--- a/NEWS Fri Nov 06 14:16:57 2009 +0100
+++ b/NEWS Fri Nov 06 16:59:17 2009 +0100
@@ -227,7 +227,9 @@
* Maclauren.DERIV_tac and Maclauren.deriv_tac was removed, they are
replaced by: (auto intro!: DERIV_intros). INCOMPATIBILITY.
-* Renamed method "sizechange" to "size_change".
+* Renamed methods:
+ sizechange -> size_change
+ induct_scheme -> induction_schema
*** ML ***
--- a/src/HOL/FunDef.thy Fri Nov 06 14:16:57 2009 +0100
+++ b/src/HOL/FunDef.thy Fri Nov 06 16:59:17 2009 +0100
@@ -22,7 +22,7 @@
("Tools/Function/lexicographic_order.ML")
("Tools/Function/pat_completeness.ML")
("Tools/Function/fun.ML")
- ("Tools/Function/induction_scheme.ML")
+ ("Tools/Function/induction_schema.ML")
("Tools/Function/termination.ML")
("Tools/Function/decompose.ML")
("Tools/Function/descent.ML")
@@ -114,13 +114,13 @@
use "Tools/Function/function.ML"
use "Tools/Function/pat_completeness.ML"
use "Tools/Function/fun.ML"
-use "Tools/Function/induction_scheme.ML"
+use "Tools/Function/induction_schema.ML"
setup {*
Function.setup
#> Pat_Completeness.setup
#> Function_Fun.setup
- #> Induction_Scheme.setup
+ #> Induction_Schema.setup
*}
subsection {* Measure Functions *}
--- a/src/HOL/IsaMakefile Fri Nov 06 14:16:57 2009 +0100
+++ b/src/HOL/IsaMakefile Fri Nov 06 16:59:17 2009 +0100
@@ -180,7 +180,7 @@
Tools/Function/function_lib.ML \
Tools/Function/function.ML \
Tools/Function/fun.ML \
- Tools/Function/induction_scheme.ML \
+ Tools/Function/induction_schema.ML \
Tools/Function/lexicographic_order.ML \
Tools/Function/measure_functions.ML \
Tools/Function/mutual.ML \
@@ -939,7 +939,7 @@
ex/Efficient_Nat_examples.thy ex/Eval_Examples.thy ex/Fundefs.thy \
ex/Groebner_Examples.thy ex/Guess.thy ex/HarmonicSeries.thy \
ex/Hebrew.thy ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy \
- ex/Hilbert_Classical.thy ex/Induction_Scheme.thy \
+ ex/Hilbert_Classical.thy ex/Induction_Schema.thy \
ex/InductiveInvariant.thy ex/InductiveInvariant_examples.thy \
ex/Intuitionistic.thy ex/Lagrange.thy ex/LocaleTest2.thy ex/MT.thy \
ex/MergeSort.thy ex/Meson_Test.thy ex/MonoidGroup.thy \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Function/induction_schema.ML Fri Nov 06 16:59:17 2009 +0100
@@ -0,0 +1,405 @@
+(* Title: HOL/Tools/Function/induction_schema.ML
+ Author: Alexander Krauss, TU Muenchen
+
+A method to prove induction schemas.
+*)
+
+signature INDUCTION_SCHEMA =
+sig
+ val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
+ -> Proof.context -> thm list -> tactic
+ val induction_schema_tac : Proof.context -> thm list -> tactic
+ val setup : theory -> theory
+end
+
+
+structure Induction_Schema : INDUCTION_SCHEMA =
+struct
+
+open Function_Lib
+
+
+type rec_call_info = int * (string * typ) list * term list * term list
+
+datatype scheme_case =
+ SchemeCase of
+ {
+ bidx : int,
+ qs: (string * typ) list,
+ oqnames: string list,
+ gs: term list,
+ lhs: term list,
+ rs: rec_call_info list
+ }
+
+datatype scheme_branch =
+ SchemeBranch of
+ {
+ P : term,
+ xs: (string * typ) list,
+ ws: (string * typ) list,
+ Cs: term list
+ }
+
+datatype ind_scheme =
+ IndScheme of
+ {
+ T: typ, (* sum of products *)
+ branches: scheme_branch list,
+ cases: scheme_case list
+ }
+
+val ind_atomize = MetaSimplifier.rewrite true @{thms induct_atomize}
+val ind_rulify = MetaSimplifier.rewrite true @{thms induct_rulify}
+
+fun meta thm = thm RS eq_reflection
+
+val sum_prod_conv = MetaSimplifier.rewrite true
+ (map meta (@{thm split_conv} :: @{thms sum.cases}))
+
+fun term_conv thy cv t =
+ cv (cterm_of thy t)
+ |> prop_of |> Logic.dest_equals |> snd
+
+fun mk_relT T = HOLogic.mk_setT (HOLogic.mk_prodT (T, T))
+
+fun dest_hhf ctxt t =
+ let
+ val (ctxt', vars, imp) = dest_all_all_ctx ctxt t
+ in
+ (ctxt', vars, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
+ end
+
+
+fun mk_scheme' ctxt cases concl =
+ let
+ fun mk_branch concl =
+ let
+ val (ctxt', ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
+ val (P, xs) = strip_comb Pxs
+ in
+ SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
+ end
+
+ val (branches, cases') = (* correction *)
+ case Logic.dest_conjunction_list concl of
+ [conc] =>
+ let
+ val _ $ Pxs = Logic.strip_assums_concl conc
+ val (P, _) = strip_comb Pxs
+ val (cases', conds) = take_prefix (Term.exists_subterm (curry op aconv P)) cases
+ val concl' = fold_rev (curry Logic.mk_implies) conds conc
+ in
+ ([mk_branch concl'], cases')
+ end
+ | concls => (map mk_branch concls, cases)
+
+ fun mk_case premise =
+ let
+ val (ctxt', qs, prems, _ $ Plhs) = dest_hhf ctxt premise
+ val (P, lhs) = strip_comb Plhs
+
+ fun bidx Q = find_index (fn SchemeBranch {P=P',...} => Q aconv P') branches
+
+ fun mk_rcinfo pr =
+ let
+ val (ctxt'', Gvs, Gas, _ $ Phyp) = dest_hhf ctxt' pr
+ val (P', rcs) = strip_comb Phyp
+ in
+ (bidx P', Gvs, Gas, rcs)
+ end
+
+ fun is_pred v = exists (fn SchemeBranch {P,...} => v aconv P) branches
+
+ val (gs, rcprs) =
+ take_prefix (not o Term.exists_subterm is_pred) prems
+ in
+ SchemeCase {bidx=bidx P, qs=qs, oqnames=map fst qs(*FIXME*), gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
+ end
+
+ fun PT_of (SchemeBranch { xs, ...}) =
+ foldr1 HOLogic.mk_prodT (map snd xs)
+
+ val ST = Balanced_Tree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
+ in
+ IndScheme {T=ST, cases=map mk_case cases', branches=branches }
+ end
+
+
+
+fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
+ let
+ val SchemeBranch { xs, ws, Cs, ... } = nth branches bidx
+ val relevant_cases = filter (fn SchemeCase {bidx=bidx', ...} => bidx' = bidx) cases
+
+ val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) relevant_cases []
+ val (Pbool :: xs') = map Free (Variable.variant_frees ctxt allqnames (("P", HOLogic.boolT) :: xs))
+ val Cs' = map (Pattern.rewrite_term (ProofContext.theory_of ctxt) (filter_out (op aconv) (map Free xs ~~ xs')) []) Cs
+
+ fun mk_case (SchemeCase {qs, oqnames, gs, lhs, ...}) =
+ HOLogic.mk_Trueprop Pbool
+ |> fold_rev (fn x_l => curry Logic.mk_implies (HOLogic.mk_Trueprop(HOLogic.mk_eq x_l)))
+ (xs' ~~ lhs)
+ |> fold_rev (curry Logic.mk_implies) gs
+ |> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
+ in
+ HOLogic.mk_Trueprop Pbool
+ |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
+ |> fold_rev (curry Logic.mk_implies) Cs'
+ |> fold_rev (Logic.all o Free) ws
+ |> fold_rev mk_forall_rename (map fst xs ~~ xs')
+ |> mk_forall_rename ("P", Pbool)
+ end
+
+fun mk_wf ctxt R (IndScheme {T, ...}) =
+ HOLogic.Trueprop $ (Const (@{const_name wf}, mk_relT T --> HOLogic.boolT) $ R)
+
+fun mk_ineqs R (IndScheme {T, cases, branches}) =
+ let
+ fun inject i ts =
+ SumTree.mk_inj T (length branches) (i + 1) (foldr1 HOLogic.mk_prod ts)
+
+ val thesis = Free ("thesis", HOLogic.boolT) (* FIXME *)
+
+ fun mk_pres bdx args =
+ let
+ val SchemeBranch { xs, ws, Cs, ... } = nth branches bdx
+ fun replace (x, v) t = betapply (lambda (Free x) t, v)
+ val Cs' = map (fold replace (xs ~~ args)) Cs
+ val cse =
+ HOLogic.mk_Trueprop thesis
+ |> fold_rev (curry Logic.mk_implies) Cs'
+ |> fold_rev (Logic.all o Free) ws
+ in
+ Logic.mk_implies (cse, HOLogic.mk_Trueprop thesis)
+ end
+
+ fun f (SchemeCase {bidx, qs, oqnames, gs, lhs, rs, ...}) =
+ let
+ fun g (bidx', Gvs, Gas, rcarg) =
+ let val export =
+ fold_rev (curry Logic.mk_implies) Gas
+ #> fold_rev (curry Logic.mk_implies) gs
+ #> fold_rev (Logic.all o Free) Gvs
+ #> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
+ in
+ (HOLogic.mk_mem (HOLogic.mk_prod (inject bidx' rcarg, inject bidx lhs), R)
+ |> HOLogic.mk_Trueprop
+ |> export,
+ mk_pres bidx' rcarg
+ |> export
+ |> Logic.all thesis)
+ end
+ in
+ map g rs
+ end
+ in
+ map f cases
+ end
+
+
+fun mk_hol_imp a b = HOLogic.imp $ a $ b
+
+fun mk_ind_goal thy branches =
+ let
+ fun brnch (SchemeBranch { P, xs, ws, Cs, ... }) =
+ HOLogic.mk_Trueprop (list_comb (P, map Free xs))
+ |> fold_rev (curry Logic.mk_implies) Cs
+ |> fold_rev (Logic.all o Free) ws
+ |> term_conv thy ind_atomize
+ |> ObjectLogic.drop_judgment thy
+ |> tupled_lambda (foldr1 HOLogic.mk_prod (map Free xs))
+ in
+ SumTree.mk_sumcases HOLogic.boolT (map brnch branches)
+ end
+
+
+fun mk_induct_rule ctxt R x complete_thms wf_thm ineqss (IndScheme {T, cases=scases, branches}) =
+ let
+ val n = length branches
+
+ val scases_idx = map_index I scases
+
+ fun inject i ts =
+ SumTree.mk_inj T n (i + 1) (foldr1 HOLogic.mk_prod ts)
+ val P_of = nth (map (fn (SchemeBranch { P, ... }) => P) branches)
+
+ val thy = ProofContext.theory_of ctxt
+ val cert = cterm_of thy
+
+ val P_comp = mk_ind_goal thy branches
+
+ (* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
+ val ihyp = Term.all T $ Abs ("z", T,
+ Logic.mk_implies
+ (HOLogic.mk_Trueprop (
+ Const ("op :", HOLogic.mk_prodT (T, T) --> mk_relT T --> HOLogic.boolT)
+ $ (HOLogic.pair_const T T $ Bound 0 $ x)
+ $ R),
+ HOLogic.mk_Trueprop (P_comp $ Bound 0)))
+ |> cert
+
+ val aihyp = assume ihyp
+
+ (* Rule for case splitting along the sum types *)
+ val xss = map (fn (SchemeBranch { xs, ... }) => map Free xs) branches
+ val pats = map_index (uncurry inject) xss
+ val sum_split_rule = Pat_Completeness.prove_completeness thy [x] (P_comp $ x) xss (map single pats)
+
+ fun prove_branch (bidx, (SchemeBranch { P, xs, ws, Cs, ... }, (complete_thm, pat))) =
+ let
+ val fxs = map Free xs
+ val branch_hyp = assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))
+
+ val C_hyps = map (cert #> assume) Cs
+
+ val (relevant_cases, ineqss') = filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx) (scases_idx ~~ ineqss)
+ |> split_list
+
+ fun prove_case (cidx, SchemeCase {qs, oqnames, gs, lhs, rs, ...}) ineq_press =
+ let
+ val case_hyps = map (assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)
+
+ val cqs = map (cert o Free) qs
+ val ags = map (assume o cert) gs
+
+ val replace_x_ss = HOL_basic_ss addsimps (branch_hyp :: case_hyps)
+ val sih = full_simplify replace_x_ss aihyp
+
+ fun mk_Prec (idx, Gvs, Gas, rcargs) (ineq, pres) =
+ let
+ val cGas = map (assume o cert) Gas
+ val cGvs = map (cert o Free) Gvs
+ val import = fold forall_elim (cqs @ cGvs)
+ #> fold Thm.elim_implies (ags @ cGas)
+ val ipres = pres
+ |> forall_elim (cert (list_comb (P_of idx, rcargs)))
+ |> import
+ in
+ sih |> forall_elim (cert (inject idx rcargs))
+ |> Thm.elim_implies (import ineq) (* Psum rcargs *)
+ |> Conv.fconv_rule sum_prod_conv
+ |> Conv.fconv_rule ind_rulify
+ |> (fn th => th COMP ipres) (* P rs *)
+ |> fold_rev (implies_intr o cprop_of) cGas
+ |> fold_rev forall_intr cGvs
+ end
+
+ val P_recs = map2 mk_Prec rs ineq_press (* [P rec1, P rec2, ... ] *)
+
+ val step = HOLogic.mk_Trueprop (list_comb (P, lhs))
+ |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
+ |> fold_rev (curry Logic.mk_implies) gs
+ |> fold_rev (Logic.all o Free) qs
+ |> cert
+
+ val Plhs_to_Pxs_conv =
+ foldl1 (uncurry Conv.combination_conv)
+ (Conv.all_conv :: map (fn ch => K (Thm.symmetric (ch RS eq_reflection))) case_hyps)
+
+ val res = assume step
+ |> fold forall_elim cqs
+ |> fold Thm.elim_implies ags
+ |> fold Thm.elim_implies P_recs (* P lhs *)
+ |> Conv.fconv_rule (Conv.arg_conv Plhs_to_Pxs_conv) (* P xs *)
+ |> fold_rev (implies_intr o cprop_of) (ags @ case_hyps)
+ |> fold_rev forall_intr cqs (* !!qs. Gas ==> xs = lhss ==> P xs *)
+ in
+ (res, (cidx, step))
+ end
+
+ val (cases, steps) = split_list (map2 prove_case relevant_cases ineqss')
+
+ val bstep = complete_thm
+ |> forall_elim (cert (list_comb (P, fxs)))
+ |> fold (forall_elim o cert) (fxs @ map Free ws)
+ |> fold Thm.elim_implies C_hyps (* FIXME: optimization using rotate_prems *)
+ |> fold Thm.elim_implies cases (* P xs *)
+ |> fold_rev (implies_intr o cprop_of) C_hyps
+ |> fold_rev (forall_intr o cert o Free) ws
+
+ val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
+ |> Goal.init
+ |> (MetaSimplifier.rewrite_goals_tac (map meta (branch_hyp :: @{thm split_conv} :: @{thms sum.cases}))
+ THEN CONVERSION ind_rulify 1)
+ |> Seq.hd
+ |> Thm.elim_implies (Conv.fconv_rule Drule.beta_eta_conversion bstep)
+ |> Goal.finish ctxt
+ |> implies_intr (cprop_of branch_hyp)
+ |> fold_rev (forall_intr o cert) fxs
+ in
+ (Pxs, steps)
+ end
+
+ val (branches, steps) = split_list (map_index prove_branch (branches ~~ (complete_thms ~~ pats)))
+ |> apsnd flat
+
+ val istep = sum_split_rule
+ |> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
+ |> implies_intr ihyp
+ |> forall_intr (cert x) (* "!!x. (!!y<x. P y) ==> P x" *)
+
+ val induct_rule =
+ @{thm "wf_induct_rule"}
+ |> (curry op COMP) wf_thm
+ |> (curry op COMP) istep
+
+ val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
+ in
+ (steps_sorted, induct_rule)
+ end
+
+
+fun mk_ind_tac comp_tac pres_tac term_tac ctxt facts = (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL
+(SUBGOAL (fn (t, i) =>
+ let
+ val (ctxt', _, cases, concl) = dest_hhf ctxt t
+ val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
+(* val _ = tracing (makestring scheme)*)
+ val ([Rn,xn], ctxt'') = Variable.variant_fixes ["R","x"] ctxt'
+ val R = Free (Rn, mk_relT ST)
+ val x = Free (xn, ST)
+ val cert = cterm_of (ProofContext.theory_of ctxt)
+
+ val ineqss = mk_ineqs R scheme
+ |> map (map (pairself (assume o cert)))
+ val complete = map_range (mk_completeness ctxt scheme #> cert #> assume) (length branches)
+ val wf_thm = mk_wf ctxt R scheme |> cert |> assume
+
+ val (descent, pres) = split_list (flat ineqss)
+ val newgoals = complete @ pres @ wf_thm :: descent
+
+ val (steps, indthm) = mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme
+
+ fun project (i, SchemeBranch {xs, ...}) =
+ let
+ val inst = cert (SumTree.mk_inj ST (length branches) (i + 1) (foldr1 HOLogic.mk_prod (map Free xs)))
+ in
+ indthm |> Drule.instantiate' [] [SOME inst]
+ |> simplify SumTree.sumcase_split_ss
+ |> Conv.fconv_rule ind_rulify
+(* |> (fn thm => (tracing (makestring thm); thm))*)
+ end
+
+ val res = Conjunction.intr_balanced (map_index project branches)
+ |> fold_rev implies_intr (map cprop_of newgoals @ steps)
+ |> (fn thm => Thm.generalize ([], [Rn]) (Thm.maxidx_of thm + 1) thm)
+
+ val nbranches = length branches
+ val npres = length pres
+ in
+ Thm.compose_no_flatten false (res, length newgoals) i
+ THEN term_tac (i + nbranches + npres)
+ THEN (EVERY (map (TRY o pres_tac) ((i + nbranches + npres - 1) downto (i + nbranches))))
+ THEN (EVERY (map (TRY o comp_tac) ((i + nbranches - 1) downto i)))
+ end))
+
+
+fun induction_schema_tac ctxt =
+ mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt;
+
+val setup =
+ Method.setup @{binding induction_schema} (Scan.succeed (RAW_METHOD o induction_schema_tac))
+ "proves an induction principle"
+
+end
--- a/src/HOL/Tools/Function/induction_scheme.ML Fri Nov 06 14:16:57 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,405 +0,0 @@
-(* Title: HOL/Tools/Function/induction_scheme.ML
- Author: Alexander Krauss, TU Muenchen
-
-A method to prove induction schemes.
-*)
-
-signature INDUCTION_SCHEME =
-sig
- val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
- -> Proof.context -> thm list -> tactic
- val induct_scheme_tac : Proof.context -> thm list -> tactic
- val setup : theory -> theory
-end
-
-
-structure Induction_Scheme : INDUCTION_SCHEME =
-struct
-
-open Function_Lib
-
-
-type rec_call_info = int * (string * typ) list * term list * term list
-
-datatype scheme_case =
- SchemeCase of
- {
- bidx : int,
- qs: (string * typ) list,
- oqnames: string list,
- gs: term list,
- lhs: term list,
- rs: rec_call_info list
- }
-
-datatype scheme_branch =
- SchemeBranch of
- {
- P : term,
- xs: (string * typ) list,
- ws: (string * typ) list,
- Cs: term list
- }
-
-datatype ind_scheme =
- IndScheme of
- {
- T: typ, (* sum of products *)
- branches: scheme_branch list,
- cases: scheme_case list
- }
-
-val ind_atomize = MetaSimplifier.rewrite true @{thms induct_atomize}
-val ind_rulify = MetaSimplifier.rewrite true @{thms induct_rulify}
-
-fun meta thm = thm RS eq_reflection
-
-val sum_prod_conv = MetaSimplifier.rewrite true
- (map meta (@{thm split_conv} :: @{thms sum.cases}))
-
-fun term_conv thy cv t =
- cv (cterm_of thy t)
- |> prop_of |> Logic.dest_equals |> snd
-
-fun mk_relT T = HOLogic.mk_setT (HOLogic.mk_prodT (T, T))
-
-fun dest_hhf ctxt t =
- let
- val (ctxt', vars, imp) = dest_all_all_ctx ctxt t
- in
- (ctxt', vars, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
- end
-
-
-fun mk_scheme' ctxt cases concl =
- let
- fun mk_branch concl =
- let
- val (ctxt', ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
- val (P, xs) = strip_comb Pxs
- in
- SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
- end
-
- val (branches, cases') = (* correction *)
- case Logic.dest_conjunction_list concl of
- [conc] =>
- let
- val _ $ Pxs = Logic.strip_assums_concl conc
- val (P, _) = strip_comb Pxs
- val (cases', conds) = take_prefix (Term.exists_subterm (curry op aconv P)) cases
- val concl' = fold_rev (curry Logic.mk_implies) conds conc
- in
- ([mk_branch concl'], cases')
- end
- | concls => (map mk_branch concls, cases)
-
- fun mk_case premise =
- let
- val (ctxt', qs, prems, _ $ Plhs) = dest_hhf ctxt premise
- val (P, lhs) = strip_comb Plhs
-
- fun bidx Q = find_index (fn SchemeBranch {P=P',...} => Q aconv P') branches
-
- fun mk_rcinfo pr =
- let
- val (ctxt'', Gvs, Gas, _ $ Phyp) = dest_hhf ctxt' pr
- val (P', rcs) = strip_comb Phyp
- in
- (bidx P', Gvs, Gas, rcs)
- end
-
- fun is_pred v = exists (fn SchemeBranch {P,...} => v aconv P) branches
-
- val (gs, rcprs) =
- take_prefix (not o Term.exists_subterm is_pred) prems
- in
- SchemeCase {bidx=bidx P, qs=qs, oqnames=map fst qs(*FIXME*), gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
- end
-
- fun PT_of (SchemeBranch { xs, ...}) =
- foldr1 HOLogic.mk_prodT (map snd xs)
-
- val ST = Balanced_Tree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
- in
- IndScheme {T=ST, cases=map mk_case cases', branches=branches }
- end
-
-
-
-fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
- let
- val SchemeBranch { xs, ws, Cs, ... } = nth branches bidx
- val relevant_cases = filter (fn SchemeCase {bidx=bidx', ...} => bidx' = bidx) cases
-
- val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) relevant_cases []
- val (Pbool :: xs') = map Free (Variable.variant_frees ctxt allqnames (("P", HOLogic.boolT) :: xs))
- val Cs' = map (Pattern.rewrite_term (ProofContext.theory_of ctxt) (filter_out (op aconv) (map Free xs ~~ xs')) []) Cs
-
- fun mk_case (SchemeCase {qs, oqnames, gs, lhs, ...}) =
- HOLogic.mk_Trueprop Pbool
- |> fold_rev (fn x_l => curry Logic.mk_implies (HOLogic.mk_Trueprop(HOLogic.mk_eq x_l)))
- (xs' ~~ lhs)
- |> fold_rev (curry Logic.mk_implies) gs
- |> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
- in
- HOLogic.mk_Trueprop Pbool
- |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
- |> fold_rev (curry Logic.mk_implies) Cs'
- |> fold_rev (Logic.all o Free) ws
- |> fold_rev mk_forall_rename (map fst xs ~~ xs')
- |> mk_forall_rename ("P", Pbool)
- end
-
-fun mk_wf ctxt R (IndScheme {T, ...}) =
- HOLogic.Trueprop $ (Const (@{const_name wf}, mk_relT T --> HOLogic.boolT) $ R)
-
-fun mk_ineqs R (IndScheme {T, cases, branches}) =
- let
- fun inject i ts =
- SumTree.mk_inj T (length branches) (i + 1) (foldr1 HOLogic.mk_prod ts)
-
- val thesis = Free ("thesis", HOLogic.boolT) (* FIXME *)
-
- fun mk_pres bdx args =
- let
- val SchemeBranch { xs, ws, Cs, ... } = nth branches bdx
- fun replace (x, v) t = betapply (lambda (Free x) t, v)
- val Cs' = map (fold replace (xs ~~ args)) Cs
- val cse =
- HOLogic.mk_Trueprop thesis
- |> fold_rev (curry Logic.mk_implies) Cs'
- |> fold_rev (Logic.all o Free) ws
- in
- Logic.mk_implies (cse, HOLogic.mk_Trueprop thesis)
- end
-
- fun f (SchemeCase {bidx, qs, oqnames, gs, lhs, rs, ...}) =
- let
- fun g (bidx', Gvs, Gas, rcarg) =
- let val export =
- fold_rev (curry Logic.mk_implies) Gas
- #> fold_rev (curry Logic.mk_implies) gs
- #> fold_rev (Logic.all o Free) Gvs
- #> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
- in
- (HOLogic.mk_mem (HOLogic.mk_prod (inject bidx' rcarg, inject bidx lhs), R)
- |> HOLogic.mk_Trueprop
- |> export,
- mk_pres bidx' rcarg
- |> export
- |> Logic.all thesis)
- end
- in
- map g rs
- end
- in
- map f cases
- end
-
-
-fun mk_hol_imp a b = HOLogic.imp $ a $ b
-
-fun mk_ind_goal thy branches =
- let
- fun brnch (SchemeBranch { P, xs, ws, Cs, ... }) =
- HOLogic.mk_Trueprop (list_comb (P, map Free xs))
- |> fold_rev (curry Logic.mk_implies) Cs
- |> fold_rev (Logic.all o Free) ws
- |> term_conv thy ind_atomize
- |> ObjectLogic.drop_judgment thy
- |> tupled_lambda (foldr1 HOLogic.mk_prod (map Free xs))
- in
- SumTree.mk_sumcases HOLogic.boolT (map brnch branches)
- end
-
-
-fun mk_induct_rule ctxt R x complete_thms wf_thm ineqss (IndScheme {T, cases=scases, branches}) =
- let
- val n = length branches
-
- val scases_idx = map_index I scases
-
- fun inject i ts =
- SumTree.mk_inj T n (i + 1) (foldr1 HOLogic.mk_prod ts)
- val P_of = nth (map (fn (SchemeBranch { P, ... }) => P) branches)
-
- val thy = ProofContext.theory_of ctxt
- val cert = cterm_of thy
-
- val P_comp = mk_ind_goal thy branches
-
- (* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
- val ihyp = Term.all T $ Abs ("z", T,
- Logic.mk_implies
- (HOLogic.mk_Trueprop (
- Const ("op :", HOLogic.mk_prodT (T, T) --> mk_relT T --> HOLogic.boolT)
- $ (HOLogic.pair_const T T $ Bound 0 $ x)
- $ R),
- HOLogic.mk_Trueprop (P_comp $ Bound 0)))
- |> cert
-
- val aihyp = assume ihyp
-
- (* Rule for case splitting along the sum types *)
- val xss = map (fn (SchemeBranch { xs, ... }) => map Free xs) branches
- val pats = map_index (uncurry inject) xss
- val sum_split_rule = Pat_Completeness.prove_completeness thy [x] (P_comp $ x) xss (map single pats)
-
- fun prove_branch (bidx, (SchemeBranch { P, xs, ws, Cs, ... }, (complete_thm, pat))) =
- let
- val fxs = map Free xs
- val branch_hyp = assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))
-
- val C_hyps = map (cert #> assume) Cs
-
- val (relevant_cases, ineqss') = filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx) (scases_idx ~~ ineqss)
- |> split_list
-
- fun prove_case (cidx, SchemeCase {qs, oqnames, gs, lhs, rs, ...}) ineq_press =
- let
- val case_hyps = map (assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)
-
- val cqs = map (cert o Free) qs
- val ags = map (assume o cert) gs
-
- val replace_x_ss = HOL_basic_ss addsimps (branch_hyp :: case_hyps)
- val sih = full_simplify replace_x_ss aihyp
-
- fun mk_Prec (idx, Gvs, Gas, rcargs) (ineq, pres) =
- let
- val cGas = map (assume o cert) Gas
- val cGvs = map (cert o Free) Gvs
- val import = fold forall_elim (cqs @ cGvs)
- #> fold Thm.elim_implies (ags @ cGas)
- val ipres = pres
- |> forall_elim (cert (list_comb (P_of idx, rcargs)))
- |> import
- in
- sih |> forall_elim (cert (inject idx rcargs))
- |> Thm.elim_implies (import ineq) (* Psum rcargs *)
- |> Conv.fconv_rule sum_prod_conv
- |> Conv.fconv_rule ind_rulify
- |> (fn th => th COMP ipres) (* P rs *)
- |> fold_rev (implies_intr o cprop_of) cGas
- |> fold_rev forall_intr cGvs
- end
-
- val P_recs = map2 mk_Prec rs ineq_press (* [P rec1, P rec2, ... ] *)
-
- val step = HOLogic.mk_Trueprop (list_comb (P, lhs))
- |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
- |> fold_rev (curry Logic.mk_implies) gs
- |> fold_rev (Logic.all o Free) qs
- |> cert
-
- val Plhs_to_Pxs_conv =
- foldl1 (uncurry Conv.combination_conv)
- (Conv.all_conv :: map (fn ch => K (Thm.symmetric (ch RS eq_reflection))) case_hyps)
-
- val res = assume step
- |> fold forall_elim cqs
- |> fold Thm.elim_implies ags
- |> fold Thm.elim_implies P_recs (* P lhs *)
- |> Conv.fconv_rule (Conv.arg_conv Plhs_to_Pxs_conv) (* P xs *)
- |> fold_rev (implies_intr o cprop_of) (ags @ case_hyps)
- |> fold_rev forall_intr cqs (* !!qs. Gas ==> xs = lhss ==> P xs *)
- in
- (res, (cidx, step))
- end
-
- val (cases, steps) = split_list (map2 prove_case relevant_cases ineqss')
-
- val bstep = complete_thm
- |> forall_elim (cert (list_comb (P, fxs)))
- |> fold (forall_elim o cert) (fxs @ map Free ws)
- |> fold Thm.elim_implies C_hyps (* FIXME: optimization using rotate_prems *)
- |> fold Thm.elim_implies cases (* P xs *)
- |> fold_rev (implies_intr o cprop_of) C_hyps
- |> fold_rev (forall_intr o cert o Free) ws
-
- val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
- |> Goal.init
- |> (MetaSimplifier.rewrite_goals_tac (map meta (branch_hyp :: @{thm split_conv} :: @{thms sum.cases}))
- THEN CONVERSION ind_rulify 1)
- |> Seq.hd
- |> Thm.elim_implies (Conv.fconv_rule Drule.beta_eta_conversion bstep)
- |> Goal.finish ctxt
- |> implies_intr (cprop_of branch_hyp)
- |> fold_rev (forall_intr o cert) fxs
- in
- (Pxs, steps)
- end
-
- val (branches, steps) = split_list (map_index prove_branch (branches ~~ (complete_thms ~~ pats)))
- |> apsnd flat
-
- val istep = sum_split_rule
- |> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
- |> implies_intr ihyp
- |> forall_intr (cert x) (* "!!x. (!!y<x. P y) ==> P x" *)
-
- val induct_rule =
- @{thm "wf_induct_rule"}
- |> (curry op COMP) wf_thm
- |> (curry op COMP) istep
-
- val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
- in
- (steps_sorted, induct_rule)
- end
-
-
-fun mk_ind_tac comp_tac pres_tac term_tac ctxt facts = (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL
-(SUBGOAL (fn (t, i) =>
- let
- val (ctxt', _, cases, concl) = dest_hhf ctxt t
- val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
-(* val _ = tracing (makestring scheme)*)
- val ([Rn,xn], ctxt'') = Variable.variant_fixes ["R","x"] ctxt'
- val R = Free (Rn, mk_relT ST)
- val x = Free (xn, ST)
- val cert = cterm_of (ProofContext.theory_of ctxt)
-
- val ineqss = mk_ineqs R scheme
- |> map (map (pairself (assume o cert)))
- val complete = map_range (mk_completeness ctxt scheme #> cert #> assume) (length branches)
- val wf_thm = mk_wf ctxt R scheme |> cert |> assume
-
- val (descent, pres) = split_list (flat ineqss)
- val newgoals = complete @ pres @ wf_thm :: descent
-
- val (steps, indthm) = mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme
-
- fun project (i, SchemeBranch {xs, ...}) =
- let
- val inst = cert (SumTree.mk_inj ST (length branches) (i + 1) (foldr1 HOLogic.mk_prod (map Free xs)))
- in
- indthm |> Drule.instantiate' [] [SOME inst]
- |> simplify SumTree.sumcase_split_ss
- |> Conv.fconv_rule ind_rulify
-(* |> (fn thm => (tracing (makestring thm); thm))*)
- end
-
- val res = Conjunction.intr_balanced (map_index project branches)
- |> fold_rev implies_intr (map cprop_of newgoals @ steps)
- |> (fn thm => Thm.generalize ([], [Rn]) (Thm.maxidx_of thm + 1) thm)
-
- val nbranches = length branches
- val npres = length pres
- in
- Thm.compose_no_flatten false (res, length newgoals) i
- THEN term_tac (i + nbranches + npres)
- THEN (EVERY (map (TRY o pres_tac) ((i + nbranches + npres - 1) downto (i + nbranches))))
- THEN (EVERY (map (TRY o comp_tac) ((i + nbranches - 1) downto i)))
- end))
-
-
-fun induct_scheme_tac ctxt =
- mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt;
-
-val setup =
- Method.setup @{binding induct_scheme} (Scan.succeed (RAW_METHOD o induct_scheme_tac))
- "proves an induction principle"
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Induction_Schema.thy Fri Nov 06 16:59:17 2009 +0100
@@ -0,0 +1,48 @@
+(* Title: HOL/ex/Induction_Schema.thy
+ Author: Alexander Krauss, TU Muenchen
+*)
+
+header {* Examples of automatically derived induction rules *}
+
+theory Induction_Schema
+imports Main
+begin
+
+subsection {* Some simple induction principles on nat *}
+
+lemma nat_standard_induct: (* cf. Nat.thy *)
+ "\<lbrakk>P 0; \<And>n. P n \<Longrightarrow> P (Suc n)\<rbrakk> \<Longrightarrow> P x"
+by induction_schema (pat_completeness, lexicographic_order)
+
+lemma nat_induct2:
+ "\<lbrakk> P 0; P (Suc 0); \<And>k. P k ==> P (Suc k) ==> P (Suc (Suc k)) \<rbrakk>
+ \<Longrightarrow> P n"
+by induction_schema (pat_completeness, lexicographic_order)
+
+lemma minus_one_induct:
+ "\<lbrakk>\<And>n::nat. (n \<noteq> 0 \<Longrightarrow> P (n - 1)) \<Longrightarrow> P n\<rbrakk> \<Longrightarrow> P x"
+by induction_schema (pat_completeness, lexicographic_order)
+
+theorem diff_induct: (* cf. Nat.thy *)
+ "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
+ (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
+by induction_schema (pat_completeness, lexicographic_order)
+
+lemma list_induct2': (* cf. List.thy *)
+ "\<lbrakk> P [] [];
+ \<And>x xs. P (x#xs) [];
+ \<And>y ys. P [] (y#ys);
+ \<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
+ \<Longrightarrow> P xs ys"
+by induction_schema (pat_completeness, lexicographic_order)
+
+theorem even_odd_induct:
+ assumes "R 0"
+ assumes "Q 0"
+ assumes "\<And>n. Q n \<Longrightarrow> R (Suc n)"
+ assumes "\<And>n. R n \<Longrightarrow> Q (Suc n)"
+ shows "R n" "Q n"
+ using assms
+by induction_schema (pat_completeness+, lexicographic_order)
+
+end
\ No newline at end of file
--- a/src/HOL/ex/Induction_Scheme.thy Fri Nov 06 14:16:57 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,49 +0,0 @@
-(* Title: HOL/ex/Induction_Scheme.thy
- ID: $Id$
- Author: Alexander Krauss, TU Muenchen
-*)
-
-header {* Examples of automatically derived induction rules *}
-
-theory Induction_Scheme
-imports Main
-begin
-
-subsection {* Some simple induction principles on nat *}
-
-lemma nat_standard_induct: (* cf. Nat.thy *)
- "\<lbrakk>P 0; \<And>n. P n \<Longrightarrow> P (Suc n)\<rbrakk> \<Longrightarrow> P x"
-by induct_scheme (pat_completeness, lexicographic_order)
-
-lemma nat_induct2:
- "\<lbrakk> P 0; P (Suc 0); \<And>k. P k ==> P (Suc k) ==> P (Suc (Suc k)) \<rbrakk>
- \<Longrightarrow> P n"
-by induct_scheme (pat_completeness, lexicographic_order)
-
-lemma minus_one_induct:
- "\<lbrakk>\<And>n::nat. (n \<noteq> 0 \<Longrightarrow> P (n - 1)) \<Longrightarrow> P n\<rbrakk> \<Longrightarrow> P x"
-by induct_scheme (pat_completeness, lexicographic_order)
-
-theorem diff_induct: (* cf. Nat.thy *)
- "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
- (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
-by induct_scheme (pat_completeness, lexicographic_order)
-
-lemma list_induct2': (* cf. List.thy *)
- "\<lbrakk> P [] [];
- \<And>x xs. P (x#xs) [];
- \<And>y ys. P [] (y#ys);
- \<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
- \<Longrightarrow> P xs ys"
-by induct_scheme (pat_completeness, lexicographic_order)
-
-theorem even_odd_induct:
- assumes "R 0"
- assumes "Q 0"
- assumes "\<And>n. Q n \<Longrightarrow> R (Suc n)"
- assumes "\<And>n. R n \<Longrightarrow> Q (Suc n)"
- shows "R n" "Q n"
- using assms
-by induct_scheme (pat_completeness+, lexicographic_order)
-
-end
\ No newline at end of file
--- a/src/HOL/ex/ROOT.ML Fri Nov 06 14:16:57 2009 +0100
+++ b/src/HOL/ex/ROOT.ML Fri Nov 06 16:59:17 2009 +0100
@@ -24,7 +24,7 @@
"Binary",
"Recdefs",
"Fundefs",
- "Induction_Scheme",
+ "Induction_Schema",
"InductiveInvariant_examples",
"LocaleTest2",
"Records",