--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real_Asymp/multiseries_expansion.ML Sun Jul 15 14:46:57 2018 +0200
@@ -0,0 +1,2374 @@
+signature MULTISERIES_EXPANSION = sig
+
+type expansion_thm = thm
+type trimmed_thm = thm
+type expr = Exp_Log_Expression.expr
+type basis = Asymptotic_Basis.basis
+
+datatype trim_mode = Simple_Trim | Pos_Trim | Neg_Trim | Sgn_Trim
+
+datatype zeroness = IsZero | IsNonZero | IsPos | IsNeg
+
+datatype intyness = Nat of thm | Neg_Nat of thm | No_Nat
+datatype parity = Even of thm | Odd of thm | Unknown_Parity
+
+datatype limit =
+ Zero_Limit of bool option
+ | Finite_Limit of term
+ | Infinite_Limit of bool option
+
+datatype trim_result =
+ Trimmed of zeroness * trimmed_thm option
+ | Aborted of order
+
+val get_intyness : Proof.context -> cterm -> intyness
+val get_parity : cterm -> parity
+
+val get_expansion : thm -> term
+val get_coeff : term -> term
+val get_exponent : term -> term
+val get_expanded_fun : thm -> term
+val get_eval : term -> term
+val expands_to_hd : thm -> thm
+
+val mk_eval_ctxt : Proof.context -> Lazy_Eval.eval_ctxt
+val expand : Lazy_Eval.eval_ctxt -> expr -> basis -> expansion_thm * basis
+val expand_term : Lazy_Eval.eval_ctxt -> term -> basis -> expansion_thm * basis
+val expand_terms : Lazy_Eval.eval_ctxt -> term list -> basis -> expansion_thm list * basis
+
+val limit_of_expansion : bool * bool -> Lazy_Eval.eval_ctxt -> thm * basis -> limit * thm
+val compute_limit : Lazy_Eval.eval_ctxt -> term -> limit * thm
+val compare_expansions :
+ Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis ->
+ order * thm * expansion_thm * expansion_thm
+
+(* TODO DEBUG *)
+datatype comparison_result =
+ Cmp_Dominated of order * thm list * zeroness * trimmed_thm * expansion_thm * expansion_thm
+| Cmp_Asymp_Equiv of thm * thm
+val compare_expansions' :
+ Lazy_Eval.eval_ctxt ->
+ thm * thm * Asymptotic_Basis.basis ->
+ comparison_result
+
+val prove_at_infinity : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_at_top : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_at_bot : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_nhds : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_at_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_at_left_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+val prove_at_right_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+
+val prove_smallo : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+val prove_bigo : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+val prove_bigtheta : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+val prove_asymp_equiv : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+
+val prove_asymptotic_relation : Lazy_Eval.eval_ctxt -> thm * thm * basis -> order * thm
+val prove_eventually_less : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+val prove_eventually_greater : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
+val prove_eventually_nonzero : Lazy_Eval.eval_ctxt -> thm * basis -> thm
+
+val extract_terms : int * bool -> Lazy_Eval.eval_ctxt -> basis -> term -> term * term option
+
+(* Internal functions *)
+val check_expansion : Exp_Log_Expression.expr -> expansion_thm -> expansion_thm
+
+val zero_expansion : basis -> expansion_thm
+val const_expansion : Lazy_Eval.eval_ctxt -> basis -> term -> expansion_thm
+val ln_expansion :
+ Lazy_Eval.eval_ctxt -> trimmed_thm -> expansion_thm -> basis -> expansion_thm * basis
+val exp_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> basis -> expansion_thm * basis
+val powr_expansion :
+ Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm * basis
+val powr_const_expansion :
+ Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
+val powr_nat_expansion :
+ Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm * basis
+val power_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
+val root_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
+
+val sgn_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis -> expansion_thm
+val min_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm
+val max_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm
+val arctan_expansion : Lazy_Eval.eval_ctxt -> basis -> expansion_thm -> expansion_thm
+
+val ev_zeroness_oracle : Lazy_Eval.eval_ctxt -> term -> thm option
+val zeroness_oracle : bool -> trim_mode option -> Lazy_Eval.eval_ctxt -> term -> zeroness * thm option
+
+val whnf_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> term option * expansion_thm * thm
+val simplify_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> expansion_thm
+val simplify_term : Lazy_Eval.eval_ctxt -> term -> term
+
+val trim_expansion_while_greater :
+ bool -> term list option -> bool -> trim_mode option -> Lazy_Eval.eval_ctxt ->
+ thm * Asymptotic_Basis.basis -> thm * trim_result * (zeroness * thm) list
+val trim_expansion : bool -> trim_mode option -> Lazy_Eval.eval_ctxt -> expansion_thm * basis ->
+ expansion_thm * zeroness * trimmed_thm option
+val try_drop_leading_term_ex : bool -> Lazy_Eval.eval_ctxt -> expansion_thm -> expansion_thm option
+
+val try_prove_real_eq : bool -> Lazy_Eval.eval_ctxt -> term * term -> thm option
+val try_prove_ev_eq : Lazy_Eval.eval_ctxt -> term * term -> thm option
+val prove_compare_expansions : order -> thm list -> thm
+
+val simplify_trimmed_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * trimmed_thm ->
+ expansion_thm * trimmed_thm
+val retrim_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis -> expansion_thm * thm
+val retrim_pos_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis * trimmed_thm ->
+ expansion_thm * thm * trimmed_thm
+
+val register_sign_oracle :
+ binding * (Proof.context -> int -> tactic) -> Context.generic -> Context.generic
+val get_sign_oracles :
+ Context.generic -> (string * (Proof.context -> int -> tactic)) list
+
+val solve_eval_eq : thm -> thm
+
+end
+
+structure Multiseries_Expansion : MULTISERIES_EXPANSION = struct
+
+open Asymptotic_Basis
+open Exp_Log_Expression
+open Lazy_Eval
+
+structure Data = Generic_Data
+(
+ type T = (Proof.context -> int -> tactic) Name_Space.table;
+ val empty : T = Name_Space.empty_table "sign oracle tactics";
+ val extend = I;
+ fun merge (tactics1, tactics2) : T = Name_Space.merge_tables (tactics1, tactics2);
+);
+
+fun register_sign_oracle (s, tac) ctxt =
+ Data.map (Name_Space.define ctxt false (s, tac) #> snd) ctxt
+
+fun get_sign_oracles ctxt = Name_Space.fold_table cons (Data.get ctxt) []
+
+fun apply_sign_oracles ctxt tac =
+ let
+ val oracles = get_sign_oracles (Context.Proof ctxt)
+ fun tac' {context = ctxt, concl, ...} =
+ if Thm.term_of concl = @{term "HOL.Trueprop HOL.False"} then
+ no_tac
+ else
+ FIRST (map (fn tac => HEADGOAL (snd tac ctxt)) oracles)
+ in
+ tac THEN_ALL_NEW (Subgoal.FOCUS_PREMS tac' ctxt)
+ end
+
+
+type expansion_thm = thm
+type trimmed_thm = thm
+
+val dest_fun = dest_comb #> fst
+val dest_arg = dest_comb #> snd
+val concl_of' = Thm.concl_of #> HOLogic.dest_Trueprop
+
+fun get_expansion thm =
+ thm |> Thm.concl_of |> HOLogic.dest_Trueprop |> Term.dest_comb |> fst |> Term.dest_comb |> snd
+
+fun get_expanded_fun thm = thm |> concl_of' |> dest_fun |> dest_fun |> dest_arg
+
+(*
+ The following function is useful in order to detect whether a given real constant is
+ an integer, which allows us to use the "f(x) ^ n" operation instead of "f(x) powr n".
+ This usually leads to nicer results.
+*)
+datatype intyness = Nat of thm | Neg_Nat of thm | No_Nat
+
+fun get_intyness ctxt ct =
+ if Thm.typ_of_cterm ct = @{typ Real.real} then
+ let
+ val ctxt' = put_simpset HOL_basic_ss ctxt addsimps @{thms intyness_simps}
+ val conv =
+ Simplifier.rewrite ctxt then_conv Simplifier.rewrite ctxt'
+ fun flip (Nat thm) = Neg_Nat (thm RS @{thm intyness_uminus})
+ | flip _ = No_Nat
+ fun get_intyness' ct =
+ case Thm.term_of ct of
+ @{term "0::real"} => Nat @{thm intyness_0}
+ | @{term "1::real"} => Nat @{thm intyness_1}
+ | Const (@{const_name numeral}, _) $ _ =>
+ Nat (Thm.reflexive (Thm.dest_arg ct) RS @{thm intyness_numeral})
+ | Const (@{const_name uminus}, _) $ _ => flip (get_intyness' (Thm.dest_arg ct))
+ | Const (@{const_name of_nat}, _) $ _ =>
+ Nat (Thm.reflexive (Thm.dest_arg ct) RS @{thm intyness_of_nat})
+ | _ => No_Nat
+ val thm = conv ct
+ val ct' = thm |> Thm.cprop_of |> Thm.dest_equals_rhs
+ in
+ case get_intyness' ct' of
+ Nat thm' => Nat (Thm.transitive thm thm' RS @{thm HOL.meta_eq_to_obj_eq})
+ | Neg_Nat thm' => Neg_Nat (Thm.transitive thm thm' RS @{thm HOL.meta_eq_to_obj_eq})
+ | No_Nat => No_Nat
+ end
+ handle CTERM _ => No_Nat
+ else
+ No_Nat
+
+datatype parity = Even of thm | Odd of thm | Unknown_Parity
+
+(* TODO: powers *)
+fun get_parity ct =
+ let
+ fun inst thm cts =
+ let
+ val tvs = Term.add_tvars (Thm.concl_of thm) []
+ in
+ case tvs of
+ [v] =>
+ let
+ val thm' = Thm.instantiate ([(v, Thm.ctyp_of_cterm ct)], []) thm
+ val vs = take (length cts) (rev (Term.add_vars (Thm.concl_of thm') []))
+ in
+ Thm.instantiate ([], vs ~~ cts) thm'
+ end
+ | _ => raise THM ("get_parity", 0, [thm])
+ end
+ val get_num = Thm.dest_arg o Thm.dest_arg
+ in
+ case Thm.term_of ct of
+ Const (@{const_name Groups.zero}, _) => Even (inst @{thm even_zero} [])
+ | Const (@{const_name Groups.one}, _) => Odd (inst @{thm odd_one} [])
+ | Const (@{const_name Num.numeral_class.numeral}, _) $ @{term "Num.One"} =>
+ Odd (inst @{thm odd_Numeral1} [])
+ | Const (@{const_name Num.numeral_class.numeral}, _) $ (@{term "Num.Bit0"} $ _) =>
+ Even (inst @{thm even_numeral} [get_num ct])
+ | Const (@{const_name Num.numeral_class.numeral}, _) $ (@{term "Num.Bit1"} $ _) =>
+ Odd (inst @{thm odd_numeral} [get_num ct])
+ | Const (@{const_name Groups.uminus}, _) $ _ => (
+ case get_parity (Thm.dest_arg ct) of
+ Even thm => Even (@{thm even_uminusI} OF [thm])
+ | Odd thm => Odd (@{thm odd_uminusI} OF [thm])
+ | _ => Unknown_Parity)
+ | Const (@{const_name Groups.plus}, _) $ _ $ _ => (
+ case apply2 get_parity (Thm.dest_binop ct) of
+ (Even thm1, Even thm2) => Even (@{thm even_addI(1)} OF [thm1, thm2])
+ | (Odd thm1, Odd thm2) => Even (@{thm even_addI(2)} OF [thm1, thm2])
+ | (Even thm1, Odd thm2) => Odd (@{thm odd_addI(1)} OF [thm1, thm2])
+ | (Odd thm1, Even thm2) => Odd (@{thm odd_addI(2)} OF [thm1, thm2])
+ | _ => Unknown_Parity)
+ | Const (@{const_name Groups.minus}, _) $ _ $ _ => (
+ case apply2 get_parity (Thm.dest_binop ct) of
+ (Even thm1, Even thm2) => Even (@{thm even_diffI(1)} OF [thm1, thm2])
+ | (Odd thm1, Odd thm2) => Even (@{thm even_diffI(2)} OF [thm1, thm2])
+ | (Even thm1, Odd thm2) => Odd (@{thm odd_diffI(1)} OF [thm1, thm2])
+ | (Odd thm1, Even thm2) => Odd (@{thm odd_diffI(2)} OF [thm1, thm2])
+ | _ => Unknown_Parity)
+ | Const (@{const_name Groups.times}, _) $ _ $ _ => (
+ case apply2 get_parity (Thm.dest_binop ct) of
+ (Even thm1, _) => Even (@{thm even_multI(1)} OF [thm1])
+ | (_, Even thm2) => Even (@{thm even_multI(2)} OF [thm2])
+ | (Odd thm1, Odd thm2) => Odd (@{thm odd_multI} OF [thm1, thm2])
+ | _ => Unknown_Parity)
+ | Const (@{const_name Power.power}, _) $ _ $ _ =>
+ let
+ val (a, n) = Thm.dest_binop ct
+ in
+ case get_parity a of
+ Odd thm => Odd (inst @{thm odd_powerI} [a, n] OF [thm])
+ | _ => Unknown_Parity
+ end
+ | _ => Unknown_Parity
+ end
+
+fun simplify_term' facts ctxt =
+ let
+ val ctxt = Simplifier.add_prems facts ctxt
+ in
+ Thm.cterm_of ctxt #> Simplifier.rewrite ctxt #>
+ Thm.concl_of #> Logic.dest_equals #> snd
+ end
+
+fun simplify_term ectxt = simplify_term' (get_facts ectxt) (get_ctxt ectxt)
+
+fun simplify_eval ctxt =
+ simplify_term' [] (put_simpset HOL_basic_ss ctxt addsimps @{thms eval_simps})
+
+datatype zeroness = IsZero | IsNonZero | IsPos | IsNeg
+
+
+(* Caution: The following functions assume that the given expansion is in normal form already
+ as far as needed. *)
+
+(* Returns the leading coefficient of the given expansion. This coefficient is a multiseries. *)
+fun try_get_coeff expr =
+ case expr of
+ Const (@{const_name MS}, _) $ (
+ Const (@{const_name MSLCons}, _) $ (
+ Const (@{const_name Pair}, _) $ c $ _) $ _) $ _ =>
+ SOME c
+ | _ => NONE
+
+fun get_coeff expr =
+ expr |> dest_comb |> fst |> dest_comb |> snd |> dest_comb |> fst |> dest_comb |> snd
+ |> dest_comb |> fst |> dest_comb |> snd
+
+(* Returns the coefficient of the leading term in the expansion (i.e. a real number) *)
+fun get_lead_coeff expr =
+ case try_get_coeff expr of
+ NONE => expr
+ | SOME c => get_lead_coeff c
+
+(* Returns the exponent (w.r.t. the fastest-growing basis element) of the leading term *)
+fun get_exponent expr =
+ expr |> dest_comb |> fst |> dest_comb |> snd |> dest_comb |> fst |> dest_comb |> snd
+ |> dest_comb |> snd
+
+(* Returns the list of exponents of the leading term *)
+fun get_exponents exp =
+ if fastype_of exp = @{typ real} then
+ []
+ else
+ get_exponent exp :: get_exponents (get_coeff exp)
+
+(* Returns the function that the expansion corresponds to *)
+fun get_eval expr =
+ if fastype_of expr = @{typ real} then
+ Abs ("x", @{typ real}, expr)
+ else
+ expr |> dest_comb |> snd
+
+val eval_simps = @{thms eval_simps [THEN eq_reflection]}
+
+(* Tries to prove that the given function is eventually zero *)
+fun ev_zeroness_oracle ectxt t =
+ let
+ val ctxt = Lazy_Eval.get_ctxt ectxt
+ val goal =
+ betapply (@{term "\<lambda>f::real \<Rightarrow> real. eventually (\<lambda>x. f x = 0) at_top"}, t)
+ |> HOLogic.mk_Trueprop
+ fun tac {context = ctxt, ...} =
+ HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
+ THEN Local_Defs.unfold_tac ctxt eval_simps
+ THEN HEADGOAL (Simplifier.asm_full_simp_tac ctxt)
+ in
+ try (Goal.prove ctxt [] [] goal) tac
+ end
+
+(*
+ Encodes the kind of trimming/zeroness checking operation to be performed.
+ Simple_Trim only checks for zeroness/non-zeroness. Pos_Trim/Neg_Trim try to prove either
+ zeroness or positivity (resp. negativity). Sgn_Trim tries all three possibilities (positive,
+ negative, zero). *)
+datatype trim_mode = Simple_Trim | Pos_Trim | Neg_Trim | Sgn_Trim
+
+(*
+ Checks (and proves) whether the given term (assumed to be a real number) is zero, positive,
+ or negative, depending on given flags. The "fail" flag determines whether an exception is
+ thrown if this fails.
+*)
+fun zeroness_oracle fail mode ectxt exp =
+ let
+ val ctxt = Lazy_Eval.get_ctxt ectxt
+ val eq = (exp, @{term "0::real"}) |> HOLogic.mk_eq
+ val goal1 = (IsZero, eq |> HOLogic.mk_Trueprop)
+ val goal2 =
+ case mode of
+ SOME Pos_Trim =>
+ (IsPos, @{term "(<) (0::real)"} $ exp |> HOLogic.mk_Trueprop)
+ | SOME Sgn_Trim =>
+ (IsPos, @{term "(<) (0::real)"} $ exp |> HOLogic.mk_Trueprop)
+ | SOME Neg_Trim =>
+ (IsNeg, betapply (@{term "\<lambda>x. x < (0::real)"}, exp) |> HOLogic.mk_Trueprop)
+ | _ =>
+ (IsNonZero, eq |> HOLogic.mk_not |> HOLogic.mk_Trueprop)
+ val goals =
+ (if mode = SOME Sgn_Trim then
+ [(IsNeg, betapply (@{term "\<lambda>x. x < (0::real)"}, exp) |> HOLogic.mk_Trueprop)]
+ else
+ [])
+ val goals = goal2 :: goals
+ fun tac {context = ctxt, ...} =
+ HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
+ THEN Local_Defs.unfold_tac ctxt eval_simps
+ THEN HEADGOAL (apply_sign_oracles ctxt (Simplifier.asm_full_simp_tac ctxt))
+ fun prove (res, goal) = try (fn goal => (res, SOME (Goal.prove ctxt [] [] goal tac))) goal
+ fun err () =
+ let
+ val mode_msg =
+ case mode of
+ SOME Simple_Trim => "whether the following constant is zero"
+ | SOME Pos_Trim => "whether the following constant is zero or positive"
+ | SOME Neg_Trim => "whether the following constant is zero or negative"
+ | SOME Sgn_Trim => "the sign of the following constant"
+ | _ => raise Match
+ val t = simplify_term' (get_facts ectxt) ctxt exp
+ val _ =
+ if #verbose (#ctxt ectxt) then
+ let
+ val p = Pretty.str ("real_asymp failed to determine " ^ mode_msg ^ ":")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
+ in
+ Pretty.writeln p
+ end else ()
+ in
+ raise TERM ("zeroness_oracle", [t])
+ end
+ in
+ case prove goal1 of
+ SOME res => res
+ | NONE =>
+ if mode = NONE then
+ (IsNonZero, NONE)
+ else
+ case get_first prove (goal2 :: goals) of
+ NONE => if fail then err () else (IsNonZero, NONE)
+ | SOME res => res
+ end
+
+(* Tries to prove a given equality of real numbers. *)
+fun try_prove_real_eq fail ectxt (lhs, rhs) =
+ case zeroness_oracle false NONE ectxt (@{term "(-) :: real => _"} $ lhs $ rhs) of
+ (IsZero, SOME thm) => SOME (thm RS @{thm real_eqI})
+ | _ =>
+ if not fail then NONE else
+ let
+ val ctxt = get_ctxt ectxt
+ val ts = map (simplify_term' (get_facts ectxt) ctxt) [lhs, rhs]
+ val _ =
+ if #verbose (#ctxt ectxt) then
+ let
+ val p =
+ Pretty.str ("real_asymp failed to prove that the following two numbers are equal:")
+ val p = Pretty.chunks (p :: map (Pretty.indent 2 o Syntax.pretty_term ctxt) ts)
+ in
+ Pretty.writeln p
+ end else ()
+ in
+ raise TERM ("try_prove_real_eq", [lhs, rhs])
+ end
+
+(* Tries to prove a given eventual equality of real functions. *)
+fun try_prove_ev_eq ectxt (f, g) =
+ let
+ val t = Envir.beta_eta_contract (@{term "\<lambda>(f::real=>real) g x. f x - g x"} $ f $ g)
+ in
+ Option.map (fn thm => thm RS @{thm eventually_diff_zero_imp_eq}) (ev_zeroness_oracle ectxt t)
+ end
+
+fun real_less a b = @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
+fun real_eq a b = @{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
+fun real_neq a b = @{term "(\<noteq>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
+
+(* The hook that is called by the Lazy_Eval module whenever two real numbers have to be compared *)
+fun real_sgn_hook ({pctxt = ctxt, facts, verbose, ...}) t =
+ let
+ val get_rhs = Thm.concl_of #> Logic.dest_equals #> snd
+ fun tac {context = ctxt, ...} =
+ HEADGOAL (Method.insert_tac ctxt (Net.content facts)
+ THEN' (apply_sign_oracles ctxt (Simplifier.asm_full_simp_tac ctxt)))
+ fun prove_first err [] [] =
+ if not verbose then raise TERM ("real_sgn_hook", [t])
+ else let val _ = err () in raise TERM ("real_sgn_hook", [t]) end
+ | prove_first err (goal :: goals) (thm :: thms) =
+ (case try (Goal.prove ctxt [] [] goal) tac of
+ SOME thm' =>
+ let val thm'' = thm' RS thm in SOME (get_rhs thm'', Conv.rewr_conv thm'') end
+ | NONE => prove_first err goals thms)
+ | prove_first _ _ _ = raise Match
+ in
+ case t of
+ @{term "(=) :: real => _"} $ a $ @{term "0 :: real"} =>
+ let
+ val goals =
+ map (fn c => HOLogic.mk_Trueprop (c a @{term "0 :: real"})) [real_neq, real_eq]
+ fun err () =
+ let
+ val facts' = Net.content facts
+ val a' = simplify_term' facts' ctxt a
+ val p = Pretty.str ("real_asymp failed to determine whether the following " ^
+ "constant is zero: ")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt a')]
+ in
+ Pretty.writeln p
+ end
+ in
+ prove_first err goals @{thms Eq_FalseI Eq_TrueI}
+ end
+ | Const (@{const_name COMPARE}, _) $ a $ b =>
+ let
+ val goals = map HOLogic.mk_Trueprop [real_less a b, real_less b a, real_eq a b]
+ fun err () =
+ let
+ val facts' = Net.content facts
+ val (a', b') = apply2 (simplify_term' facts' ctxt) (a, b)
+ val p = Pretty.str ("real_asymp failed to compare" ^
+ "the following two constants: ")
+ val p = Pretty.chunks (p :: map (Pretty.indent 2 o Syntax.pretty_term ctxt) [a', b'])
+ in
+ Pretty.writeln p
+ end
+ in
+ prove_first err goals @{thms COMPARE_intros}
+ end
+ | _ => NONE
+ end
+
+(*
+ Returns the datatype constructors registered for use with the Lazy_Eval package.
+ All constructors on which pattern matching is performed need to be registered for evaluation
+ to work. It should be rare for users to add additional ones.
+*)
+fun get_constructors ctxt =
+ let
+ val thms = Named_Theorems.get ctxt @{named_theorems exp_log_eval_constructor}
+ fun go _ [] acc = rev acc
+ | go f (x :: xs) acc =
+ case f x of
+ NONE => go f xs acc
+ | SOME y => go f xs (y :: acc)
+ fun map_option f xs = go f xs []
+ fun dest_constructor thm =
+ case Thm.concl_of thm of
+ Const (@{const_name HOL.Trueprop}, _) $
+ (Const (@{const_name REAL_ASYMP_EVAL_CONSTRUCTOR}, _) $ Const (c, T)) =>
+ SOME (c, length (fst (strip_type T)))
+ | _ => NONE
+ in
+ thms |> map_option dest_constructor
+ end
+
+(*
+ Creates an evaluation context with the correct setup of constructors, equations, and hooks.
+*)
+fun mk_eval_ctxt ctxt =
+ let
+ val eval_eqs = (Named_Theorems.get ctxt @{named_theorems real_asymp_eval_eqs})
+ val constructors = get_constructors ctxt
+ in
+ Lazy_Eval.mk_eval_ctxt ctxt constructors eval_eqs
+ |> add_hook real_sgn_hook
+ end
+
+(* A pattern for determining the leading coefficient of a multiseries *)
+val exp_pat =
+ let
+ val anypat = AnyPat ("_", 0)
+ in
+ ConsPat (@{const_name MS},
+ [ConsPat (@{const_name MSLCons},
+ [ConsPat (@{const_name Pair}, [anypat, anypat]), anypat]), anypat])
+ end
+
+(*
+ Evaluates an expansion to (weak) head normal form, so that the leading coefficient and
+ exponent can be read off.
+*)
+fun whnf_expansion ectxt thm =
+ let
+ val ctxt = get_ctxt ectxt
+ val exp = get_expansion thm
+ val (_, _, conv) = match ectxt exp_pat exp (SOME [])
+ val eq_thm = conv (Thm.cterm_of ctxt exp)
+ val exp' = eq_thm |> Thm.concl_of |> Logic.dest_equals |> snd
+ in
+ case exp' of
+ Const (@{const_name MS}, _) $ (Const (@{const_name MSLCons}, _) $
+ (Const (@{const_name Pair}, _) $ c $ _) $ _) $ _ =>
+ (SOME c, @{thm expands_to_meta_eq_cong} OF [thm, eq_thm], eq_thm)
+ | Const (@{const_name MS}, _) $ Const (@{const_name MSLNil}, _) $ _ =>
+ (NONE, @{thm expands_to_meta_eq_cong} OF [thm, eq_thm], eq_thm)
+ | _ => raise TERM ("whnf_expansion", [exp'])
+ end
+
+fun try_lift_function ectxt (thm, SEmpty) _ = (NONE, thm)
+ | try_lift_function ectxt (thm, basis) cont =
+ case whnf_expansion ectxt thm of
+ (SOME c, thm, _) =>
+ let
+ val f = get_expanded_fun thm
+ val T = fastype_of c
+ val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
+ val t = Term.betapply (Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) g x. f x - g x"}, f), t)
+ in
+ case ev_zeroness_oracle ectxt t of
+ NONE => (NONE, thm)
+ | SOME zero_thm =>
+ let
+ val thm' = cont ectxt (thm RS @{thm expands_to_hd''}, tl_basis basis)
+ val thm'' = @{thm expands_to_lift_function} OF [zero_thm, thm']
+ in
+ (SOME (lift basis thm''), thm)
+ end
+ end
+ | _ => (NONE, thm)
+
+(* Turns an expansion theorem into an expansion theorem for the leading coefficient. *)
+fun expands_to_hd thm = thm RS
+ (if fastype_of (get_expansion thm) = @{typ "real ms"} then
+ @{thm expands_to_hd'}
+ else
+ @{thm expands_to_hd})
+
+fun simplify_expansion ectxt thm =
+ let
+ val exp = get_expansion thm
+ val ctxt = get_ctxt ectxt
+ val eq_thm = Simplifier.rewrite ctxt (Thm.cterm_of ctxt exp)
+ in
+ @{thm expands_to_meta_eq_cong} OF [thm, eq_thm]
+ end
+
+(*
+ Simplifies a trimmed expansion and returns the simplified expansion theorem and
+ the trimming theorem for that simplified expansion.
+*)
+fun simplify_trimmed_expansion ectxt (thm, trimmed_thm) =
+ let
+ val exp = get_expansion thm
+ val ctxt = get_ctxt ectxt
+ val eq_thm = Simplifier.rewrite ctxt (Thm.cterm_of ctxt exp)
+ val trimmed_cong_thm =
+ case trimmed_thm |> concl_of' |> dest_fun of
+ Const (@{const_name trimmed}, _) => @{thm trimmed_eq_cong}
+ | Const (@{const_name trimmed_pos}, _) => @{thm trimmed_pos_eq_cong}
+ | Const (@{const_name trimmed_neg}, _) => @{thm trimmed_neg_eq_cong}
+ | _ => raise THM ("simplify_trimmed_expansion", 2, [thm, trimmed_thm])
+ in
+ (@{thm expands_to_meta_eq_cong} OF [thm, eq_thm],
+ trimmed_cong_thm OF [trimmed_thm, eq_thm])
+ end
+
+(*
+ Re-normalises a trimmed expansion (so that the leading term with its (real) coefficient and
+ all exponents can be read off. This may be necessary after lifting a trimmed expansion to
+ a larger basis.
+*)
+fun retrim_expansion ectxt (thm, basis) =
+ let
+ val (c, thm, eq_thm) = whnf_expansion ectxt thm
+ in
+ case c of
+ NONE => (thm, eq_thm)
+ | SOME c =>
+ if fastype_of c = @{typ real} then
+ (thm, eq_thm)
+ else
+ let
+ val c_thm = thm RS @{thm expands_to_hd''}
+ val (c_thm', eq_thm') = retrim_expansion ectxt (c_thm, tl_basis basis)
+ val thm = @{thm expands_to_trim_cong} OF [thm, c_thm']
+ in
+ (thm, @{thm trim_lift_eq} OF [eq_thm, eq_thm'])
+ end
+ end
+
+fun retrim_pos_expansion ectxt (thm, basis, trimmed_thm) =
+ let
+ val (thm', eq_thm) = retrim_expansion ectxt (thm, basis)
+ in
+ (thm', eq_thm, @{thm trimmed_pos_eq_cong} OF [trimmed_thm, eq_thm])
+ end
+
+(*
+ Tries to determine whether the leading term is (identically) zero and drops it if it is.
+ If "fail" is set, an exception is thrown when that term is a real number and zeroness cannot
+ be determined. (Which typically indicates missing facts or case distinctions)
+*)
+fun try_drop_leading_term_ex fail ectxt thm =
+ let
+ val exp = get_expansion thm
+ in
+ if fastype_of exp = @{typ real} then
+ NONE
+ else if fastype_of (get_coeff exp) = @{typ real} then
+ case zeroness_oracle fail (SOME Simple_Trim) ectxt (get_coeff exp) of
+ (IsZero, SOME zero_thm) => SOME (@{thm drop_zero_ms'} OF [zero_thm, thm])
+ | _ => NONE
+ else
+ let
+ val c = get_coeff exp
+ val T = fastype_of c
+ val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
+ in
+ case ev_zeroness_oracle ectxt t of
+ SOME zero_thm => SOME (@{thm expands_to_drop_zero} OF [zero_thm, thm])
+ | _ => NONE
+ end
+ end
+
+(*
+ Tries to drop the leading term of an expansion. If this is not possible, an exception
+ is thrown and an informative error message is printed.
+*)
+fun try_drop_leading_term ectxt thm =
+ let
+ fun err () =
+ let
+ val ctxt = get_ctxt ectxt
+ val exp = get_expansion thm
+ val c = get_coeff exp
+ val t =
+ if fastype_of c = @{typ real} then c else c |> dest_arg
+ val t = simplify_term' (get_facts ectxt) ctxt t
+ val _ =
+ if #verbose (#ctxt ectxt) then
+ let
+ val p = Pretty.str ("real_asymp failed to prove that the following term is zero: ")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
+ in
+ Pretty.writeln p
+ end else ()
+ in
+ raise TERM ("try_drop_leading_term", [t])
+ end
+ in
+ case try_drop_leading_term_ex true ectxt thm of
+ NONE => err ()
+ | SOME thm => thm
+ end
+
+
+datatype trim_result =
+ Trimmed of zeroness * trimmed_thm option
+ | Aborted of order
+
+fun cstrip_assms ct =
+ case Thm.term_of ct of
+ @{term "(==>)"} $ _ $ _ => cstrip_assms (snd (Thm.dest_implies ct))
+ | _ => ct
+
+(*
+ Trims an expansion (i.e. drops leading zero terms) and provides a trimmedness theorem.
+ Optionally, a list of exponents can be given to instruct the function to only trim until
+ the exponents of the leading term are lexicographically less than (or less than or equal) than
+ the given ones. This is useful to avoid unnecessary trimming.
+
+ The "strict" flag indicates whether the trimming should already be aborted when the
+ exponents are lexicographically equal or not.
+
+ The "fail" flag is passed on to the zeroness oracle and determines whether a failure to determine
+ the sign of a real number leads to an exception.
+
+ "mode" indicates what kind of trimmedness theorem will be returned: Simple_Trim only gives the
+ default trimmedness theorem, whereas Pos_Trim/Neg_Trim/Sgn_Trim will give trimmed_pos or
+ trimmed_neg. Giving "None" as mode will produce no trimmedness theorem; it will only drop
+ leading zero terms until zeroness cannot be proven anymore, upon which it will stop.
+
+ The main result of the function is the trimmed expansion theorem.
+
+ The function returns whether the trimming has been aborted or not. If was aborted, either
+ LESS or EQUAL will be returned, indicating whether the exponents of the leading term are
+ now lexicographically smaller or equal to the given ones. In the other case, the zeroness
+ of the leading coefficient is returned (zero, non-zero, positive, negative) together with a
+ trimmedness theorem.
+
+ Lastly, a list of the exponent comparison results and associated theorems is also returned, so
+ that the caller can reconstruct the result of the lexicographic ordering without doing the
+ exponent comparisons again.
+*)
+fun trim_expansion_while_greater strict es fail mode ectxt (thm, basis) =
+ let
+ val (_, thm, _) = whnf_expansion ectxt thm
+ val thm = simplify_expansion ectxt thm
+ val cexp = thm |> Thm.cprop_of |> cstrip_assms |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_arg
+ val c = try_get_coeff (get_expansion thm)
+ fun lift_trimmed_thm nz thm =
+ let
+ val cexp = thm |> Thm.cprop_of |> cstrip_assms |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_arg
+ val lift_thm =
+ case nz of
+ IsNonZero => @{thm trimmed_eq_cong[rotated, OF _ lift_trimmed]}
+ | IsPos => @{thm trimmed_pos_eq_cong[rotated, OF _ lift_trimmed_pos]}
+ | IsNeg => @{thm trimmed_neg_eq_cong[rotated, OF _ lift_trimmed_neg]}
+ | _ => raise TERM ("Unexpected zeroness result in trim_expansion", [])
+ in
+ Thm.reflexive cexp RS lift_thm
+ end
+ fun trimmed_real_thm nz = Thm.reflexive cexp RS (
+ case nz of
+ IsNonZero => @{thm trimmed_eq_cong[rotated, OF _ lift_trimmed[OF trimmed_realI]]}
+ | IsPos => @{thm trimmed_pos_eq_cong[rotated, OF _ lift_trimmed_pos[OF trimmed_pos_realI]]}
+ | IsNeg => @{thm trimmed_neg_eq_cong[rotated, OF _ lift_trimmed_neg[OF trimmed_neg_realI]]}
+ | _ => raise TERM ("Unexpected zeroness result in trim_expansion", []))
+ fun do_trim es =
+ let
+ val c = the c
+ val T = fastype_of c
+ val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
+ in
+ if T = @{typ real} then (
+ case zeroness_oracle fail mode ectxt c of
+ (IsZero, SOME zero_thm) =>
+ trim_expansion_while_greater strict es fail mode ectxt
+ (@{thm drop_zero_ms'} OF [zero_thm, thm], basis)
+ | (nz, SOME nz_thm) => (thm, Trimmed (nz, SOME (nz_thm RS trimmed_real_thm nz)), [])
+ | (nz, NONE) => (thm, Trimmed (nz, NONE), []))
+ else
+ case trim_expansion_while_greater strict (Option.map tl es) fail mode ectxt
+ (thm RS @{thm expands_to_hd''}, tl_basis basis) of
+ (c_thm', Aborted ord, thms) =>
+ (@{thm expands_to_trim_cong} OF [thm, c_thm'], Aborted ord, thms)
+ | (c_thm', Trimmed (nz, trimmed_thm), thms) =>
+ let
+ val thm = (@{thm expands_to_trim_cong} OF [thm, c_thm'])
+ fun err () =
+ raise TERM ("trim_expansion: zero coefficient should have been trimmed", [c])
+ in
+ case (nz, trimmed_thm) of
+ (IsZero, _) =>
+ if #verbose (#ctxt ectxt) then
+ let
+ val ctxt = get_ctxt ectxt
+ val t' = t |> simplify_eval ctxt |> simplify_term' (get_facts ectxt) ctxt
+ val p = Pretty.str ("trim_expansion failed to recognise zeroness of " ^
+ "the following term:")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t')]
+ val _ = Pretty.writeln p
+ in
+ err ()
+ end
+ else err ()
+ | (_, SOME trimmed_thm) =>
+ (thm, Trimmed (nz, SOME (trimmed_thm RS lift_trimmed_thm nz thm)), thms)
+ | (_, NONE) => (thm, Trimmed (nz, NONE), thms)
+ end
+ end
+ val minus = @{term "(-) :: real => real => real"}
+ in
+ case (c, es) of
+ (NONE, _) => (thm, Trimmed (IsZero, NONE), [])
+ | (SOME c, SOME (e' :: _)) =>
+ let
+ val e = get_exponent (get_expansion thm)
+ in
+ case zeroness_oracle true (SOME Sgn_Trim) ectxt (minus $ e $ e') of
+ (IsPos, SOME pos_thm) => (
+ case try_drop_leading_term_ex false ectxt thm of
+ SOME thm =>
+ trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
+ | NONE => do_trim NONE |> @{apply 3(3)} (fn thms => (IsPos, pos_thm) :: thms))
+ | (IsNeg, SOME neg_thm) => (thm, Aborted LESS, [(IsNeg, neg_thm)])
+ | (IsZero, SOME zero_thm) =>
+ if not strict andalso fastype_of c = @{typ real} then
+ (thm, Aborted EQUAL, [(IsZero, zero_thm)])
+ else (
+ case try_drop_leading_term_ex false ectxt thm of
+ SOME thm => trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
+ | NONE => (do_trim es |> @{apply 3(3)} (fn thms => (IsZero, zero_thm) :: thms)))
+ | _ => do_trim NONE
+ end
+ | _ => (
+ case try_drop_leading_term_ex false ectxt thm of
+ SOME thm => trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
+ | NONE => do_trim NONE)
+ end
+
+(*
+ Trims an expansion without any stopping criterion.
+*)
+fun trim_expansion fail mode ectxt (thm, basis) =
+ case trim_expansion_while_greater false NONE fail mode ectxt (thm, basis) of
+ (thm, Trimmed (zeroness, trimmed_thm), _) => (thm, zeroness, trimmed_thm)
+ | _ => raise Match
+
+(*
+ Determines the sign of an expansion that has already been trimmed.
+*)
+fun determine_trimmed_sgn ectxt exp =
+ if fastype_of exp = @{typ real} then
+ (case zeroness_oracle true (SOME Sgn_Trim) ectxt exp of
+ (IsPos, SOME thm) => (IsPos, thm RS @{thm trimmed_pos_realI})
+ | (IsNeg, SOME thm) => (IsNeg, thm RS @{thm trimmed_neg_realI})
+ | _ => raise TERM ("determine_trimmed_sgn", []))
+ else
+ let
+ val ct = Thm.cterm_of (get_ctxt ectxt) exp
+ in
+ (case determine_trimmed_sgn ectxt (get_coeff exp) of
+ (IsPos, thm) => (IsPos, @{thm lift_trimmed_pos'} OF [thm, Thm.reflexive ct])
+ | (IsNeg, thm) => (IsNeg, @{thm lift_trimmed_neg'} OF [thm, Thm.reflexive ct])
+ | _ => raise TERM ("determine_trimmed_sgn", []))
+ end
+
+fun mk_compare_expansions_const T =
+ Const (@{const_name compare_expansions},
+ T --> T --> @{typ "cmp_result \<times> real \<times> real"})
+
+datatype comparison_result =
+ Cmp_Dominated of order * thm list * zeroness * trimmed_thm * expansion_thm * expansion_thm
+| Cmp_Asymp_Equiv of thm * thm
+
+fun compare_expansions' _ (thm1, thm2, SEmpty) = Cmp_Asymp_Equiv (thm1, thm2)
+ | compare_expansions' ectxt (thm1, thm2, basis) =
+ let
+ fun lift_trimmed_thm nz =
+ case nz of
+ IsPos => @{thm lift_trimmed_pos}
+ | IsNeg => @{thm lift_trimmed_neg}
+ | _ => raise TERM ("Unexpected zeroness result in compare_expansions'", [])
+ val (e1, e2) = apply2 (get_expansion #> get_exponent) (thm1, thm2)
+ val e = @{term "(-) :: real => _"} $ e1 $ e2
+ fun trim thm = trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis)
+ val try_drop = Option.map (whnf_expansion ectxt #> #2) o try_drop_leading_term_ex false ectxt
+ fun handle_result ord zeroness trimmed_thm thm1 thm2 =
+ let
+ val (e1, e2) = apply2 (get_expansion #> get_exponent) (thm1, thm2)
+ val e = @{term "(-) :: real => _"} $ e1 $ e2
+ val mode = if ord = LESS then Neg_Trim else Pos_Trim
+ in
+ case zeroness_oracle true (SOME mode) ectxt e of
+ (_, SOME e_thm) => Cmp_Dominated (ord, [e_thm], zeroness, trimmed_thm, thm1, thm2)
+ | _ => raise Match
+ end
+ fun recurse e_zero_thm =
+ case basis of
+ SNE (SSng _) => Cmp_Asymp_Equiv (thm1, thm2)
+ | _ =>
+ let
+ val (thm1', thm2') = apply2 (fn thm => thm RS @{thm expands_to_hd''}) (thm1, thm2)
+ val (thm1', thm2') = apply2 (whnf_expansion ectxt #> #2) (thm1', thm2')
+ in
+ case compare_expansions' ectxt (thm1', thm2', tl_basis basis) of
+ Cmp_Dominated (order, e_thms, zeroness, trimmed_thm, thm1', thm2') =>
+ Cmp_Dominated (order, e_zero_thm :: e_thms, zeroness,
+ trimmed_thm RS lift_trimmed_thm zeroness,
+ @{thm expands_to_trim_cong} OF [thm1, thm1'],
+ @{thm expands_to_trim_cong} OF [thm2, thm2'])
+ | Cmp_Asymp_Equiv (thm1', thm2') => Cmp_Asymp_Equiv
+ (@{thm expands_to_trim_cong} OF [thm1, thm1'],
+ @{thm expands_to_trim_cong} OF [thm2, thm2'])
+ end
+ in
+ case zeroness_oracle false (SOME Sgn_Trim) ectxt e of
+ (IsPos, SOME _) => (
+ case try_drop thm1 of
+ SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => (
+ case trim thm1 of
+ (thm1, zeroness, SOME trimmed_thm) =>
+ handle_result GREATER zeroness trimmed_thm thm1 thm2
+ | _ => raise TERM ("compare_expansions", map get_expansion [thm1, thm2])))
+ | (IsNeg, SOME _) => (
+ case try_drop thm2 of
+ SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => (
+ case trim thm2 of
+ (thm2, zeroness, SOME trimmed_thm) =>
+ handle_result LESS zeroness trimmed_thm thm1 thm2
+ | _ => raise TERM ("compare_expansions", map get_expansion [thm1, thm2])))
+ | (IsZero, SOME e_zero_thm) => (
+ case try_drop thm1 of
+ SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => (
+ case try_drop thm2 of
+ SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => recurse e_zero_thm))
+ | _ =>
+ case try_drop thm1 of
+ SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => (
+ case try_drop thm2 of
+ SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
+ | NONE => raise TERM ("compare_expansions", [e1, e2]))
+ end
+
+(* Uses a list of exponent comparison results to show that compare_expansions has a given result.*)
+fun prove_compare_expansions ord [thm] = (
+ case ord of
+ LESS => @{thm compare_expansions_LT_I} OF [thm]
+ | GREATER => @{thm compare_expansions_GT_I} OF [thm]
+ | EQUAL => @{thm compare_expansions_same_exp[OF _ compare_expansions_real]} OF [thm])
+ | prove_compare_expansions ord (thm :: thms) =
+ @{thm compare_expansions_same_exp} OF [thm, prove_compare_expansions ord thms]
+ | prove_compare_expansions _ [] = raise Match
+
+val ev_zero_pos_thm = Eventuallize.eventuallize @{context}
+ @{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x > 0 \<longrightarrow> f x < g x" by auto} NONE
+ OF @{thms _ expands_to_imp_eventually_pos}
+
+val ev_zero_neg_thm = Eventuallize.eventuallize @{context}
+ @{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x < 0 \<longrightarrow> f x > g x" by auto} NONE
+ OF @{thms _ expands_to_imp_eventually_neg}
+
+val ev_zero_zero_thm = Eventuallize.eventuallize @{context}
+ @{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x = 0 \<longrightarrow> f x = g x" by auto} NONE
+
+fun compare_expansions_trivial ectxt (thm1, thm2, basis) =
+ case try_prove_ev_eq ectxt (apply2 get_expanded_fun (thm1, thm2)) of
+ SOME thm => SOME (EQUAL, thm, thm1, thm2)
+ | NONE =>
+ case apply2 (ev_zeroness_oracle ectxt o get_expanded_fun) (thm1, thm2) of
+ (NONE, NONE) => NONE
+ | (SOME zero1_thm, NONE) => (
+ case trim_expansion true (SOME Sgn_Trim) ectxt (thm2, basis) of
+ (thm2, IsPos, SOME trimmed2_thm) =>
+ SOME (LESS, ev_zero_pos_thm OF
+ [zero1_thm, get_basis_wf_thm basis, thm2, trimmed2_thm], thm1, thm2)
+ | (thm2, IsNeg, SOME trimmed2_thm) =>
+ SOME (GREATER, ev_zero_neg_thm OF
+ [zero1_thm, get_basis_wf_thm basis, thm2, trimmed2_thm], thm1, thm2)
+ | _ => raise TERM ("Unexpected zeroness result in compare_expansions", []))
+ | (NONE, SOME zero2_thm) => (
+ case trim_expansion true (SOME Sgn_Trim) ectxt (thm1, basis) of
+ (thm1, IsPos, SOME trimmed1_thm) =>
+ SOME (GREATER, ev_zero_pos_thm OF
+ [zero2_thm, get_basis_wf_thm basis, thm1, trimmed1_thm], thm1, thm2)
+ | (thm1, IsNeg, SOME trimmed1_thm) =>
+ SOME (LESS, ev_zero_neg_thm OF
+ [zero2_thm, get_basis_wf_thm basis, thm1, trimmed1_thm], thm1, thm2)
+ | _ => raise TERM ("Unexpected zeroness result in compare_expansions", []))
+ | (SOME zero1_thm, SOME zero2_thm) =>
+ SOME (EQUAL, ev_zero_zero_thm OF [zero1_thm, zero2_thm] , thm1, thm2)
+
+fun compare_expansions ectxt (thm1, thm2, basis) =
+ case compare_expansions_trivial ectxt (thm1, thm2, basis) of
+ SOME res => res
+ | NONE =>
+ let
+ val (_, thm1, _) = whnf_expansion ectxt thm1
+ val (_, thm2, _) = whnf_expansion ectxt thm2
+ in
+ case compare_expansions' ectxt (thm1, thm2, basis) of
+ Cmp_Dominated (order, e_thms, zeroness, trimmed_thm, thm1, thm2) =>
+ let
+ val wf_thm = get_basis_wf_thm basis
+ val cmp_thm = prove_compare_expansions order e_thms
+ val trimmed_thm' = trimmed_thm RS
+ (if zeroness = IsPos then @{thm trimmed_pos_imp_trimmed}
+ else @{thm trimmed_neg_imp_trimmed})
+ val smallo_thm =
+ (if order = LESS then @{thm compare_expansions_LT} else @{thm compare_expansions_GT}) OF
+ [cmp_thm, trimmed_thm', thm1, thm2, wf_thm]
+ val thm' =
+ if zeroness = IsPos then @{thm smallo_trimmed_imp_eventually_less}
+ else @{thm smallo_trimmed_imp_eventually_greater}
+ val result_thm =
+ thm' OF [smallo_thm, if order = LESS then thm2 else thm1, wf_thm, trimmed_thm]
+ in
+ (order, result_thm, thm1, thm2)
+ end
+ | Cmp_Asymp_Equiv (thm1, thm2) =>
+ let
+ val thm = @{thm expands_to_minus} OF [get_basis_wf_thm basis, thm1, thm2]
+ val (order, result_thm) =
+ case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
+ (thm, IsPos, SOME pos_thm) => (GREATER,
+ @{thm expands_to_imp_eventually_gt} OF [get_basis_wf_thm basis, thm, pos_thm])
+ | (thm, IsNeg, SOME neg_thm) => (LESS,
+ @{thm expands_to_imp_eventually_lt} OF [get_basis_wf_thm basis, thm, neg_thm])
+ | _ => raise TERM ("Unexpected zeroness result in prove_eventually_less", [])
+ in
+ (order, result_thm, thm1, thm2)
+ end
+ end
+
+
+
+(*
+ Throws an exception and prints an error message indicating that the leading term could
+ not be determined to be either zero or non-zero.
+*)
+fun raise_trimming_error ectxt thm =
+ let
+ val ctxt = get_ctxt ectxt
+ fun lead_coeff exp =
+ if fastype_of exp = @{typ real} then exp else lead_coeff (get_coeff exp)
+ val c = lead_coeff (get_expansion thm)
+ fun err () =
+ let
+ val t = simplify_term' (get_facts ectxt) ctxt c
+ val _ =
+ if #verbose (#ctxt ectxt) then
+ let
+ val p = Pretty.str
+ ("real_asymp failed to determine whether the following constant is zero:")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
+ in
+ Pretty.writeln p
+ end else ()
+ in
+ raise TERM ("zeroness_oracle", [t])
+ end
+ in
+ err ()
+ end
+
+
+(* TODO Here be dragons *)
+fun solve_eval_eq thm =
+ case try (fn _ => @{thm refl} RS thm) () of
+ SOME thm' => thm'
+ | NONE =>
+ case try (fn _ => @{thm eval_real_def} RS thm) () of
+ SOME thm' => thm'
+ | NONE => @{thm eval_ms.simps} RS thm
+
+(*
+ Returns an expansion theorem for the logarithm of the given expansion.
+ May add one additional element to the basis at the end.
+*)
+fun ln_expansion _ _ _ SEmpty = raise TERM ("ln_expansion: empty basis", [])
+ | ln_expansion ectxt trimmed_thm thm (SNE basis) =
+ let
+ fun trailing_exponent expr (SSng _) = get_exponent expr
+ | trailing_exponent expr (SCons (_, _, tl)) = trailing_exponent (get_coeff expr) tl
+ val e = trailing_exponent (get_expansion thm) basis
+ fun ln_expansion_aux trimmed_thm zero_thm thm basis =
+ let
+ val t = betapply (@{term "\<lambda>(f::real \<Rightarrow> real) x. f x - 1 :: real"}, get_expanded_fun thm)
+ in
+ case ev_zeroness_oracle ectxt t of
+ NONE => ln_expansion_aux' trimmed_thm zero_thm thm basis
+ | SOME zero_thm =>
+ @{thm expands_to_ln_eventually_1} OF
+ [get_basis_wf_thm' basis, mk_expansion_level_eq_thm' basis, zero_thm]
+ end
+ and ln_expansion_aux' trimmed_thm zero_thm thm (SSng {wf_thm, ...}) =
+ ( @{thm expands_to_ln} OF
+ [trimmed_thm, wf_thm, thm,
+ @{thm expands_to_ln_aux_0} OF [zero_thm, @{thm expands_to_ln_const}]])
+ |> solve_eval_eq
+ | ln_expansion_aux' trimmed_thm zero_thm thm (SCons ({wf_thm, ...}, {ln_thm, ...}, basis')) =
+ let
+ val c_thm =
+ ln_expansion_aux (trimmed_thm RS @{thm trimmed_pos_hd_coeff}) zero_thm
+ (expands_to_hd thm) basis'
+ val e = get_exponent (get_expansion thm)
+ val c_thm' =
+ case zeroness_oracle true NONE ectxt e of
+ (IsZero, SOME thm) =>
+ @{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux_0]} OF [thm,c_thm]
+ | _ =>
+ case try_prove_real_eq false ectxt (e, @{term "1::real"}) of
+ SOME thm =>
+ @{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux_1]}
+ OF [thm, wf_thm, c_thm, ln_thm]
+ | NONE =>
+ @{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux]}
+ OF [wf_thm, c_thm, ln_thm]
+ in
+ (@{thm expands_to_ln} OF [trimmed_thm, wf_thm, thm, c_thm'])
+ |> solve_eval_eq
+ end
+ in
+ case zeroness_oracle true NONE ectxt e of
+ (IsZero, SOME zero_thm) => (ln_expansion_aux trimmed_thm zero_thm thm basis, SNE basis)
+ | _ =>
+ let
+ val basis' = insert_ln (SNE basis)
+ val lifting = mk_lifting (get_basis_list' basis) basis'
+ val thm' = lift_expands_to_thm lifting thm
+ val trimmed_thm' = lift_trimmed_pos_thm lifting trimmed_thm
+ val (thm'', eq_thm) = retrim_expansion ectxt (thm', basis')
+ val trimmed_thm'' = @{thm trimmed_pos_eq_cong} OF [trimmed_thm', eq_thm]
+ in
+ ln_expansion ectxt trimmed_thm'' thm'' basis'
+ end
+ end
+
+(*
+ Handles a possible basis change after expanding exp(c(x)) for an expansion of the form
+ f(x) = c(x) + g(x). Expanding exp(c(x)) may have inserted an additional basis element. If the
+ old basis was b :: bs (i.e. c is an expansion w.r.t. bs) and the updated one is bs' (which
+ agrees with bs except for one additional element b'), we need to argue that b :: bs' is still
+ well-formed. This may require us to show that ln(b') is o(ln(b)), which the function takes
+ as an argument.
+*)
+fun adjust_exp_basis basis basis' ln_smallo_thm =
+ if length (get_basis_list basis) = length (get_basis_list basis') + 1 then
+ basis
+ else
+ let
+ val SNE (SCons (info, ln_info, tail)) = basis
+ val SNE tail' = basis'
+ val wf_thms = map get_basis_wf_thm [basis, basis']
+ val wf_thm' =
+ case
+ get_first (fn f => try f ())
+ [fn _ => @{thm basis_wf_lift_modification} OF wf_thms,
+ fn _ => @{thm basis_wf_insert_exp_near} OF (wf_thms @ [ln_smallo_thm]),
+ fn _ => @{thm basis_wf_insert_exp_near} OF (wf_thms @
+ [ln_smallo_thm RS @{thm basis_wf_insert_exp_uminus'}])] of
+ SOME wf_thm => wf_thm
+ | _ => raise TERM ("Lifting basis modification in exp_expansion failed.", map Thm.concl_of (wf_thms @ [ln_smallo_thm]))
+ val info' = {wf_thm = wf_thm', head = #head info}
+ val lifting = mk_lifting (get_basis_list' tail) basis'
+ val ln_info' =
+ {trimmed_thm = lift_trimmed_pos_thm lifting (#trimmed_thm ln_info),
+ ln_thm = lift_expands_to_thm lifting (#ln_thm ln_info)}
+ in
+ SNE (SCons (info', ln_info', tail'))
+ end
+
+(* inserts the exponential of a given function at the beginning of the given basis *)
+fun insert_exp _ _ _ _ _ SEmpty = raise TERM ("insert_exp", [])
+ | insert_exp t ln_thm ln_smallo_thm ln_trimmed_thm lim_thm (SNE basis) =
+ let
+ val head = Envir.beta_eta_contract (@{term "\<lambda>(f::real\<Rightarrow>real) x. exp (f x)"} $ t)
+ val ln_smallo_thm = ln_smallo_thm RS @{thm ln_smallo_ln_exp}
+ val wf_thm = @{thm basis_wf_manyI} OF [lim_thm, ln_smallo_thm, get_basis_wf_thm' basis]
+ val basis' = SNE (SCons ({wf_thm = wf_thm, head = head},
+ {ln_thm = ln_thm, trimmed_thm = ln_trimmed_thm} , basis))
+ in
+ check_basis basis'
+ end
+
+(*
+ Returns an expansion of the exponential of the given expansion. This may add several
+ new basis elements at any position of the basis (except at the very end
+*)
+fun exp_expansion _ thm SEmpty = (thm RS @{thm expands_to_exp_real}, SEmpty)
+ | exp_expansion ectxt thm basis =
+ let
+ val (_, thm, _) = whnf_expansion ectxt thm
+ in
+ case ev_zeroness_oracle ectxt (get_eval (get_expansion thm)) of
+ SOME zero_thm =>
+ (@{thm expands_to_exp_zero} OF
+ [thm, zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)
+ | NONE =>
+ let
+ val ln =
+ Option.map (fn x => (#ln_thm x, #trimmed_thm x)) (get_ln_info basis)
+ val ln = Option.map (fn (x, y) => retrim_pos_expansion ectxt (x, basis, y)) ln
+ val es' = @{term "0::real"} :: (
+ case ln of
+ NONE => []
+ | SOME (ln_thm, _, _) => get_exponents (get_expansion ln_thm))
+ val trim_result =
+ trim_expansion_while_greater true (SOME es') false (SOME Simple_Trim) ectxt (thm, basis)
+ in
+ exp_expansion' ectxt trim_result ln basis
+ end
+ end
+and exp_expansion' _ (thm, _, _) _ SEmpty = (thm RS @{thm expands_to_exp_real}, SEmpty)
+ | exp_expansion' ectxt (thm, trim_result, e_thms) ln basis =
+ let
+ val exp = get_expansion thm
+ val wf_thm = get_basis_wf_thm basis
+ val f = get_expanded_fun thm
+ fun exp_expansion_insert ln_smallo_thm = (
+ case determine_trimmed_sgn ectxt exp of
+ (IsPos, trimmed_thm) =>
+ let
+ val [lim_thm, ln_thm', thm'] =
+ @{thms expands_to_exp_insert_pos}
+ |> map (fn thm' => thm' OF [thm, wf_thm, trimmed_thm, ln_smallo_thm])
+ val basis' = insert_exp f ln_thm' ln_smallo_thm trimmed_thm lim_thm basis
+ in
+ (thm', basis')
+ end
+ | (IsNeg, trimmed_thm) =>
+ let
+ val [lim_thm, ln_thm', ln_trimmed_thm, thm'] =
+ @{thms expands_to_exp_insert_neg}
+ |> map (fn thm' => thm' OF [thm, wf_thm, trimmed_thm, ln_smallo_thm])
+ val ln_smallo_thm = ln_smallo_thm RS @{thm basis_wf_insert_exp_uminus}
+ val f' = Envir.beta_eta_contract (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"} $ f)
+ val basis' = insert_exp f' ln_thm' ln_smallo_thm ln_trimmed_thm lim_thm basis
+ in
+ (thm', basis')
+ end
+ | _ => raise TERM ("Unexpected zeroness result in exp_expansion", []))
+ fun lexord (IsNeg :: _) = LESS
+ | lexord (IsPos :: _) = GREATER
+ | lexord (IsZero :: xs) = lexord xs
+ | lexord [] = EQUAL
+ | lexord _ = raise Match
+ val compare_result = lexord (map fst e_thms)
+ in
+ case (trim_result, e_thms, compare_result) of
+ (Aborted _, (IsNeg, e_neg_thm) :: _, _) =>
+ (* leading exponent is negative; we can simply Taylor-expand exp(x) around 0 *)
+ (@{thm expands_to_exp_neg} OF [thm, get_basis_wf_thm basis, e_neg_thm], basis)
+ | (Trimmed (_, SOME trimmed_thm), (IsPos, e_pos_thm) :: _, GREATER) =>
+ (* leading exponent is positive; exp(f(x)) or exp(-f(x)) is new basis element *)
+ let
+ val ln_smallo_thm =
+ @{thm basis_wf_insert_exp_pos} OF [thm, get_basis_wf_thm basis, trimmed_thm, e_pos_thm]
+ in
+ exp_expansion_insert ln_smallo_thm
+ end
+ | (Trimmed (_, SOME trimmed_thm), _, GREATER) =>
+ (* leading exponent is zero, but f(x) grows faster than ln(b(x)), so
+ exp(f(x)) or exp(-f(x)) must still be new basis elements *)
+ let
+ val ln_thm =
+ case ln of
+ SOME (ln_thm, _, _) => ln_thm
+ | NONE => raise TERM ("TODO blubb", [])
+ val ln_thm = @{thm expands_to_lift''} OF [get_basis_wf_thm basis, ln_thm]
+ val ln_smallo_thm =
+ @{thm compare_expansions_GT} OF [prove_compare_expansions GREATER (map snd e_thms),
+ trimmed_thm, thm, ln_thm, get_basis_wf_thm basis]
+ in
+ exp_expansion_insert ln_smallo_thm
+ end
+ | (Aborted LESS, (IsZero, e_zero_thm) :: e_thms', _) =>
+ (* leading exponent is zero and f(x) grows more slowly than ln(b(x)), so
+ we can write f(x) = c(x) + g(x) and therefore exp(f(x)) = exp(c(x)) * exp(g(x)).
+ The former is treated by a recursive call; the latter by Taylor expansion. *)
+ let
+ val (ln_thm, trimmed_thm) =
+ case ln of
+ SOME (ln_thm, _, trimmed_thm) =>
+ (ln_thm, trimmed_thm RS @{thm trimmed_pos_imp_trimmed})
+ | NONE => raise TERM ("TODO foo", [])
+ val c_thm = expands_to_hd thm
+ val ln_smallo_thm =
+ @{thm compare_expansions_LT} OF [prove_compare_expansions LESS (map snd e_thms'),
+ trimmed_thm, c_thm, ln_thm, get_basis_wf_thm (tl_basis basis)]
+ val (c_thm, c_basis) = exp_expansion ectxt c_thm (tl_basis basis)
+ val basis' = adjust_exp_basis basis c_basis ln_smallo_thm
+ val wf_thm = get_basis_wf_thm basis'
+ val thm' = lift basis' thm
+ val (thm'', _) = retrim_expansion ectxt (thm', basis')
+ in
+ (@{thm expands_to_exp_0} OF [thm'', wf_thm, e_zero_thm, c_thm], basis')
+ end
+ | (Trimmed _, [(IsZero, e_zero_thm)], EQUAL) =>
+ (* f(x) can be written as c + g(x) where c is just a real constant.
+ We can therefore write exp(f(x)) = exp(c) * exp(g(x)), where the latter is
+ a simple Taylor expansion. *)
+ (@{thm expands_to_exp_0_real} OF [thm, wf_thm, e_zero_thm], basis)
+ | (Trimmed _, (_, e_zero_thm) :: _, EQUAL) =>
+ (* f(x) is asymptotically equivalent to c * ln(b(x)), so we can write f(x) as
+ c * ln(b(x)) + g(x) and therefore exp(f(x)) = b(x)^c * exp(g(x)). The second
+ factor is handled by a recursive call *)
+ let
+ val ln_thm =
+ case ln of
+ SOME (ln_thm, _, _) => ln_thm
+ | NONE => raise TERM ("TODO blargh", [])
+ val c =
+ case (thm, ln_thm) |> apply2 (get_expansion #> get_lead_coeff) of
+ (c1, c2) => @{term "(/) :: real => _"} $ c1 $ c2
+ val c = Thm.cterm_of (get_ctxt ectxt) c
+
+ val thm' =
+ @{thm expands_to_exp_0_pull_out1}
+ OF [thm, ln_thm, wf_thm, e_zero_thm, Thm.reflexive c]
+ val (thm'', basis') = exp_expansion ectxt thm' basis
+ val pat = ConsPat ("MS", [AnyPat ("_", 0), AnyPat ("_", 0)])
+ val (_, _, conv) = match ectxt pat (get_expansion thm'') (SOME [])
+ val eq_thm = conv (Thm.cterm_of (get_ctxt ectxt) (get_expansion thm''))
+ val thm''' = @{thm expands_to_meta_eq_cong} OF [thm'', eq_thm]
+ val thm'''' =
+ case get_intyness (get_ctxt ectxt) c of
+ No_Nat =>
+ @{thm expands_to_exp_0_pull_out2} OF [thm''', get_basis_wf_thm basis']
+ | Nat nat_thm =>
+ @{thm expands_to_exp_0_pull_out2_nat} OF
+ [thm''', get_basis_wf_thm basis', nat_thm]
+ | Neg_Nat nat_thm =>
+ @{thm expands_to_exp_0_pull_out2_neg_nat} OF
+ [thm''', get_basis_wf_thm basis', nat_thm]
+ in
+ (thm'''', basis')
+ end
+ | (Trimmed (IsZero, _), [], _) =>
+ (* Expansion is empty, i.e. f(x) is identically zero *)
+ (@{thm expands_to_exp_MSLNil} OF [thm, get_basis_wf_thm basis], basis)
+ | (Trimmed (_, NONE), _, GREATER) =>
+ (* We could not determine whether f(x) grows faster than ln(b(x)) or not. *)
+ raise_trimming_error ectxt thm
+ | _ => raise Match
+ end
+
+fun powr_expansion ectxt (thm1, thm2, basis) =
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm1) of
+ SOME zero_thm =>
+ (@{thm expands_to_powr_0} OF
+ [zero_thm, Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) (get_expanded_fun thm2)),
+ get_basis_wf_thm basis, mk_expansion_level_eq_thm basis],
+ basis)
+ | NONE =>
+ let
+ val (thm1, _, SOME trimmed_thm) =
+ trim_expansion true (SOME Pos_Trim) ectxt (thm1, basis)
+ val (ln_thm, basis') = ln_expansion ectxt trimmed_thm thm1 basis
+ val thm2' = lift basis' thm2 |> simplify_expansion ectxt
+ val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis', ln_thm, thm2']
+ val (exp_thm, basis'') = exp_expansion ectxt mult_thm basis'
+ val thm = @{thm expands_to_powr} OF
+ [trimmed_thm, get_basis_wf_thm basis, thm1, exp_thm]
+ in
+ (thm, basis'')
+ end
+
+fun powr_nat_expansion ectxt (thm1, thm2, basis) =
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm1) of
+ SOME zero_thm => (
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm2) of
+ SOME zero'_thm => (@{thm expands_to_powr_nat_0_0} OF
+ [zero_thm, zero'_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)
+ | NONE => (
+ case trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis) of
+ (thm2, _, SOME trimmed_thm) =>
+ (@{thm expands_to_powr_nat_0} OF [zero_thm, thm2, trimmed_thm,
+ get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)))
+ | NONE =>
+ let
+ val (thm1, _, SOME trimmed_thm) =
+ trim_expansion true (SOME Pos_Trim) ectxt (thm1, basis)
+ val (ln_thm, basis') = ln_expansion ectxt trimmed_thm thm1 basis
+ val thm2' = lift basis' thm2 |> simplify_expansion ectxt
+ val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis', ln_thm, thm2']
+ val (exp_thm, basis'') = exp_expansion ectxt mult_thm basis'
+ val thm = @{thm expands_to_powr_nat} OF
+ [trimmed_thm, get_basis_wf_thm basis, thm1, exp_thm]
+ in
+ (thm, basis'')
+ end
+
+fun is_numeral t =
+ let
+ val _ = HOLogic.dest_number t
+ in
+ true
+ end
+ handle TERM _ => false
+
+fun power_expansion ectxt (thm, n, basis) =
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => @{thm expands_to_power_0} OF
+ [zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis,
+ Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) n)]
+ | NONE => (
+ case trim_expansion true (SOME Simple_Trim) ectxt (thm, basis) of
+ (thm', _, SOME trimmed_thm) =>
+ let
+ val ctxt = get_ctxt ectxt
+ val thm =
+ if is_numeral n then @{thm expands_to_power[where abort = True]}
+ else @{thm expands_to_power[where abort = False]}
+ val thm =
+ Drule.infer_instantiate' ctxt [NONE, NONE, NONE, SOME (Thm.cterm_of ctxt n)] thm
+ in
+ thm OF [trimmed_thm, get_basis_wf_thm basis, thm']
+ end
+ | _ => raise TERM ("Unexpected zeroness result in power_expansion", []))
+
+fun powr_const_expansion ectxt (thm, p, basis) =
+ let
+ val pthm = Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) p)
+ in
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => @{thm expands_to_powr_const_0} OF
+ [zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis, pthm]
+ | NONE =>
+ case trim_expansion true (SOME Pos_Trim) ectxt (thm, basis) of
+ (_, _, NONE) => raise TERM ("Unexpected zeroness result for powr", [])
+ | (thm, _, SOME trimmed_thm) =>
+ (if is_numeral p then @{thm expands_to_powr_const[where abort = True]}
+ else @{thm expands_to_powr_const[where abort = False]})
+ OF [trimmed_thm, get_basis_wf_thm basis, thm, pthm]
+ end
+
+fun sgn_expansion ectxt (thm, basis) =
+ let
+ val thms = [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
+ in
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => @{thm expands_to_sgn_zero} OF (zero_thm :: thms)
+ | NONE =>
+ case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
+ (thm, IsPos, SOME trimmed_thm) =>
+ @{thm expands_to_sgn_pos} OF ([trimmed_thm, thm] @ thms)
+ | (thm, IsNeg, SOME trimmed_thm) =>
+ @{thm expands_to_sgn_neg} OF ([trimmed_thm, thm] @ thms)
+ | _ => raise TERM ("Unexpected zeroness result in sgn_expansion", [])
+ end
+
+(*
+ Returns an expansion of the sine and cosine of the given expansion. Fails if that function
+ goes to infinity.
+*)
+fun sin_cos_expansion _ thm SEmpty =
+ (thm RS @{thm expands_to_sin_real}, thm RS @{thm expands_to_cos_real})
+ | sin_cos_expansion ectxt thm basis =
+ let
+ val exp = get_expansion thm
+ val e = get_exponent exp
+ in
+ case zeroness_oracle true (SOME Sgn_Trim) ectxt e of
+ (IsPos, _) => raise THM ("sin_cos_expansion", 0, [thm])
+ | (IsNeg, SOME e_thm) =>
+ let
+ val [thm1, thm2] =
+ map (fn thm' => thm' OF [e_thm, get_basis_wf_thm basis, thm])
+ @{thms expands_to_sin_ms_neg_exp expands_to_cos_ms_neg_exp}
+ in
+ (thm1, thm2)
+ end
+ | (IsZero, SOME e_thm) =>
+ let
+ val (sin_thm, cos_thm) = (sin_cos_expansion ectxt (expands_to_hd thm) (tl_basis basis))
+ fun mk_thm thm' =
+ (thm' OF [e_thm, get_basis_wf_thm basis, thm, sin_thm, cos_thm]) |> solve_eval_eq
+ val [thm1, thm2] =
+ map mk_thm @{thms expands_to_sin_ms_zero_exp expands_to_cos_ms_zero_exp}
+ in
+ (thm1, thm2)
+ end
+ | _ => raise TERM ("Unexpected zeroness result in sin_exp_expansion", [])
+ end
+
+fun abconv (t, t') = Envir.beta_eta_contract t aconv Envir.beta_eta_contract t'
+
+(*
+ Makes sure that an expansion theorem really talks about the right function.
+ This is basically a sanity check to make things fail early and in the right place.
+*)
+fun check_expansion e thm =
+ if abconv (expr_to_term e, get_expanded_fun thm) then
+ thm
+ else
+(* TODO Remove Debugging stuff *)
+let val _ = @{print} e
+val _ = @{print} (get_expanded_fun thm)
+in
+ raise TERM ("check_expansion", [Thm.concl_of thm, expr_to_term e])
+end
+
+fun minmax_expansion max [less_thm, eq_thm, gt_thm] ectxt (thm1, thm2, basis) = (
+ case compare_expansions ectxt (thm1, thm2, basis) of
+ (LESS, less_thm', thm1, thm2) => less_thm OF [if max then thm2 else thm1, less_thm']
+ | (GREATER, gt_thm', thm1, thm2) => gt_thm OF [if max then thm1 else thm2, gt_thm']
+ | (EQUAL, eq_thm', thm1, _) => eq_thm OF [thm1, eq_thm'])
+ | minmax_expansion _ _ _ _ = raise Match
+
+val min_expansion =
+ minmax_expansion false @{thms expands_to_min_lt expands_to_min_eq expands_to_min_gt}
+val max_expansion =
+ minmax_expansion true @{thms expands_to_max_lt expands_to_max_eq expands_to_max_gt}
+
+fun zero_expansion basis =
+ @{thm expands_to_zero} OF [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
+
+fun const_expansion _ basis @{term "0 :: real"} = zero_expansion basis
+ | const_expansion ectxt basis t =
+ let
+ val ctxt = get_ctxt ectxt
+ val thm = Drule.infer_instantiate' ctxt [NONE, SOME (Thm.cterm_of ctxt t)]
+ @{thm expands_to_const}
+ in
+ thm OF [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
+ end
+
+fun root_expansion ectxt (thm, n, basis) =
+ let
+ val ctxt = get_ctxt ectxt
+ fun tac {context = ctxt, ...} =
+ HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
+ THEN Local_Defs.unfold_tac ctxt eval_simps
+ THEN HEADGOAL (Simplifier.asm_full_simp_tac ctxt)
+ fun prove goal =
+ try (Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (Term.betapply (goal, n)))) tac
+ fun err () =
+ let
+ val t = simplify_term' (get_facts ectxt) ctxt n
+ val _ =
+ if #verbose (#ctxt ectxt) then
+ let
+ val p = Pretty.str ("real_asymp failed to determine whether the following constant " ^
+ "is zero or not:")
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
+ in
+ Pretty.writeln p
+ end else ()
+ in
+ raise TERM ("zeroness_oracle", [n])
+ end
+ fun aux nz_thm =
+ case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
+ (thm, IsPos, SOME trimmed_thm) =>
+ @{thm expands_to_root} OF [nz_thm, trimmed_thm, get_basis_wf_thm basis, thm]
+ | (thm, IsNeg, SOME trimmed_thm) =>
+ @{thm expands_to_root_neg} OF [nz_thm, trimmed_thm, get_basis_wf_thm basis, thm]
+ | _ => raise TERM ("Unexpected zeroness result in root_expansion", [])
+ in
+ case prove @{term "\<lambda>n::nat. n = 0"} of
+ SOME zero_thm =>
+ @{thm expands_to_0th_root} OF
+ [zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis,
+ Thm.reflexive (Thm.cterm_of ctxt (get_expanded_fun thm))]
+ | NONE =>
+ case prove @{term "\<lambda>n::nat. n > 0"} of
+ NONE => err ()
+ | SOME nz_thm =>
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => @{thm expands_to_root_0} OF
+ [nz_thm, zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
+ | NONE => aux nz_thm
+ end
+
+
+fun arctan_expansion _ SEmpty thm =
+ @{thm expands_to_real_compose[where g = arctan]} OF [thm]
+ | arctan_expansion ectxt basis thm =
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => @{thm expands_to_arctan_zero} OF [zero_expansion basis, zero_thm]
+ | NONE =>
+ let
+ val (thm, _, _) = trim_expansion true (SOME Simple_Trim) ectxt (thm, basis)
+ val e = get_exponent (get_expansion thm)
+ fun cont ectxt (thm, basis) = arctan_expansion ectxt basis thm
+ in
+ case zeroness_oracle true (SOME Sgn_Trim) ectxt e of
+ (IsNeg, SOME neg_thm) =>
+ @{thm expands_to_arctan_ms_neg_exp} OF [neg_thm, get_basis_wf_thm basis, thm]
+ | (IsPos, SOME e_pos_thm) => (
+ case determine_trimmed_sgn ectxt (get_expansion thm) of
+ (IsPos, trimmed_thm) =>
+ @{thm expands_to_arctan_ms_pos_exp_pos} OF
+ [e_pos_thm, trimmed_thm, get_basis_wf_thm basis, thm]
+ | (IsNeg, trimmed_thm) =>
+ @{thm expands_to_arctan_ms_pos_exp_neg} OF
+ [e_pos_thm, trimmed_thm, get_basis_wf_thm basis, thm]
+ | _ => raise TERM ("Unexpected trim result during expansion of arctan", []))
+ | (IsZero, _) => (
+ case try_lift_function ectxt (thm, basis) cont of
+ (SOME thm', _) => thm'
+ | _ =>
+ let
+ val _ = if get_verbose ectxt then
+ writeln "Unsupported occurrence of arctan" else ()
+ in
+ raise TERM ("Unsupported occurence of arctan", [])
+ end)
+ | _ => raise TERM ("Unexpected trim result during expansion of arctan", [])
+ end
+
+(* Returns an expansion theorem for a function that is already a basis element *)
+fun expand_basic _ t SEmpty = raise TERM ("expand_basic", [t])
+ | expand_basic thm t basis =
+ if abconv (get_basis_head basis, t) then
+ thm (get_basis_wf_thm basis) (mk_expansion_level_eq_thm (tl_basis basis))
+ else
+ @{thm expands_to_lift'} OF [get_basis_wf_thm basis, expand_basic thm t (tl_basis basis)]
+
+fun expand_unary ectxt thm e basis =
+ let
+ val (thm', basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
+ in
+ (thm OF [get_basis_wf_thm basis', thm'], basis')
+ end
+and expand_binary ectxt thm (e1, e2) basis =
+ let
+ val (thm1, basis') = expand' ectxt e1 basis |> apfst (simplify_expansion ectxt)
+ val (thm2, basis'') = expand' ectxt e2 basis' |> apfst (simplify_expansion ectxt)
+ val thm1 = lift basis'' thm1 |> simplify_expansion ectxt
+ in
+ (thm OF [get_basis_wf_thm basis'', thm1, thm2], basis'')
+ end
+and trim_nz mode ectxt e basis =
+ let
+ val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
+ val (thm', nz, trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis')
+ in
+ case trimmed_thm of
+ NONE => raise TERM ("expand: zero denominator", [get_expansion thm])
+ | SOME trimmed_thm => (thm', basis', nz, trimmed_thm)
+ end
+and expand'' ectxt (ConstExpr c) basis = (const_expansion ectxt basis c, basis)
+ | expand'' _ X basis = (lift basis @{thm expands_to_X}, basis)
+ | expand'' ectxt (Uminus e) basis = expand_unary ectxt @{thm expands_to_uminus} e basis
+ | expand'' ectxt (Add e12) basis = expand_binary ectxt @{thm expands_to_add} e12 basis
+ | expand'' ectxt (Minus e12) basis = expand_binary ectxt @{thm expands_to_minus} e12 basis
+ | expand'' ectxt (Mult e12) basis = expand_binary ectxt @{thm expands_to_mult} e12 basis
+ | expand'' ectxt (Powr' (e, p)) basis = (* TODO zero basis *)
+ let
+ val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
+ in
+ (powr_const_expansion ectxt (thm, p, basis'), basis')
+ end
+ | expand'' ectxt (Powr (e1, e2)) basis =
+ let
+ val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
+ val (thm1, basis2) = expand' ectxt e1 basis1 |> apfst (simplify_expansion ectxt)
+ in
+ powr_expansion ectxt (thm1, thm2, basis2)
+ end
+ | expand'' ectxt (Powr_Nat (e1, e2)) basis =
+ let
+ val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
+ val (thm1, basis2) = expand' ectxt e1 basis1 |> apfst (simplify_expansion ectxt)
+ in
+ powr_nat_expansion ectxt (thm1, thm2, basis2)
+ end
+ | expand'' ectxt (LnPowr (e1, e2)) basis =
+ let (* TODO zero base *)
+ val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
+ val (thm1, basis2, _, trimmed_thm) = trim_nz Pos_Trim ectxt e1 basis1
+ val (ln_thm, basis3) = ln_expansion ectxt trimmed_thm thm1 basis2
+ val thm2' = lift basis3 thm2 |> simplify_expansion ectxt
+ val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis3, ln_thm, thm2']
+ val thm = @{thm expands_to_ln_powr} OF
+ [trimmed_thm, get_basis_wf_thm basis2, thm1, mult_thm]
+ in
+ (thm, basis3)
+ end
+ | expand'' ectxt (ExpLn e) basis =
+ let
+ val (thm, basis', _, trimmed_thm) = trim_nz Pos_Trim ectxt e basis
+ val thm = @{thm expands_to_exp_ln} OF [trimmed_thm, get_basis_wf_thm basis', thm]
+ in
+ (thm, basis')
+ end
+ | expand'' ectxt (Power (e, n)) basis =
+ let
+ val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
+ in
+ (power_expansion ectxt (thm, n, basis'), basis')
+ end
+ | expand'' ectxt (Root (e, n)) basis =
+ let
+ val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
+ in
+ (root_expansion ectxt (thm, n, basis'), basis')
+ end
+ | expand'' ectxt (Inverse e) basis =
+ (case trim_nz Simple_Trim ectxt e basis of
+ (thm, basis', _, trimmed_thm) =>
+ (@{thm expands_to_inverse} OF [trimmed_thm, get_basis_wf_thm basis', thm], basis'))
+ | expand'' ectxt (Div (e1, e2)) basis =
+ let
+ val (thm1, basis') = expand' ectxt e1 basis
+ val (thm2, basis'', _, trimmed_thm) = trim_nz Simple_Trim ectxt e2 basis'
+ val thm1 = lift basis'' thm1
+ in
+ (@{thm expands_to_divide} OF [trimmed_thm, get_basis_wf_thm basis'', thm1, thm2], basis'')
+ end
+ | expand'' ectxt (Ln e) basis =
+ let
+ val (thm, basis', _, trimmed_thm) = trim_nz Pos_Trim ectxt e basis
+ in
+ ln_expansion ectxt trimmed_thm thm basis'
+ end
+ | expand'' ectxt (Exp e) basis =
+ let
+ val (thm, basis') = expand' ectxt e basis
+ in
+ exp_expansion ectxt thm basis'
+ end
+ | expand'' ectxt (Absolute e) basis =
+ let
+ val (thm, basis', nz, trimmed_thm) = trim_nz Sgn_Trim ectxt e basis
+ val thm' =
+ case nz of
+ IsPos => @{thm expands_to_abs_pos}
+ | IsNeg => @{thm expands_to_abs_neg}
+ | _ => raise TERM ("Unexpected trim result during expansion of abs", [])
+ in
+ (thm' OF [trimmed_thm, get_basis_wf_thm basis', thm], basis')
+ end
+ | expand'' ectxt (Sgn e) basis =
+ let
+ val (thm, basis') = expand' ectxt e basis
+ in
+ (sgn_expansion ectxt (thm, basis'), basis')
+ end
+ | expand'' ectxt (Min (e1, e2)) basis = (
+ case try_prove_ev_eq ectxt (apply2 expr_to_term (e1, e2)) of
+ SOME eq_thm =>
+ expand' ectxt e1 basis
+ |> apfst (fn thm => @{thm expands_to_min_eq} OF [thm, eq_thm])
+ | NONE =>
+ let
+ val (thm1, basis') = expand' ectxt e1 basis
+ val (thm2, basis'') = expand' ectxt e2 basis'
+ val thm1' = lift basis'' thm1
+ in
+ (min_expansion ectxt (thm1', thm2, basis''), basis'')
+ end)
+ | expand'' ectxt (Max (e1, e2)) basis = (
+ case try_prove_ev_eq ectxt (apply2 expr_to_term (e1, e2)) of
+ SOME eq_thm =>
+ expand' ectxt e1 basis
+ |> apfst (fn thm => @{thm expands_to_max_eq} OF [thm, eq_thm])
+ | NONE =>
+ let
+ val (thm1, basis') = expand' ectxt e1 basis
+ val (thm2, basis'') = expand' ectxt e2 basis'
+ val thm1' = lift basis'' thm1
+ in
+ (max_expansion ectxt (thm1', thm2, basis''), basis'')
+ end)
+ | expand'' ectxt (Sin e) basis =
+ let
+ val (thm, basis', _, _) = trim_nz Simple_Trim ectxt e basis (* TODO could be relaxed *)
+ in
+ (sin_cos_expansion ectxt thm basis' |> fst, basis')
+ end
+ | expand'' ectxt (Cos e) basis =
+ let
+ val (thm, basis', _, _) = trim_nz Simple_Trim ectxt e basis (* TODO could be relaxed *)
+ in
+ (sin_cos_expansion ectxt thm basis' |> snd, basis')
+ end
+ | expand'' _ (Floor _) _ =
+ raise TERM ("floor not supported.", [])
+ | expand'' _ (Ceiling _) _ =
+ raise TERM ("ceiling not supported.", [])
+ | expand'' _ (Frac _) _ =
+ raise TERM ("frac not supported.", [])
+ | expand'' _ (NatMod _) _ =
+ raise TERM ("mod not supported.", [])
+ | expand'' ectxt (ArcTan e) basis =
+ let
+ (* TODO: what if it's zero *)
+ val (thm, basis') = expand' ectxt e basis
+ in
+ (arctan_expansion ectxt basis' thm, basis')
+ end
+ | expand'' ectxt (Custom (name, t, args)) basis =
+ let
+ fun expand_args acc basis [] = (rev acc, basis)
+ | expand_args acc basis (arg :: args) =
+ case expand' ectxt arg basis of
+ (thm, basis') => expand_args (thm :: acc) basis' args
+ in
+ case expand_custom (get_ctxt ectxt) name of
+ NONE => raise TERM ("Unsupported custom function: " ^ name, [t])
+ | SOME e => e ectxt t (expand_args [] basis args)
+ end
+
+and expand' ectxt (e' as (Inverse e)) basis =
+ let
+ val t = expr_to_term e
+ fun thm wf_thm len_thm =
+ @{thm expands_to_basic_inverse} OF [wf_thm, len_thm]
+ in
+ if member abconv (get_basis_list basis) t then
+ (expand_basic thm t basis, basis) |> apfst (check_expansion e')
+ else
+ expand'' ectxt e' basis |> apfst (check_expansion e')
+ end
+ | expand' ectxt (Div (e1, e2)) basis =
+ let
+ val (thm1, basis') = expand' ectxt e1 basis
+ val t = expr_to_term e2
+ fun thm wf_thm len_thm =
+ @{thm expands_to_basic_inverse} OF [wf_thm, len_thm]
+ in
+ if member abconv (get_basis_list basis') t then
+ (@{thm expands_to_div'} OF [get_basis_wf_thm basis', thm1, expand_basic thm t basis'],
+ basis')
+ else
+ let
+ val (thm2, basis'', _, trimmed_thm) = trim_nz Simple_Trim ectxt e2 basis'
+ val thm1 = lift basis'' thm1
+ in
+ (@{thm expands_to_divide} OF [trimmed_thm, get_basis_wf_thm basis'', thm1, thm2],
+ basis'')
+ end
+ end
+ | expand' ectxt (e' as (Powr' (e, p))) basis =
+ let
+ val t = expr_to_term e
+ val ctxt = get_ctxt ectxt
+ fun thm wf_thm len_thm =
+ (Drule.infer_instantiate' ctxt [NONE, NONE, SOME (Thm.cterm_of ctxt p)]
+ @{thm expands_to_basic_powr}) OF [wf_thm, len_thm]
+ in
+ if member abconv (get_basis_list basis) t then
+ (expand_basic thm t basis, basis) |> apfst (check_expansion e')
+ else
+ expand'' ectxt e' basis |> apfst (check_expansion e')
+ end
+ | expand' ectxt e basis =
+ let
+ val t = expr_to_term e
+ fun thm wf_thm len_thm = @{thm expands_to_basic} OF [wf_thm, len_thm]
+ in
+ if member abconv (get_basis_list basis) t then
+ (expand_basic thm t basis, basis) |> apfst (check_expansion e)
+ else
+ expand'' ectxt e basis |> apfst (check_expansion e)
+ end
+
+fun expand ectxt e basis =
+ expand' ectxt e basis
+ |> apfst (simplify_expansion ectxt)
+ |> apfst (check_expansion e)
+
+fun expand_term ectxt t basis =
+ let
+ val ctxt = get_ctxt ectxt
+ val (e, eq_thm) = reify ctxt t
+ val (thm, basis) = expand ectxt e basis
+ in
+ (@{thm expands_to_meta_eq_cong'} OF [thm, eq_thm], basis)
+ end
+
+fun expand_terms ectxt ts basis =
+ let
+ val ctxt = get_ctxt ectxt
+ val e_eq_thms = map (reify ctxt) ts
+ fun step (e, eq_thm) (thms, basis) =
+ let
+ val (thm, basis) = expand' ectxt e basis
+ val thm = @{thm expands_to_meta_eq_cong'} OF [simplify_expansion ectxt thm, eq_thm]
+ in
+ (thm :: thms, basis)
+ end
+ val (thms, basis) = fold step e_eq_thms ([], basis)
+ fun lift thm = lift_expands_to_thm (mk_lifting (extract_basis_list thm) basis) thm
+ in
+ (map lift (rev thms), basis)
+ end
+
+datatype limit =
+ Zero_Limit of bool option
+ | Finite_Limit of term
+ | Infinite_Limit of bool option
+
+fun is_empty_expansion (Const (@{const_name MS}, _) $ Const (@{const_name MSLNil}, _) $ _) = true
+ | is_empty_expansion _ = false
+
+fun limit_of_expansion_aux ectxt basis thm =
+ let
+ val n = length (get_basis_list basis)
+ val (thm, res, e_thms) =
+ trim_expansion_while_greater false (SOME (replicate n @{term "0::real"})) true
+ (SOME Simple_Trim) ectxt (thm, basis)
+ val trimmed_thm = case res of Trimmed (_, trimmed_thm) => trimmed_thm | _ => NONE
+ val res = case res of Trimmed _ => GREATER | Aborted res => res
+ val exp = get_expansion thm
+ val _ = if res = GREATER andalso is_none trimmed_thm andalso not (is_empty_expansion exp) then
+ raise TERM ("limit_of_expansion", [get_expansion thm]) else ()
+ fun go thm _ _ [] = (
+ case zeroness_oracle false (SOME Simple_Trim) ectxt (get_expansion thm) of
+ (IsZero, _) => (Zero_Limit NONE, @{thm expands_to_real_imp_filterlim} OF [thm])
+ | _ => (Finite_Limit @{term "0::real"}, @{thm expands_to_real_imp_filterlim} OF [thm]))
+ | go thm _ basis ((IsNeg, neg_thm) :: _) = (Zero_Limit NONE,
+ @{thm expands_to_neg_exponent_imp_filterlim} OF
+ [thm, get_basis_wf_thm basis, neg_thm RS @{thm compare_reals_diff_sgnD(1)}])
+ | go thm trimmed_thm basis ((IsPos, pos_thm) :: _) = (Infinite_Limit NONE,
+ @{thm expands_to_pos_exponent_imp_filterlim} OF
+ [thm, the trimmed_thm, get_basis_wf_thm basis,
+ pos_thm RS @{thm compare_reals_diff_sgnD(3)}])
+ | go thm trimmed_thm basis ((IsZero, zero_thm) :: e_thms) =
+ let
+ val thm' = thm RS @{thm expands_to_hd''}
+ val trimmed_thm' = Option.map (fn thm => thm RS @{thm trimmed_hd}) trimmed_thm
+ val (lim, lim_thm) = go thm' trimmed_thm' (tl_basis basis) e_thms
+ val lim_lift_thm =
+ case lim of
+ Infinite_Limit _ => @{thm expands_to_zero_exponent_imp_filterlim(1)}
+ | _ => @{thm expands_to_zero_exponent_imp_filterlim(2)}
+ val lim_thm' =
+ lim_lift_thm OF [thm, get_basis_wf_thm basis,
+ zero_thm RS @{thm compare_reals_diff_sgnD(2)}, lim_thm]
+ in
+ (lim, lim_thm')
+ end
+ | go _ _ _ _ = raise Match
+ in
+ if is_empty_expansion exp then
+ (Zero_Limit NONE, thm RS @{thm expands_to_MSLNil_imp_filterlim}, thm)
+ else
+ case go thm trimmed_thm basis e_thms of
+ (lim, lim_thm) => (lim, lim_thm, thm)
+ end
+
+(*
+ Determines the limit of a function from its expansion. The two flags control whether the
+ the sign of the approach should be determined for the infinite case (i.e. at_top/at_bot instead
+ of just at_infinity) and the zero case (i.e. at_right 0/at_left 0 instead of just nhds 0)
+*)
+fun limit_of_expansion (sgn_zero, sgn_inf) ectxt (thm, basis) =
+ let
+ val (lim, lim_thm, thm) = limit_of_expansion_aux ectxt basis thm
+ in
+ case lim of
+ Zero_Limit _ => (
+ if sgn_zero then
+ case trim_expansion false (SOME Sgn_Trim) ectxt (thm, basis) of
+ (thm, IsPos, SOME pos_thm) => (Zero_Limit (SOME true),
+ @{thm tendsto_imp_filterlim_at_right[OF _ expands_to_imp_eventually_pos]} OF
+ [lim_thm, get_basis_wf_thm basis, thm, pos_thm])
+ | (thm, IsNeg, SOME neg_thm) => (Zero_Limit (SOME false),
+ @{thm tendsto_imp_filterlim_at_left[OF _ expands_to_imp_eventually_neg]} OF
+ [lim_thm, get_basis_wf_thm basis, thm, neg_thm])
+ | _ => (Zero_Limit NONE, lim_thm)
+ else (Zero_Limit NONE, lim_thm))
+ | Infinite_Limit _ => (
+ if sgn_inf then
+ case trim_expansion false (SOME Sgn_Trim) ectxt (thm, basis) of
+ (thm, IsPos, SOME pos_thm) => (Infinite_Limit (SOME true),
+ (@{thm filterlim_at_infinity_imp_filterlim_at_top[OF _ expands_to_imp_eventually_pos]} OF
+ [lim_thm, get_basis_wf_thm basis, thm, pos_thm]))
+ | (thm, IsNeg, SOME neg_thm) => (Infinite_Limit (SOME false),
+ @{thm filterlim_at_infinity_imp_filterlim_at_bot[OF _ expands_to_imp_eventually_neg]} OF
+ [lim_thm, get_basis_wf_thm basis, thm, neg_thm])
+ | _ => (Infinite_Limit NONE, lim_thm)
+ else (Infinite_Limit NONE, lim_thm))
+ | Finite_Limit c => (Finite_Limit c, lim_thm)
+ end
+
+fun compute_limit ectxt t =
+ case expand_term ectxt t default_basis of
+ (thm, basis) => limit_of_expansion (true, true) ectxt (thm, basis)
+
+fun prove_at_infinity ectxt (thm, basis) =
+ let
+ fun err () = raise TERM ("prove_at_infinity", [get_expanded_fun thm])
+ val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm, basis)
+ fun go basis thm trimmed_thm =
+ if fastype_of (get_expansion thm) = @{typ "real"} then
+ err ()
+ else
+ case zeroness_oracle true (SOME Pos_Trim) ectxt (get_exponent (get_expansion thm)) of
+ (IsPos, SOME pos_thm) =>
+ @{thm expands_to_pos_exponent_imp_filterlim} OF
+ [thm, trimmed_thm, get_basis_wf_thm basis, pos_thm]
+ | (IsZero, SOME zero_thm) =>
+ @{thm expands_to_zero_exponent_imp_filterlim(1)} OF
+ [thm, get_basis_wf_thm basis, zero_thm,
+ go (tl_basis basis) (thm RS @{thm expands_to_hd''})
+ (trimmed_thm RS @{thm trimmed_hd})]
+ | _ => err ()
+ in
+ go basis thm trimmed_thm
+ end
+
+fun prove_at_top_at_bot mode ectxt (thm, basis) =
+ let
+ val s = if mode = Pos_Trim then "prove_at_top" else "prove_at_bot"
+ fun err () = raise TERM (s, [get_expanded_fun thm])
+ val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis)
+ val trimmed_thm' = trimmed_thm RS
+ (if mode = Pos_Trim then @{thm trimmed_pos_imp_trimmed} else @{thm trimmed_neg_imp_trimmed})
+ fun go basis thm trimmed_thm =
+ if fastype_of (get_expansion thm) = @{typ "real"} then
+ err ()
+ else
+ case zeroness_oracle true (SOME Pos_Trim) ectxt (get_exponent (get_expansion thm)) of
+ (IsPos, SOME pos_thm) =>
+ @{thm expands_to_pos_exponent_imp_filterlim} OF
+ [thm, trimmed_thm, get_basis_wf_thm basis, pos_thm]
+ | (IsZero, SOME zero_thm) =>
+ @{thm expands_to_zero_exponent_imp_filterlim(1)} OF
+ [thm, get_basis_wf_thm basis, zero_thm,
+ go (tl_basis basis) (thm RS @{thm expands_to_hd''})
+ (trimmed_thm RS @{thm trimmed_hd})]
+ | _ => err ()
+ val lim_thm = go basis thm trimmed_thm'
+ val add_sign_thm =
+ if mode = Pos_Trim then
+ @{thm filterlim_at_infinity_imp_filterlim_at_top[OF _ expands_to_imp_eventually_pos]}
+ else
+ @{thm filterlim_at_infinity_imp_filterlim_at_bot[OF _ expands_to_imp_eventually_neg]}
+ in
+ add_sign_thm OF [lim_thm, get_basis_wf_thm basis, thm, trimmed_thm]
+ end
+
+val prove_at_top = prove_at_top_at_bot Pos_Trim
+val prove_at_bot = prove_at_top_at_bot Neg_Trim
+
+
+fun prove_at_aux mode ectxt (thm, basis) =
+ let
+ val (s, add_sign_thm) =
+ case mode of
+ Simple_Trim =>
+ ("prove_at_0", @{thm Topological_Spaces.filterlim_atI[OF _ expands_to_imp_eventually_nz]})
+ | Pos_Trim =>
+ ("prove_at_right_0",
+ @{thm tendsto_imp_filterlim_at_right[OF _ expands_to_imp_eventually_pos]})
+ | Neg_Trim =>
+ ("prove_at_left_0",
+ @{thm tendsto_imp_filterlim_at_left[OF _ expands_to_imp_eventually_neg]})
+ fun err () = raise TERM (s, [get_expanded_fun thm])
+ val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis)
+ fun go basis thm =
+ if fastype_of (get_expansion thm) = @{typ "real"} then
+ err ()
+ else
+ case zeroness_oracle true (SOME Neg_Trim) ectxt (get_exponent (get_expansion thm)) of
+ (IsNeg, SOME neg_thm) =>
+ @{thm expands_to_neg_exponent_imp_filterlim} OF
+ [thm, get_basis_wf_thm basis, neg_thm]
+ | (IsZero, SOME zero_thm) =>
+ @{thm expands_to_zero_exponent_imp_filterlim(2)} OF
+ [thm, get_basis_wf_thm basis, zero_thm,
+ go (tl_basis basis) (thm RS @{thm expands_to_hd''})]
+ | _ => err ()
+ val lim_thm = go basis thm
+ in
+ add_sign_thm OF [lim_thm, get_basis_wf_thm basis, thm, trimmed_thm]
+ end
+
+val prove_at_0 = prove_at_aux Simple_Trim
+val prove_at_left_0 = prove_at_aux Neg_Trim
+val prove_at_right_0 = prove_at_aux Pos_Trim
+
+
+fun prove_nhds ectxt (thm, basis) =
+ let
+ fun simplify (a, b, c) = (a, simplify_expansion ectxt b, c)
+ fun go thm basis =
+ if fastype_of (get_expansion thm) = @{typ "real"} then
+ @{thm expands_to_real_imp_filterlim} OF [thm]
+ else
+ case whnf_expansion ectxt thm |> simplify of
+ (NONE, thm, _) => @{thm expands_to_MSLNil_imp_filterlim} OF [thm]
+ | (SOME _, thm, _) => (
+ case zeroness_oracle true (SOME Sgn_Trim) ectxt (get_exponent (get_expansion thm)) of
+ (IsZero, SOME zero_thm) =>
+ @{thm expands_to_zero_exponent_imp_filterlim(2)} OF
+ [thm, get_basis_wf_thm basis, zero_thm,
+ go (thm RS @{thm expands_to_hd''}) (tl_basis basis)]
+ | (IsNeg, SOME neg_thm) =>
+ @{thm expands_to_neg_exponent_imp_filterlim} OF
+ [thm, get_basis_wf_thm basis, neg_thm]
+ | (IsPos, _) =>
+ go (try_drop_leading_term ectxt thm) basis
+ | _ => raise TERM ("Unexpected zeroness result in prove_nhds",
+ [get_exponent (get_expansion thm)]))
+ in
+ go thm basis
+ end
+
+fun prove_equivalent theta ectxt (thm1, thm2, basis) =
+ let
+ val ((thm1, _, SOME trimmed_thm1), (thm2, _, SOME trimmed_thm2)) =
+ apply2 (trim_expansion true (SOME Simple_Trim) ectxt) ((thm1, basis), (thm2, basis))
+ val pat = ConsPat (@{const_name Pair}, [ConsPat (@{const_name Lazy_Eval.cmp_result.EQ}, []),
+ ConsPat (@{const_name Pair}, [AnyPat ("_", 0), AnyPat ("_", 0)])])
+ val (exp1, exp2) = apply2 get_expansion (thm1, thm2)
+ val T = fastype_of exp1
+ val t = mk_compare_expansions_const T $ exp1 $ exp2
+ fun eq_thm conv = HOLogic.mk_obj_eq (conv (Thm.cterm_of (get_ctxt ectxt) t))
+ val imp_thm =
+ if theta then @{thm compare_expansions_EQ_imp_bigtheta}
+ else @{thm compare_expansions_EQ_same}
+ in
+ case match ectxt pat t (SOME []) of
+ (SOME _, t, conv) =>
+ let
+ val [_, c1, c2] = HOLogic.strip_tuple t
+ val c12_thm = if theta then [] else [the (try_prove_real_eq true ectxt (c1, c2))]
+ in
+ imp_thm OF ([eq_thm conv, trimmed_thm1, trimmed_thm2, thm1, thm2, get_basis_wf_thm basis]
+ @ c12_thm)
+ end
+ | _ => raise TERM ("prove_equivalent", map get_expanded_fun [thm1, thm2])
+ end
+
+val prove_bigtheta = prove_equivalent true
+val prove_asymp_equiv = prove_equivalent false
+
+fun print_trimming_error s ectxt exp =
+ let
+ val c = get_coeff exp
+ val t = if fastype_of c = @{typ real} then c else get_eval c
+ in
+ if #verbose (#ctxt ectxt) then
+ let
+ val ctxt = get_ctxt ectxt
+ val p = Pretty.str "real_asymp failed to show zeroness of the following expression:"
+ val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
+ val _ = Pretty.writeln p
+ in
+ raise TERM (s, [t])
+ end
+ else
+ raise TERM (s, [t])
+ end
+
+fun prove_smallo ectxt (thm1, thm2, basis) =
+ let
+ val (thm2, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis)
+ val es = get_exponents (get_expansion thm2)
+ in
+ case trim_expansion_while_greater true (SOME es) false NONE ectxt (thm1, basis) of
+ (thm1, Aborted LESS, thms) =>
+ @{thm compare_expansions_LT} OF [prove_compare_expansions LESS (map snd thms),
+ trimmed_thm, thm1, thm2, get_basis_wf_thm basis]
+ | (thm1, Aborted _, _) =>
+ print_trimming_error "prove_smallo" ectxt (get_expansion thm1)
+ | (thm1, Trimmed _, _) =>
+ print_trimming_error "prove_smallo" ectxt (get_expansion thm1)
+ end
+
+fun prove_bigo ectxt (thm1, thm2, basis) =
+ let
+ val (thm2, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis)
+ val es = get_exponents (get_expansion thm2)
+ in
+ case trim_expansion_while_greater false (SOME es) false NONE ectxt (thm1, basis) of
+ (thm1, Aborted LESS, thms) =>
+ @{thm landau_o.small_imp_big[OF compare_expansions_LT]} OF
+ [prove_compare_expansions LESS (map snd thms), trimmed_thm, thm1, thm2,
+ get_basis_wf_thm basis]
+ | (thm1, Aborted EQ, thms) =>
+ @{thm compare_expansions_EQ_imp_bigo} OF [prove_compare_expansions EQ (map snd thms),
+ trimmed_thm, thm1, thm2, get_basis_wf_thm basis]
+ | (thm1, Trimmed _, _) =>
+ print_trimming_error "prove_bigo" ectxt (get_expansion thm1)
+ end
+
+
+fun prove_asymptotic_relation_aux mode f ectxt (thm1, thm2, basis) = f (
+ let
+ val thm = @{thm expands_to_minus} OF [get_basis_wf_thm basis, thm1, thm2]
+ in
+ case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
+ SOME zero_thm => (EQUAL, zero_thm RS @{thm eventually_diff_zero_imp_eq})
+ | _ => (
+ case trim_expansion true (SOME mode) ectxt (thm, basis) of
+ (thm, IsPos, SOME pos_thm) =>
+ (GREATER, @{thm expands_to_imp_eventually_gt} OF [get_basis_wf_thm basis, thm, pos_thm])
+ | (thm, IsNeg, SOME neg_thm) =>
+ (LESS, @{thm expands_to_imp_eventually_lt} OF [get_basis_wf_thm basis, thm, neg_thm])
+ | _ => raise TERM ("Unexpected zeroness result in prove_asymptotic_relation", []))
+ end)
+
+val prove_eventually_greater = prove_asymptotic_relation_aux Pos_Trim snd
+val prove_eventually_less = prove_asymptotic_relation_aux Neg_Trim snd
+val prove_asymptotic_relation = prove_asymptotic_relation_aux Sgn_Trim I
+
+fun prove_eventually_nonzero ectxt (thm, basis) =
+ case trim_expansion true (SOME Simple_Trim) ectxt (thm, basis) of
+ (thm, _, SOME trimmed_thm) =>
+ @{thm expands_to_imp_eventually_nz} OF [get_basis_wf_thm basis, thm, trimmed_thm]
+ | _ => raise TERM ("prove_eventually_nonzero", [get_expanded_fun thm])
+
+fun extract_terms (n, strict) ectxt basis t =
+ let
+ val bs = get_basis_list basis
+ fun mk_constfun c = (Abs ("x", @{typ real}, c))
+ val const_0 = mk_constfun @{term "0 :: real"}
+ val const_1 = mk_constfun @{term "1 :: real"}
+ fun uminus t = Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, t)
+ fun betapply2 a b c = Term.betapply (Term.betapply (a, b), c)
+
+ fun mk_sum' [] acc = acc
+ | mk_sum' ((t, sgn) :: ts) acc = mk_sum' ts (
+ if sgn then
+ betapply2 @{term "%(f::real=>real) g x. f x - g x"} acc t
+ else
+ betapply2 @{term "%(f::real=>real) g x. f x + g x"} acc t)
+ fun mk_sum [] = const_0
+ | mk_sum ((t, sgn) :: ts) = mk_sum' ts (if sgn then uminus t else t)
+
+ fun mk_mult a b =
+ if a aconv const_0 then
+ const_0
+ else if b aconv const_0 then
+ const_0
+ else if a aconv @{term "\<lambda>_::real. 1 :: real"} then
+ b
+ else if b aconv @{term "\<lambda>_::real. 1 :: real"} then
+ a
+ else if a aconv @{term "\<lambda>_::real. -1 :: real"} then
+ Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, b)
+ else if b aconv @{term "\<lambda>_::real. -1 :: real"} then
+ Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, a)
+ else
+ Abs ("x", @{typ real}, @{term "( *) :: real => _"} $
+ (Term.betapply (a, Bound 0)) $ (Term.betapply (b, Bound 0)))
+
+ fun mk_powr b e =
+ if e = @{term "0 :: real"} then
+ const_1
+ else
+ let
+ val n = HOLogic.dest_number e |> snd
+ in
+ if n >= 0 then
+ Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x ^ e"}, b),
+ HOLogic.mk_number @{typ nat} n)
+ else
+ Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x powr e"}, b), e)
+ end
+ handle TERM _ =>
+ Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x powr e"}, b), e)
+
+ fun mk_scale_elem b e acc =
+ let
+ val e = simplify_term ectxt e
+ in
+ if e = @{term "0 :: real"} then
+ acc
+ else if e = @{term "1 :: real"} then
+ mk_mult acc b
+ else
+ mk_mult acc (mk_powr b e)
+ end
+
+ fun mk_scale_elems [] _ acc = acc
+ | mk_scale_elems (b :: bs) (e :: es) acc =
+ mk_scale_elems bs es (mk_scale_elem b e acc)
+ | mk_scale_elems _ _ _ = raise Match
+
+ fun mk_summand c es =
+ let
+ val es = mk_scale_elems bs es @{term "\<lambda>_::real. 1 :: real"}
+ in
+ case c of
+ Const (@{const_name uminus}, _) $ c => ((c, true), es)
+ | _ => ((c, false), es)
+ end
+
+ fun go _ _ _ acc 0 = (acc, 0)
+ | go 0 es t acc n =
+ let
+ val c = simplify_term ectxt t
+ in
+ if strict andalso c = @{term "0 :: real"} then
+ (acc, n)
+ else
+ (mk_summand c (rev es) :: acc, n - 1)
+ end
+ | go m es t acc n =
+ case Lazy_Eval.whnf ectxt t |> fst of
+ Const (@{const_name MS}, _) $ cs $ _ =>
+ go' m es (simplify_term ectxt cs) acc n
+ | _ => raise TERM("extract_terms", [t])
+ and go' _ _ _ acc 0 = (acc, 0)
+ | go' m es cs acc n =
+ case Lazy_Eval.whnf ectxt cs |> fst of
+ Const (@{const_name MSLNil}, _) => (acc, n)
+ | Const (@{const_name MSLCons}, _) $ c $ cs => (
+ case Lazy_Eval.whnf ectxt c |> fst |> HOLogic.dest_prod of
+ (c, e) =>
+ case go (m - 1) (e :: es) c acc n of
+ (acc, n) => go' m es (simplify_term ectxt cs) acc n)
+ | _ => raise TERM("extract_terms", [t])
+ val (summands, remaining) = go (basis_size basis) [] t [] (n + 1)
+ val (summands, error) =
+ if remaining = 0 then (rev (tl summands), SOME (snd (hd summands))) else (rev summands, NONE)
+ val summands = map (fn ((c, sgn), es) => (mk_mult (mk_constfun c) es, sgn)) summands
+ val error = Option.map (fn err => Term.betapply (@{term "\<lambda>f::real\<Rightarrow>real. O(f)"}, err)) error
+ val expansion = mk_sum summands
+ in
+ (expansion, error)
+ end
+
+end
+
+
+structure Multiseries_Expansion_Basic : EXPANSION_INTERFACE =
+struct
+open Multiseries_Expansion;
+
+type T = expansion_thm
+
+val expand_term = expand_term
+val expand_terms = expand_terms
+
+val prove_nhds = prove_nhds
+val prove_at_infinity = prove_at_infinity
+val prove_at_top = prove_at_top
+val prove_at_bot = prove_at_bot
+val prove_at_0 = prove_at_0
+val prove_at_right_0 = prove_at_right_0
+val prove_at_left_0 = prove_at_left_0
+val prove_eventually_nonzero = prove_eventually_nonzero
+
+val prove_eventually_less = prove_eventually_less
+val prove_eventually_greater = prove_eventually_greater
+
+val prove_smallo = prove_smallo
+val prove_bigo = prove_bigo
+val prove_bigtheta = prove_bigtheta
+val prove_asymp_equiv = prove_asymp_equiv
+
+end