src/HOL/Library/comm_ring.ML
author huffman
Thu Jun 11 09:03:24 2009 -0700 (2009-06-11)
changeset 31563 ded2364d14d4
parent 31242 ed40b987a474
child 31986 a68f88d264f7
permissions -rw-r--r--
cleaned up some proofs
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(*  Author:     Amine Chaieb
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Tactic for solving equalities over commutative rings.
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*)
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signature COMM_RING =
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sig
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  val comm_ring_tac : Proof.context -> int -> tactic
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  val setup : theory -> theory
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end
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structure CommRing: COMM_RING =
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struct
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(* The Cring exception for erronous uses of cring_tac *)
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exception CRing of string;
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(* Zero and One of the commutative ring *)
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fun cring_zero T = Const (@{const_name HOL.zero}, T);
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fun cring_one T = Const (@{const_name HOL.one}, T);
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(* reification functions *)
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(* add two polynom expressions *)
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fun polT t = Type ("Commutative_Ring.pol", [t]);
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fun polexT t = Type ("Commutative_Ring.polex", [t]);
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(* pol *)
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fun pol_Pc t = Const ("Commutative_Ring.pol.Pc", t --> polT t);
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fun pol_Pinj t = Const ("Commutative_Ring.pol.Pinj", HOLogic.natT --> polT t --> polT t);
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fun pol_PX t = Const ("Commutative_Ring.pol.PX", polT t --> HOLogic.natT --> polT t --> polT t);
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(* polex *)
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fun polex_add t = Const ("Commutative_Ring.polex.Add", polexT t --> polexT t --> polexT t);
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fun polex_sub t = Const ("Commutative_Ring.polex.Sub", polexT t --> polexT t --> polexT t);
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fun polex_mul t = Const ("Commutative_Ring.polex.Mul", polexT t --> polexT t --> polexT t);
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fun polex_neg t = Const ("Commutative_Ring.polex.Neg", polexT t --> polexT t);
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fun polex_pol t = Const ("Commutative_Ring.polex.Pol", polT t --> polexT t);
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fun polex_pow t = Const ("Commutative_Ring.polex.Pow", polexT t --> HOLogic.natT --> polexT t);
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(* reification of polynoms : primitive cring expressions *)
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fun reif_pol T vs (t as Free _) =
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      let
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        val one = @{term "1::nat"};
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        val i = find_index_eq t vs
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      in if i = 0
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        then pol_PX T $ (pol_Pc T $ cring_one T)
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          $ one $ (pol_Pc T $ cring_zero T)
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        else pol_Pinj T $ HOLogic.mk_nat i
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          $ (pol_PX T $ (pol_Pc T $ cring_one T)
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            $ one $ (pol_Pc T $ cring_zero T))
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        end
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  | reif_pol T vs t = pol_Pc T $ t;
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(* reification of polynom expressions *)
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fun reif_polex T vs (Const (@{const_name HOL.plus}, _) $ a $ b) =
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      polex_add T $ reif_polex T vs a $ reif_polex T vs b
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  | reif_polex T vs (Const (@{const_name HOL.minus}, _) $ a $ b) =
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      polex_sub T $ reif_polex T vs a $ reif_polex T vs b
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  | reif_polex T vs (Const (@{const_name HOL.times}, _) $ a $ b) =
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      polex_mul T $ reif_polex T vs a $ reif_polex T vs b
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  | reif_polex T vs (Const (@{const_name HOL.uminus}, _) $ a) =
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      polex_neg T $ reif_polex T vs a
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  | reif_polex T vs (Const (@{const_name Power.power}, _) $ a $ n) =
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      polex_pow T $ reif_polex T vs a $ n
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  | reif_polex T vs t = polex_pol T $ reif_pol T vs t;
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(* reification of the equation *)
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val cr_sort = @{sort "comm_ring_1"};
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fun reif_eq thy (eq as Const("op =", Type("fun", [T, _])) $ lhs $ rhs) =
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      if Sign.of_sort thy (T, cr_sort) then
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        let
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          val fs = OldTerm.term_frees eq;
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          val cvs = cterm_of thy (HOLogic.mk_list T fs);
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          val clhs = cterm_of thy (reif_polex T fs lhs);
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          val crhs = cterm_of thy (reif_polex T fs rhs);
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          val ca = ctyp_of thy T;
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        in (ca, cvs, clhs, crhs) end
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      else raise CRing ("reif_eq: not an equation over " ^ Syntax.string_of_sort_global thy cr_sort)
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  | reif_eq _ _ = raise CRing "reif_eq: not an equation";
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(* The cring tactic *)
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(* Attention: You have to make sure that no t^0 is in the goal!! *)
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(* Use simply rewriting t^0 = 1 *)
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val cring_simps =
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  [@{thm mkPX_def}, @{thm mkPinj_def}, @{thm sub_def}, @{thm power_add},
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    @{thm even_def}, @{thm pow_if}, sym OF [@{thm power_add}]];
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fun comm_ring_tac ctxt = SUBGOAL (fn (g, i) =>
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  let
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    val thy = ProofContext.theory_of ctxt;
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    val cring_ss = Simplifier.local_simpset_of ctxt  (*FIXME really the full simpset!?*)
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      addsimps cring_simps;
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    val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g)
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    val norm_eq_th =
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      simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] @{thm norm_eq})
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  in
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    cut_rules_tac [norm_eq_th] i
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    THEN (simp_tac cring_ss i)
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    THEN (simp_tac cring_ss i)
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  end);
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val setup =
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  Method.setup @{binding comm_ring} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
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    "reflective decision procedure for equalities over commutative rings" #>
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  Method.setup @{binding algebra} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
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    "method for proving algebraic properties (same as comm_ring)";
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end;