author | haftmann |
Wed, 05 May 2010 18:25:34 +0200 | |
changeset 36692 | 54b64d4ad524 |
parent 36099 | 7e1f972df25f |
child 36862 | 952b2b102a0a |
permissions | -rw-r--r-- |
24584 | 1 |
(* Title: HOL/Tools/Qelim/ferrante_rackoff.ML |
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ID: $Id$ |
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Author: Amine Chaieb, TU Muenchen |
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Ferrante and Rackoff's algorithm for quantifier elimination in dense |
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linear orders. Proof-synthesis and tactic. |
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*) |
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signature FERRANTE_RACKOFF = |
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sig |
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val dlo_conv: Proof.context -> conv |
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val dlo_tac: Proof.context -> int -> tactic |
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end; |
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structure FerranteRackoff: FERRANTE_RACKOFF = |
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struct |
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open Ferrante_Rackoff_Data; |
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open Conv; |
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type entry = {minf: thm list, pinf: thm list, nmi: thm list, npi: thm list, |
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ld: thm list, qe: thm, atoms : cterm list} * |
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{isolate_conv: cterm list -> cterm -> thm, |
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whatis : cterm -> cterm -> ord, |
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simpset : simpset}; |
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fun get_p1 th = |
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funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE) |
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(funpow 2 Thm.dest_arg (cprop_of th)) |> Thm.dest_arg |
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fun ferrack_conv |
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(entr as ({minf = minf, pinf = pinf, nmi = nmi, npi = npi, |
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ld = ld, qe = qe, atoms = atoms}, |
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{isolate_conv = icv, whatis = wi, simpset = simpset}):entry) = |
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let |
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fun uset (vars as (x::vs)) p = case term_of p of |
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Const("op &", _)$ _ $ _ => |
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let |
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val ((b,l),r) = Thm.dest_comb p |>> Thm.dest_comb |
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val (lS,lth) = uset vars l val (rS, rth) = uset vars r |
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in (lS@rS, Drule.binop_cong_rule b lth rth) end |
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| Const("op |", _)$ _ $ _ => |
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let |
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val ((b,l),r) = Thm.dest_comb p |>> Thm.dest_comb |
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val (lS,lth) = uset vars l val (rS, rth) = uset vars r |
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in (lS@rS, Drule.binop_cong_rule b lth rth) end |
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| _ => |
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let |
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val th = icv vars p |
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val p' = Thm.rhs_of th |
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val c = wi x p' |
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val S = (if member (op =) [Lt, Le, Eq] c then single o Thm.dest_arg |
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else if member (op =) [Gt, Ge] c then single o Thm.dest_arg1 |
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else if c = NEq then single o Thm.dest_arg o Thm.dest_arg |
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else K []) p' |
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in (S,th) end |
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val ((p1_v,p2_v),(mp1_v,mp2_v)) = |
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funpow 2 (Thm.dest_arg o snd o Thm.dest_abs NONE) |
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(funpow 4 Thm.dest_arg (cprop_of (hd minf))) |
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|> Thm.dest_binop |> pairself Thm.dest_binop |> apfst (pairself Thm.dest_fun) |
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fun myfwd (th1, th2, th3, th4, th5) p1 p2 |
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[(th_1,th_2,th_3,th_4,th_5), (th_1',th_2',th_3',th_4',th_5')] = |
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let |
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val (mp1, mp2) = (get_p1 th_1, get_p1 th_1') |
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val (pp1, pp2) = (get_p1 th_2, get_p1 th_2') |
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fun fw mi th th' th'' = |
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let |
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val th0 = if mi then |
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instantiate ([],[(p1_v, p1),(p2_v, p2),(mp1_v, mp1), (mp2_v, mp2)]) th |
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else instantiate ([],[(p1_v, p1),(p2_v, p2),(mp1_v, pp1), (mp2_v, pp2)]) th |
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in implies_elim (implies_elim th0 th') th'' end |
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in (fw true th1 th_1 th_1', fw false th2 th_2 th_2', |
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fw true th3 th_3 th_3', fw false th4 th_4 th_4', fw true th5 th_5 th_5') |
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end |
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val U_v = (Thm.dest_arg o Thm.dest_arg o Thm.dest_arg1) (cprop_of qe) |
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fun main vs p = |
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let |
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val ((xn,ce),(x,fm)) = (case term_of p of |
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Const("Ex",_)$Abs(xn,xT,_) => |
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Thm.dest_comb p ||> Thm.dest_abs (SOME xn) |>> pair xn |
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| _ => raise CTERM ("main QE only treats existential quantifiers!", [p])) |
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val cT = ctyp_of_term x |
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val (u,nth) = uset (x::vs) fm |>> distinct (op aconvc) |
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val nthx = Thm.abstract_rule xn x nth |
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val q = Thm.rhs_of nth |
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val qx = Thm.rhs_of nthx |
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val enth = Drule.arg_cong_rule ce nthx |
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val [th0,th1] = map (instantiate' [SOME cT] []) @{thms "finite.intros"} |
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fun ins x th = |
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implies_elim (instantiate' [] [(SOME o Thm.dest_arg o Thm.dest_arg) |
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(Thm.cprop_of th), SOME x] th1) th |
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val fU = fold ins u th0 |
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val cU = funpow 2 Thm.dest_arg (Thm.cprop_of fU) |
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local |
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val insI1 = instantiate' [SOME cT] [] @{thm "insertI1"} |
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val insI2 = instantiate' [SOME cT] [] @{thm "insertI2"} |
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in |
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fun provein x S = |
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case term_of S of |
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Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
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changeset
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Const(@{const_name Orderings.bot}, _) => raise CTERM ("provein : not a member!", [S]) |
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set operations Int, Un, INTER, UNION, Inter, Union, empty, UNIV are now proper qualified constants with authentic syntax
haftmann
parents:
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changeset
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| Const(@{const_name insert}, _) $ y $_ => |
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let val (cy,S') = Thm.dest_binop S |
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in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1 |
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else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2) |
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(provein x S') |
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end |
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end |
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val tabU = fold (fn t => fn tab => Termtab.update (term_of t, provein t cU) tab) |
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u Termtab.empty |
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val U = the o Termtab.lookup tabU o term_of |
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val [minf_conj, minf_disj, minf_eq, minf_neq, minf_lt, |
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minf_le, minf_gt, minf_ge, minf_P] = minf |
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val [pinf_conj, pinf_disj, pinf_eq, pinf_neq, pinf_lt, |
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pinf_le, pinf_gt, pinf_ge, pinf_P] = pinf |
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val [nmi_conj, nmi_disj, nmi_eq, nmi_neq, nmi_lt, |
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nmi_le, nmi_gt, nmi_ge, nmi_P] = map (instantiate ([],[(U_v,cU)])) nmi |
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val [npi_conj, npi_disj, npi_eq, npi_neq, npi_lt, |
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npi_le, npi_gt, npi_ge, npi_P] = map (instantiate ([],[(U_v,cU)])) npi |
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val [ld_conj, ld_disj, ld_eq, ld_neq, ld_lt, |
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ld_le, ld_gt, ld_ge, ld_P] = map (instantiate ([],[(U_v,cU)])) ld |
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fun decomp_mpinf fm = |
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case term_of fm of |
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Const("op &",_)$_$_ => |
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let val (p,q) = Thm.dest_binop fm |
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in ([p,q], myfwd (minf_conj,pinf_conj, nmi_conj, npi_conj,ld_conj) |
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(Thm.cabs x p) (Thm.cabs x q)) |
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end |
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| Const("op |",_)$_$_ => |
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let val (p,q) = Thm.dest_binop fm |
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in ([p,q],myfwd (minf_disj, pinf_disj, nmi_disj, npi_disj,ld_disj) |
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(Thm.cabs x p) (Thm.cabs x q)) |
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end |
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| _ => |
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(let val c = wi x fm |
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val t = (if c=Nox then I |
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else if member (op =) [Lt, Le, Eq] c then Thm.dest_arg |
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else if member (op =) [Gt, Ge] c then Thm.dest_arg1 |
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else if c = NEq then (Thm.dest_arg o Thm.dest_arg) |
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else sys_error "decomp_mpinf: Impossible case!!") fm |
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val [mi_th, pi_th, nmi_th, npi_th, ld_th] = |
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if c = Nox then map (instantiate' [] [SOME fm]) |
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[minf_P, pinf_P, nmi_P, npi_P, ld_P] |
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else |
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let val [mi_th,pi_th,nmi_th,npi_th,ld_th] = |
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map (instantiate' [] [SOME t]) |
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(case c of Lt => [minf_lt, pinf_lt, nmi_lt, npi_lt, ld_lt] |
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| Le => [minf_le, pinf_le, nmi_le, npi_le, ld_le] |
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| Gt => [minf_gt, pinf_gt, nmi_gt, npi_gt, ld_gt] |
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| Ge => [minf_ge, pinf_ge, nmi_ge, npi_ge, ld_ge] |
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| Eq => [minf_eq, pinf_eq, nmi_eq, npi_eq, ld_eq] |
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| NEq => [minf_neq, pinf_neq, nmi_neq, npi_neq, ld_neq]) |
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val tU = U t |
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fun Ufw th = implies_elim th tU |
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in [mi_th, pi_th, Ufw nmi_th, Ufw npi_th, Ufw ld_th] |
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end |
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in ([], K (mi_th, pi_th, nmi_th, npi_th, ld_th)) end) |
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val (minf_th, pinf_th, nmi_th, npi_th, ld_th) = divide_and_conquer decomp_mpinf q |
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val qe_th = Drule.implies_elim_list |
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((fconv_rule (Thm.beta_conversion true)) |
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(instantiate' [] (map SOME [cU, qx, get_p1 minf_th, get_p1 pinf_th]) |
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qe)) |
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[fU, ld_th, nmi_th, npi_th, minf_th, pinf_th] |
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val bex_conv = |
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Simplifier.rewrite (HOL_basic_ss addsimps simp_thms@(@{thms "bex_simps" (1-5)})) |
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val result_th = fconv_rule (arg_conv bex_conv) (transitive enth qe_th) |
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in result_th |
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end |
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in main |
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end; |
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val grab_atom_bop = |
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let |
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fun h bounds tm = |
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(case term_of tm of |
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Const ("op =", T) $ _ $ _ => |
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if domain_type T = HOLogic.boolT then find_args bounds tm |
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else Thm.dest_fun2 tm |
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| Const ("Not", _) $ _ => h bounds (Thm.dest_arg tm) |
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| Const ("All", _) $ _ => find_body bounds (Thm.dest_arg tm) |
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| Const ("Ex", _) $ _ => find_body bounds (Thm.dest_arg tm) |
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| Const ("op &", _) $ _ $ _ => find_args bounds tm |
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| Const ("op |", _) $ _ $ _ => find_args bounds tm |
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| Const ("op -->", _) $ _ $ _ => find_args bounds tm |
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| Const ("==>", _) $ _ $ _ => find_args bounds tm |
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| Const ("==", _) $ _ $ _ => find_args bounds tm |
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| Const ("all", _) $ _ => find_body bounds (Thm.dest_arg tm) |
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| Const ("Trueprop", _) $ _ => h bounds (Thm.dest_arg tm) |
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| _ => Thm.dest_fun2 tm) |
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and find_args bounds tm = |
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(h bounds (Thm.dest_arg tm) handle CTERM _ => Thm.dest_arg1 tm) |
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and find_body bounds b = |
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let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b |
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in h (bounds + 1) b' end; |
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in h end; |
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fun raw_ferrack_qe_conv ctxt (thy, {isolate_conv, whatis, simpset}) tm = |
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let |
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val ss = simpset |
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val ss' = |
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merge_ss (HOL_basic_ss addsimps (simp_thms @ ex_simps @ all_simps) |
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@ [not_all,@{thm "all_not_ex"}, ex_disj_distrib], ss) |
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|> Simplifier.inherit_context ss |
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val pcv = Simplifier.rewrite ss' |
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val postcv = Simplifier.rewrite ss |
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val nnf = K (nnf_conv then_conv postcv) |
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val qe_conv = Qelim.gen_qelim_conv pcv postcv pcv cons (Thm.add_cterm_frees tm []) |
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(isolate_conv ctxt) nnf |
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(fn vs => ferrack_conv (thy,{isolate_conv = isolate_conv ctxt, |
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whatis = whatis, simpset = simpset}) vs |
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then_conv postcv) |
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in (Simplifier.rewrite ss then_conv qe_conv) tm end; |
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fun dlo_instance ctxt tm = |
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Ferrante_Rackoff_Data.match ctxt (grab_atom_bop 0 tm); |
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fun dlo_conv ctxt tm = |
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(case dlo_instance ctxt tm of |
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NONE => raise CTERM ("ferrackqe_conv: no corresponding instance in context!", [tm]) |
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| SOME instance => raw_ferrack_qe_conv ctxt instance tm); |
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fun dlo_tac ctxt = CSUBGOAL (fn (p, i) => |
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(case dlo_instance ctxt p of |
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NONE => no_tac |
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| SOME instance => |
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Object_Logic.full_atomize_tac i THEN |
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simp_tac (#simpset (snd instance)) i THEN (* FIXME already part of raw_ferrack_qe_conv? *) |
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CONVERSION (Object_Logic.judgment_conv (raw_ferrack_qe_conv ctxt instance)) i THEN |
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renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
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simp_tac (simpset_of ctxt) i)); (* FIXME really? *) |
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end; |