thm' = Thm.combination thm (Thm.reflexive cv) |
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in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end |
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fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq) |
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fun replace_lambda cvs ct (cx as (ctxt, defs)) = |
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let |
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val cvs' = used_vars cvs ct |
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val ct' = fold_rev Thm.cabs cvs' ct |
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in |
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(case Termtab.lookup defs (Thm.term_of ct') of |
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SOME eq => (apply_def cvs' eq, cx) |
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| NONE => |
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let |
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val {T, ...} = Thm.rep_cterm ct' and n = Name.uu |
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val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt |
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val cu = mk_meta_eq (cert ctxt (Free (n', T))) ct' |
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val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt' |
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val defs' = Termtab.update (Thm.term_of ct', eq) defs |
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in (apply_def cvs' eq, (ctxt'', defs')) end) |
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end |
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fun none ct cx = (Thm.reflexive ct, cx) |
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fun in_comb f g ct cx = |
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let val (cu1, cu2) = Thm.dest_comb ct |
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in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end |
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fun in_arg f = in_comb none f |
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fun in_abs f cvs ct (ctxt, defs) = |
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let |
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val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt |
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val (cv, cu) = Thm.dest_abs (SOME n) ct |
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in (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end |
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fun traverse cvs ct = |
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(case Thm.term_of ct of |
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Const (@{const_name All}, _) $ Abs _ => in_arg (in_abs traverse cvs) |
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| Const (@{const_name Ex}, _) $ Abs _ => in_arg (in_abs traverse cvs) |
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| Const (@{const_name Let}, _) $ _ $ Abs _ => |
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in_comb (in_arg (traverse cvs)) (in_abs traverse cvs) |
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| Abs _ => at_lambda cvs |
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| _ $ _ => in_comb (traverse cvs) (traverse cvs) |
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| _ => none) ct |
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and at_lambda cvs ct = |
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in_abs traverse cvs ct #-> (fn thm => |
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replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm) |
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fun has_free_lambdas t = |
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(case t of |
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Const (@{const_name All}, _) $ Abs (_, _, u) => has_free_lambdas u |
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| Const (@{const_name Ex}, _) $ Abs (_, _, u) => has_free_lambdas u |
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| Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) => |
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has_free_lambdas u1 orelse has_free_lambdas u2 |
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| Abs _ => true |
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| u1 $ u2 => has_free_lambdas u1 orelse has_free_lambdas u2 |
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| _ => false) |
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fun lift_lm f thm cx = |
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if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx) |
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else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm) |
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in |
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fun lift_lambdas thms ctxt = |
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let |
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val cx = (ctxt, Termtab.empty) |
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val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx |
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val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs [] |
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368 |
in (eqs @ thms', ctxt') end |
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369 |
end |
|
370 |
||
371 |
||
372 |
||
373 |
(* make application explicit for functions with varying number of arguments *) |
|
374 |
||
375 |
local |
|
376 |
val const = prefix "c" and free = prefix "f" |
|
377 |
fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e |
|
378 |
fun add t i = Symtab.map_default (t, (false, i)) (min i) |
|
379 |
fun traverse t = |
|
380 |
(case Term.strip_comb t of |
|
381 |
(Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts |
|
382 |
| (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts |
|
383 |
| (Abs (_, _, u), ts) => fold traverse (u :: ts) |
|
384 |
| (_, ts) => fold traverse ts) |
|
385 |
val prune = (fn (n, (true, i)) => Symtab.update (n, i) | _ => I) |
|
386 |
fun prune_tab tab = Symtab.fold prune tab Symtab.empty |
|
387 |
||
388 |
fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2 |
|
389 |
fun nary_conv conv1 conv2 ct = |
|
390 |
(Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct |
|
391 |
fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) => |
|
392 |
let val n = fst (Term.dest_Free (Thm.term_of cv)) |
|
393 |
in conv (Symtab.update (free n, 0) tb) cx end) |
|
394 |
val apply_rule = @{lemma "f x == apply f x" by (simp add: apply_def)} |
|
395 |
in |
|
396 |
fun explicit_application ctxt thms = |
|
397 |
let |
|
398 |
fun sub_conv tb ctxt ct = |
|
399 |
(case Term.strip_comb (Thm.term_of ct) of |
|
400 |
(Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt |
|
401 |
| (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt |
|
402 |
| (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt) |
|
403 |
| (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct |
|
404 |
and app_conv tb n i ctxt = |
|
405 |
(case Symtab.lookup tb n of |
|
406 |
NONE => nary_conv Conv.all_conv (sub_conv tb ctxt) |
|
407 |
| SOME j => apply_conv tb ctxt (i - j)) |
|
408 |
and apply_conv tb ctxt i ct = ( |
|
409 |
if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt) |
|
410 |
else |
|
411 |
Conv.rewr_conv apply_rule then_conv |
|
412 |
binop_conv (apply_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct |
|
413 |
||
414 |
fun needs_exp_app tab = Term.exists_subterm (fn |
|
415 |
Bound _ $ _ => true |
|
416 |
| Const (n, _) => Symtab.defined tab (const n) |
|
417 |
| Free (n, _) => Symtab.defined tab (free n) |
|
418 |
| _ => false) |
|
419 |
||
420 |
fun rewrite tab ctxt thm = |
|
421 |
if not (needs_exp_app tab (Thm.prop_of thm)) then thm |
|
422 |
else Conv.fconv_rule (sub_conv tab ctxt) thm |
|
423 |
||
424 |
val tab = prune_tab (fold (traverse o Thm.prop_of) thms Symtab.empty) |
|
425 |
in map (rewrite tab ctxt) thms end |
|
426 |
end |
|
427 |
||
428 |
||
429 |
||
430 |
(* combined normalization *) |
|
431 |
||
432 |
type extra_norm = thm list -> Proof.context -> thm list * Proof.context |
|
433 |
||
434 |
fun with_context f thms ctxt = (f ctxt thms, ctxt) |
|
435 |
||
436 |
fun normalize extra_norm thms ctxt = |
|
437 |
thms |
|
438 |
|> trivial_distinct ctxt |
|
439 |
|> rewrite_bool_cases ctxt |
|
440 |
|> normalize_numerals ctxt |
|
441 |
|> nat_as_int ctxt |
|
442 |
|> rpair ctxt |
|
443 |
|-> extra_norm |
|
444 |
|-> with_context (fn cx => map (normalize_rule cx)) |
|
445 |
|-> SMT_Monomorph.monomorph |
|
446 |
|-> lift_lambdas |
|
447 |
|-> with_context explicit_application |
|
448 |
||
449 |
end |