From 2feb56660700af107abb5a28a7120052ac405518 Mon Sep 17 00:00:00 2001 From: Josh Chen Date: Sun, 31 Jan 2021 02:54:51 +0000 Subject: rename things + some small changes --- mltt/core/MLTT.thy | 569 ++++++++++++++++++++++++++++++++++++++ mltt/core/calc.ML | 87 ++++++ mltt/core/cases.ML | 42 +++ mltt/core/comp.ML | 468 +++++++++++++++++++++++++++++++ mltt/core/context_facts.ML | 101 +++++++ mltt/core/context_tactical.ML | 256 +++++++++++++++++ mltt/core/elaborated_statement.ML | 470 +++++++++++++++++++++++++++++++ mltt/core/elaboration.ML | 91 ++++++ mltt/core/elimination.ML | 48 ++++ mltt/core/eqsubst.ML | 442 +++++++++++++++++++++++++++++ mltt/core/focus.ML | 158 +++++++++++ mltt/core/goals.ML | 213 ++++++++++++++ mltt/core/implicits.ML | 87 ++++++ mltt/core/lib.ML | 193 +++++++++++++ mltt/core/tactics.ML | 180 ++++++++++++ mltt/core/types.ML | 113 ++++++++ mltt/lib/List.thy | 191 +++++++++++++ mltt/lib/Maybe.thy | 75 +++++ mltt/lib/Prelude.thy | 153 ++++++++++ 19 files changed, 3937 insertions(+) create mode 100644 mltt/core/MLTT.thy create mode 100644 mltt/core/calc.ML create mode 100644 mltt/core/cases.ML create mode 100644 mltt/core/comp.ML create mode 100644 mltt/core/context_facts.ML create mode 100644 mltt/core/context_tactical.ML create mode 100644 mltt/core/elaborated_statement.ML create mode 100644 mltt/core/elaboration.ML create mode 100644 mltt/core/elimination.ML create mode 100644 mltt/core/eqsubst.ML create mode 100644 mltt/core/focus.ML create mode 100644 mltt/core/goals.ML create mode 100644 mltt/core/implicits.ML create mode 100644 mltt/core/lib.ML create mode 100644 mltt/core/tactics.ML create mode 100644 mltt/core/types.ML create mode 100644 mltt/lib/List.thy create mode 100644 mltt/lib/Maybe.thy create mode 100644 mltt/lib/Prelude.thy (limited to 'mltt') diff --git a/mltt/core/MLTT.thy b/mltt/core/MLTT.thy new file mode 100644 index 0000000..18bd2b7 --- /dev/null +++ b/mltt/core/MLTT.thy @@ -0,0 +1,569 @@ +theory MLTT +imports + Pure + "HOL-Eisbach.Eisbach" + "HOL-Eisbach.Eisbach_Tools" +keywords + "Theorem" "Lemma" "Corollary" "Proposition" "Definition" :: thy_goal_stmt and + "assuming" :: prf_asm % "proof" and + "focus" "\<^item>" "\<^enum>" "\" "\" "~" :: prf_script_goal % "proof" and + "calc" "print_coercions" :: thy_decl and + "rhs" "def" "vars" :: quasi_command + +begin + +section \Notation\ + +declare [[eta_contract=false]] + +text \ +Rebind notation for meta-lambdas since we want to use \\\ for the object +lambdas. Metafunctions now use the binder \fn\. +\ +setup \ +let + val typ = Simple_Syntax.read_typ + fun mixfix (sy, ps, p) = Mixfix (Input.string sy, ps, p, Position.no_range) +in + Sign.del_syntax (Print_Mode.ASCII, true) + [("_lambda", typ "pttrns \ 'a \ logic", mixfix ("(3%_./ _)", [0, 3], 3))] + #> Sign.del_syntax Syntax.mode_default + [("_lambda", typ "pttrns \ 'a \ logic", mixfix ("(3\_./ _)", [0, 3], 3))] + #> Sign.add_syntax Syntax.mode_default + [("_lambda", typ "pttrns \ 'a \ logic", mixfix ("(3fn _./ _)", [0, 3], 3))] +end +\ + +syntax "_app" :: \logic \ logic \ logic\ (infixr "$" 3) +translations "a $ b" \ "a (b)" + +abbreviation (input) K where "K x \ fn _. x" + + +section \Metalogic\ + +text \ +HOAS embedding of dependent type theory: metatype of expressions, and typing +judgment. +\ + +typedecl o + +consts has_type :: \o \ o \ prop\ ("(2_:/ _)" 999) + + +section \Axioms\ + +subsection \Universes\ + +text \\-many cumulative Russell universes.\ + +typedecl lvl + +axiomatization + O :: \lvl\ and + S :: \lvl \ lvl\ and + lt :: \lvl \ lvl \ prop\ (infix "<" 900) + where + O_min: "O < S i" and + lt_S: "i < S i" and + lt_trans: "i < j \ j < k \ i < k" + +axiomatization U :: \lvl \ o\ where + Ui_in_Uj: "i < j \ U i: U j" and + U_cumul: "A: U i \ i < j \ A: U j" + +lemma Ui_in_USi: + "U i: U (S i)" + by (rule Ui_in_Uj, rule lt_S) + +lemma U_lift: + "A: U i \ A: U (S i)" + by (erule U_cumul, rule lt_S) + +subsection \\-type\ + +axiomatization + Pi :: \o \ (o \ o) \ o\ and + lam :: \o \ (o \ o) \ o\ and + app :: \o \ o \ o\ ("(1_ `_)" [120, 121] 120) + +syntax + "_Pi" :: \idts \ o \ o \ o\ ("(2\_: _./ _)" 30) + "_Pi2" :: \idts \ o \ o \ o\ + "_lam" :: \idts \ o \ o \ o\ ("(2\_: _./ _)" 30) + "_lam2" :: \idts \ o \ o \ o\ +translations + "\x xs: A. B" \ "CONST Pi A (fn x. _Pi2 xs A B)" + "_Pi2 x A B" \ "\x: A. B" + "\x: A. B" \ "CONST Pi A (fn x. B)" + "\x xs: A. b" \ "CONST lam A (fn x. _lam2 xs A b)" + "_lam2 x A b" \ "\x: A. b" + "\x: A. b" \ "CONST lam A (fn x. b)" + +abbreviation Fn (infixr "\" 40) where "A \ B \ \_: A. B" + +axiomatization where + PiF: "\A: U i; \x. x: A \ B x: U i\ \ \x: A. B x: U i" and + + PiI: "\A: U i; \x. x: A \ b x: B x\ \ \x: A. b x: \x: A. B x" and + + PiE: "\f: \x: A. B x; a: A\ \ f `a: B a" and + + beta: "\a: A; \x. x: A \ b x: B x\ \ (\x: A. b x) `a \ b a" and + + eta: "f: \x: A. B x \ \x: A. f `x \ f" and + + Pi_cong: "\ + \x. x: A \ B x \ B' x; + A: U i; + \x. x: A \ B x: U j; + \x. x: A \ B' x: U j + \ \ \x: A. B x \ \x: A. B' x" and + + lam_cong: "\\x. x: A \ b x \ c x; A: U i\ \ \x: A. b x \ \x: A. c x" + +subsection \\-type\ + +axiomatization + Sig :: \o \ (o \ o) \ o\ and + pair :: \o \ o \ o\ ("(2<_,/ _>)") and + SigInd :: \o \ (o \ o) \ (o \ o) \ (o \ o \ o) \ o \ o\ + +syntax "_Sum" :: \idt \ o \ o \ o\ ("(2\_: _./ _)" 20) + +translations "\x: A. B" \ "CONST Sig A (fn x. B)" + +abbreviation Prod (infixl "\" 60) + where "A \ B \ \_: A. B" + +abbreviation "and" (infixl "\" 60) + where "A \ B \ A \ B" + +axiomatization where + SigF: "\A: U i; \x. x: A \ B x: U i\ \ \x: A. B x: U i" and + + SigI: "\\x. x: A \ B x: U i; a: A; b: B a\ \ : \x: A. B x" and + + SigE: "\ + p: \x: A. B x; + A: U i; + \x. x : A \ B x: U j; + \p. p: \x: A. B x \ C p: U k; + \x y. \x: A; y: B x\ \ f x y: C + \ \ SigInd A (fn x. B x) (fn p. C p) f p: C p" and + + Sig_comp: "\ + a: A; + b: B a; + \x. x: A \ B x: U i; + \p. p: \x: A. B x \ C p: U i; + \x y. \x: A; y: B x\ \ f x y: C + \ \ SigInd A (fn x. B x) (fn p. C p) f \ f a b" and + + Sig_cong: "\ + \x. x: A \ B x \ B' x; + A: U i; + \x. x : A \ B x: U j; + \x. x : A \ B' x: U j + \ \ \x: A. B x \ \x: A. B' x" + + +section \Type checking & inference\ + +ML_file \lib.ML\ +ML_file \context_facts.ML\ +ML_file \context_tactical.ML\ + +\ \Rule attributes for the typechecker\ +named_theorems form and intr and comp + +\ \Elimination/induction automation and the `elim` attribute\ +ML_file \elimination.ML\ + +lemmas + [form] = PiF SigF and + [intr] = PiI SigI and + [elim ?f] = PiE and + [elim ?p] = SigE and + [comp] = beta Sig_comp and + [cong] = Pi_cong lam_cong Sig_cong + +\ \Subsumption rule\ +lemma sub: + assumes "a: A" "A \ A'" + shows "a: A'" + using assms by simp + +\ \Basic rewriting of computational equality\ +ML_file \~~/src/Tools/misc_legacy.ML\ +ML_file \~~/src/Tools/IsaPlanner/isand.ML\ +ML_file \~~/src/Tools/IsaPlanner/rw_inst.ML\ +ML_file \~~/src/Tools/IsaPlanner/zipper.ML\ +ML_file \~~/src/Tools/eqsubst.ML\ + +\ \Term normalization, type checking & inference\ +ML_file \types.ML\ + +method_setup typechk = + \Scan.succeed (K (CONTEXT_METHOD ( + CHEADGOAL o Types.check_infer)))\ + +method_setup known = + \Scan.succeed (K (CONTEXT_METHOD ( + CHEADGOAL o Types.known_ctac)))\ + +setup \ +let val typechk = fn ctxt => + NO_CONTEXT_TACTIC ctxt o Types.check_infer + (Simplifier.prems_of ctxt @ Context_Facts.known ctxt) +in + map_theory_simpset (fn ctxt => ctxt + addSolver (mk_solver "" typechk)) +end +\ + + +section \Statements and goals\ + +ML_file \focus.ML\ +ML_file \elaboration.ML\ +ML_file \elaborated_statement.ML\ +ML_file \goals.ML\ + + +section \Proof methods\ + +named_theorems intro \ \Logical introduction rules\ + +lemmas [intro] = PiI[rotated] SigI + +\ \Case reasoning rules\ +ML_file \cases.ML\ + +ML_file \tactics.ML\ + +method_setup rule = + \Attrib.thms >> (fn ths => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 (rule_ctac ths))))\ + +method_setup dest = + \Scan.lift (Scan.option (Args.parens Parse.nat)) + -- Attrib.thms >> (fn (n_opt, ths) => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 (dest_ctac n_opt ths))))\ + +method_setup intro = + \Scan.succeed (K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 intro_ctac)))\ + +method_setup intros = + \Scan.lift (Scan.option Parse.nat) >> (fn n_opt => + K (CONTEXT_METHOD (fn facts => + case n_opt of + SOME n => CREPEAT_N n (CHEADGOAL (SIDE_CONDS 0 intro_ctac facts)) + | NONE => CCHANGED (CREPEAT (CCHANGED ( + CHEADGOAL (SIDE_CONDS 0 intro_ctac facts)))))))\ + +method_setup elim = + \Scan.repeat Args.term >> (fn tms => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 (elim_ctac tms))))\ + +method_setup cases = + \Args.term >> (fn tm => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 (cases_ctac tm))))\ + +method elims = elim+ +method facts = fact+ + + +subsection \Reflexivity\ + +named_theorems refl +method refl = (rule refl) + + +subsection \Trivial proofs (modulo automatic discharge of side conditions)\ + +method_setup this = + \Scan.succeed (K (CONTEXT_METHOD (fn facts => + CHEADGOAL (SIDE_CONDS 0 + (CONTEXT_TACTIC' (fn ctxt => resolve_tac ctxt facts)) + facts))))\ + + +subsection \Rewriting\ + +consts compute_hole :: "'a::{}" ("\") + +lemma eta_expand: + fixes f :: "'a::{} \ 'b::{}" + shows "f \ fn x. f x" . + +lemma rewr_imp: + assumes "PROP A \ PROP B" + shows "(PROP A \ PROP C) \ (PROP B \ PROP C)" + apply (Pure.rule Pure.equal_intr_rule) + apply (drule equal_elim_rule2[OF assms]; assumption) + apply (drule equal_elim_rule1[OF assms]; assumption) + done + +lemma imp_cong_eq: + "(PROP A \ (PROP B \ PROP C) \ (PROP B' \ PROP C')) \ + ((PROP B \ PROP A \ PROP C) \ (PROP B' \ PROP A \ PROP C'))" + apply (Pure.intro Pure.equal_intr_rule) + apply (drule (1) cut_rl; drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; + assumption)+ + apply (drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; assumption)+ + done + +ML_file \~~/src/HOL/Library/cconv.ML\ +ML_file \comp.ML\ + +\ \\compute\ simplifies terms via computational equalities\ +method compute uses add = + changed \repeat_new \(simp add: comp add | subst comp); typechk?\\ + + +subsection \Calculational reasoning\ + +consts "rhs" :: \'a\ ("..") + +ML_file \calc.ML\ + + +section \Implicits\ + +text \ + \{}\ is used to mark implicit arguments in definitions, while \?\ is expanded + immediately for elaboration in statements. +\ + +consts + iarg :: \'a\ ("{}") + hole :: \'b\ ("?") + +ML_file \implicits.ML\ + +attribute_setup implicit = \Scan.succeed Implicits.implicit_defs_attr\ + +ML \val _ = Context.>> (Syntax_Phases.term_check 1 "" Implicits.make_holes)\ + +text \Automatically insert inhabitation judgments where needed:\ + +syntax inhabited :: \o \ prop\ ("(_)") +translations "inhabited A" \ "CONST has_type ? A" + + +subsection \Implicit lambdas\ + +definition lam_i where [implicit]: "lam_i f \ lam {} f" + +syntax + "_lam_i" :: \idts \ o \ o\ ("(2\_./ _)" 30) + "_lam_i2" :: \idts \ o \ o\ +translations + "\x xs. b" \ "CONST lam_i (fn x. _lam_i2 xs b)" + "_lam_i2 x b" \ "\x. b" + "\x. b" \ "CONST lam_i (fn x. b)" + +translations "\x. b" \ "\x: A. b" + + +section \Lambda coercion\ + +\ \Coerce object lambdas to meta-lambdas\ +abbreviation (input) lambda :: \o \ o \ o\ + where "lambda f \ fn x. f `x" + +ML_file \~~/src/Tools/subtyping.ML\ +declare [[coercion_enabled, coercion lambda]] + +translations "f x" \ "f `x" + + +section \Functions\ + +Lemma eta_exp: + assumes "f: \x: A. B x" + shows "f \ \x: A. f x" + by (rule eta[symmetric]) + +Lemma refine_codomain: + assumes + "A: U i" + "f: \x: A. B x" + "\x. x: A \ f `x: C x" + shows "f: \x: A. C x" + by (comp eta_exp) + +Lemma lift_universe_codomain: + assumes "A: U i" "f: A \ U j" + shows "f: A \ U (S j)" + using U_lift + by (rule refine_codomain) + +subsection \Function composition\ + +definition "funcomp A g f \ \x: A. g `(f `x)" + +syntax + "_funcomp" :: \o \ o \ o \ o\ ("(2_ \\<^bsub>_\<^esub>/ _)" [111, 0, 110] 110) +translations + "g \\<^bsub>A\<^esub> f" \ "CONST funcomp A g f" + +Lemma funcompI [type]: + assumes + "A: U i" + "B: U i" + "\x. x: B \ C x: U i" + "f: A \ B" + "g: \x: B. C x" + shows + "g \\<^bsub>A\<^esub> f: \x: A. C (f x)" + unfolding funcomp_def by typechk + +Lemma funcomp_assoc [comp]: + assumes + "A: U i" + "f: A \ B" + "g: B \ C" + "h: \x: C. D x" + shows + "(h \\<^bsub>B\<^esub> g) \\<^bsub>A\<^esub> f \ h \\<^bsub>A\<^esub> g \\<^bsub>A\<^esub> f" + unfolding funcomp_def by compute + +Lemma funcomp_lambda_comp [comp]: + assumes + "A: U i" + "\x. x: A \ b x: B" + "\x. x: B \ c x: C x" + shows + "(\x: B. c x) \\<^bsub>A\<^esub> (\x: A. b x) \ \x: A. c (b x)" + unfolding funcomp_def by compute + +Lemma funcomp_apply_comp [comp]: + assumes + "A: U i" "B: U i" "\x y. x: B \ C x: U i" + "f: A \ B" "g: \x: B. C x" + "x: A" + shows "(g \\<^bsub>A\<^esub> f) x \ g (f x)" + unfolding funcomp_def by compute + +subsection \Notation\ + +definition funcomp_i (infixr "\" 120) + where [implicit]: "funcomp_i g f \ g \\<^bsub>{}\<^esub> f" + +translations "g \ f" \ "g \\<^bsub>A\<^esub> f" + +subsection \Identity function\ + +abbreviation id where "id A \ \x: A. x" + +lemma + id_type [type]: "A: U i \ id A: A \ A" and + id_comp [comp]: "x: A \ (id A) x \ x" \ \for the occasional manual rewrite\ + by compute+ + +Lemma id_left [comp]: + assumes "A: U i" "B: U i" "f: A \ B" + shows "(id B) \\<^bsub>A\<^esub> f \ f" + by (comp eta_exp[of f]) (compute, rule eta) + +Lemma id_right [comp]: + assumes "A: U i" "B: U i" "f: A \ B" + shows "f \\<^bsub>A\<^esub> (id A) \ f" + by (comp eta_exp[of f]) (compute, rule eta) + +lemma id_U [type]: + "id (U i): U i \ U i" + using Ui_in_USi by typechk + + +section \Pairs\ + +definition "fst A B \ \p: \x: A. B x. SigInd A B (fn _. A) (fn x y. x) p" +definition "snd A B \ \p: \x: A. B x. SigInd A B (fn p. B (fst A B p)) (fn x y. y) p" + +Lemma fst_type [type]: + assumes "A: U i" "\x. x: A \ B x: U i" + shows "fst A B: (\x: A. B x) \ A" + unfolding fst_def by typechk + +Lemma fst_comp [comp]: + assumes + "A: U i" "\x. x: A \ B x: U i" "a: A" "b: B a" + shows "fst A B \ a" + unfolding fst_def by compute + +Lemma snd_type [type]: + assumes "A: U i" "\x. x: A \ B x: U i" + shows "snd A B: \p: \x: A. B x. B (fst A B p)" + unfolding snd_def by typechk + +Lemma snd_comp [comp]: + assumes "A: U i" "\x. x: A \ B x: U i" "a: A" "b: B a" + shows "snd A B \ b" + unfolding snd_def by compute + +subsection \Notation\ + +definition fst_i ("fst") + where [implicit]: "fst \ MLTT.fst {} {}" + +definition snd_i ("snd") + where [implicit]: "snd \ MLTT.snd {} {}" + +translations + "fst" \ "CONST MLTT.fst A B" + "snd" \ "CONST MLTT.snd A B" + +subsection \Projections\ + +Lemma fst [type]: + assumes + "A: U i" "\x. x: A \ B x: U i" + "p: \x: A. B x" + shows "fst p: A" + by typechk + +Lemma snd [type]: + assumes + "A: U i" "\x. x: A \ B x: U i" + "p: \x: A. B x" + shows "snd p: B (fst p)" + by typechk + +method fst for p::o = rule fst[where ?p=p] +method snd for p::o = rule snd[where ?p=p] + +text \Double projections:\ + +definition [implicit]: "p\<^sub>1\<^sub>1 p \ MLTT.fst {} {} (MLTT.fst {} {} p)" +definition [implicit]: "p\<^sub>1\<^sub>2 p \ MLTT.snd {} {} (MLTT.fst {} {} p)" +definition [implicit]: "p\<^sub>2\<^sub>1 p \ MLTT.fst {} {} (MLTT.snd {} {} p)" +definition [implicit]: "p\<^sub>2\<^sub>2 p \ MLTT.snd {} {} (MLTT.snd {} {} p)" + +translations + "CONST p\<^sub>1\<^sub>1 p" \ "fst (fst p)" + "CONST p\<^sub>1\<^sub>2 p" \ "snd (fst p)" + "CONST p\<^sub>2\<^sub>1 p" \ "fst (snd p)" + "CONST p\<^sub>2\<^sub>2 p" \ "snd (snd p)" + +Lemma (def) distribute_Sig: + assumes + "A: U i" + "\x. x: A \ B x: U i" + "\x. x: A \ C x: U i" + "p: \x: A. B x \ C x" + shows "(\x: A. B x) \ (\x: A. C x)" + proof intro + have "fst p: A" and "snd p: B (fst p) \ C (fst p)" + by typechk+ + thus ": \x: A. B x" + and ": \x: A. C x" + by typechk+ + qed + + +end diff --git a/mltt/core/calc.ML b/mltt/core/calc.ML new file mode 100644 index 0000000..67dc7fc --- /dev/null +++ b/mltt/core/calc.ML @@ -0,0 +1,87 @@ +structure Calc = struct + +(* Calculational type context data + +A "calculational" type is a type expressing some congruence relation. In +particular, it has a notion of composition of terms that is often used to derive +proofs equationally. +*) + +structure RHS = Generic_Data ( + type T = (term * indexname) Termtab.table + val empty = Termtab.empty + val extend = I + val merge = Termtab.merge (Term.aconv o apply2 #1) +) + +fun register_rhs t var = + let + val key = Term.head_of t + val idxname = #1 (dest_Var var) + in + RHS.map (Termtab.update (key, (t, idxname))) + end + +fun lookup_calc ctxt t = + Termtab.lookup (RHS.get (Context.Proof ctxt)) (Term.head_of t) + + +(* Declaration *) + +local val Frees_to_Vars = + map_aterms (fn tm => + case tm of + Free (name, T) => Var (("*!"^name, 0), T) (*FIXME: Hacky naming!*) + | _ => tm) +in + +(*Declare the "right-hand side" of calculational types. Does not handle bound + variables, so no dependent RHS in declarations!*) +val _ = Outer_Syntax.local_theory \<^command_keyword>\calc\ + "declare right hand side of calculational type" + (Parse.term -- (\<^keyword>\rhs\ |-- Parse.term) >> + (fn (t_str, rhs_str) => fn lthy => + let + val (t, rhs) = apply2 (Frees_to_Vars o Syntax.read_term lthy) + (t_str, rhs_str) + in lthy |> + Local_Theory.background_theory ( + Context.theory_map (register_rhs t rhs)) + end)) + +end + + +(* Ditto "''" setup *) + +fun last_rhs ctxt = map_aterms (fn t => + case t of + Const (\<^const_name>\rhs\, _) => + let + val this_name = Name_Space.full_name (Proof_Context.naming_of ctxt) + (Binding.name Auto_Bind.thisN) + val this = #thms (the (Proof_Context.lookup_fact ctxt this_name)) + handle Option => [] + val rhs = + (case map Thm.prop_of this of + [prop] => + (let + val typ = Lib.type_of_typing (Logic.strip_assums_concl prop) + val (cong_pttrn, varname) = the (lookup_calc ctxt typ) + val unif_res = Pattern.unify (Context.Proof ctxt) + (cong_pttrn, typ) Envir.init + val rhs = #2 (the + (Vartab.lookup (Envir.term_env unif_res) varname)) + in + rhs + end handle Option => + error (".. can't match right-hand side of calculational type")) + | _ => Term.dummy) + in rhs end + | _ => t) + +val _ = Context.>> + (Syntax_Phases.term_check 5 "" (fn ctxt => map (last_rhs ctxt))) + + +end diff --git a/mltt/core/cases.ML b/mltt/core/cases.ML new file mode 100644 index 0000000..560a9f1 --- /dev/null +++ b/mltt/core/cases.ML @@ -0,0 +1,42 @@ +(* Title: cases.ML + Author: Joshua Chen + +Case reasoning. +*) + +structure Case: sig + +val rules: Proof.context -> thm list +val lookup_rule: Proof.context -> Termtab.key -> thm option +val register_rule: thm -> Context.generic -> Context.generic + +end = struct + +(* Context data *) + +(*Stores elimination rules together with a list of the indexnames of the + variables each rule eliminates. Keyed by head of the type being eliminated.*) +structure Rules = Generic_Data ( + type T = thm Termtab.table + val empty = Termtab.empty + val extend = I + val merge = Termtab.merge Thm.eq_thm_prop +) + +val rules = map #2 o Termtab.dest o Rules.get o Context.Proof +fun lookup_rule ctxt = Termtab.lookup (Rules.get (Context.Proof ctxt)) +fun register_rule rl = + let val hd = Term.head_of (Lib.type_of_typing (Thm.major_prem_of rl)) + in Rules.map (Termtab.update (hd, rl)) end + + +(* [cases] attribute *) +val _ = Theory.setup ( + Attrib.setup \<^binding>\cases\ + (Scan.succeed (Thm.declaration_attribute register_rule)) + "" + #> Global_Theory.add_thms_dynamic (\<^binding>\cases\, rules o Context.proof_of) +) + + +end diff --git a/mltt/core/comp.ML b/mltt/core/comp.ML new file mode 100644 index 0000000..2e50753 --- /dev/null +++ b/mltt/core/comp.ML @@ -0,0 +1,468 @@ +(* Title: compute.ML + Author: Christoph Traut, Lars Noschinski, TU Muenchen + Modified: Joshua Chen, University of Innsbruck + +This is a method for rewriting computational equalities that supports subterm +selection based on patterns. + +This code has been slightly modified from the original at HOL/Library/compute.ML +to incorporate automatic discharge of type-theoretic side conditions. + +Comment from the original code follows: + +The patterns accepted by compute are of the following form: + ::= | "concl" | "asm" | "for" "(" ")" + ::= (in | at ) [] + ::= [] ("to" ) + +This syntax was clearly inspired by Gonthier's and Tassi's language of +patterns but has diverged significantly during its development. + +We also allow introduction of identifiers for bound variables, +which can then be used to match arbitrary subterms inside abstractions. +*) + +infix 1 then_pconv; +infix 0 else_pconv; + +signature COMPUTE = +sig + type patconv = Proof.context -> Type.tyenv * (string * term) list -> cconv + val then_pconv: patconv * patconv -> patconv + val else_pconv: patconv * patconv -> patconv + val abs_pconv: patconv -> string option * typ -> patconv (*XXX*) + val fun_pconv: patconv -> patconv + val arg_pconv: patconv -> patconv + val imp_pconv: patconv -> patconv + val params_pconv: patconv -> patconv + val forall_pconv: patconv -> string option * typ option -> patconv + val all_pconv: patconv + val for_pconv: patconv -> (string option * typ option) list -> patconv + val concl_pconv: patconv -> patconv + val asm_pconv: patconv -> patconv + val asms_pconv: patconv -> patconv + val judgment_pconv: patconv -> patconv + val in_pconv: patconv -> patconv + val match_pconv: patconv -> term * (string option * typ) list -> patconv + val comps_pconv: term option -> thm list -> patconv + + datatype ('a, 'b) pattern = At | In | Term of 'a | Concl | Asm | For of 'b list + + val mk_hole: int -> typ -> term + + val compute_conv: Proof.context + -> (term * (string * typ) list, string * typ option) pattern list * term option + -> thm list + -> conv +end + +structure Compute : COMPUTE = +struct + +datatype ('a, 'b) pattern = At | In | Term of 'a | Concl | Asm | For of 'b list + +exception NO_TO_MATCH + +val holeN = Name.internal "_hole" + +fun prep_meta_eq ctxt = Simplifier.mksimps ctxt #> map Drule.zero_var_indexes + + +(* holes *) + +fun mk_hole i T = Var ((holeN, i), T) + +fun is_hole (Var ((name, _), _)) = (name = holeN) + | is_hole _ = false + +fun is_hole_const (Const (\<^const_name>\compute_hole\, _)) = true + | is_hole_const _ = false + +val hole_syntax = + let + (* Modified variant of Term.replace_hole *) + fun replace_hole Ts (Const (\<^const_name>\compute_hole\, T)) i = + (list_comb (mk_hole i (Ts ---> T), map_range Bound (length Ts)), i + 1) + | replace_hole Ts (Abs (x, T, t)) i = + let val (t', i') = replace_hole (T :: Ts) t i + in (Abs (x, T, t'), i') end + | replace_hole Ts (t $ u) i = + let + val (t', i') = replace_hole Ts t i + val (u', i'') = replace_hole Ts u i' + in (t' $ u', i'') end + | replace_hole _ a i = (a, i) + fun prep_holes ts = #1 (fold_map (replace_hole []) ts 1) + in + Context.proof_map (Syntax_Phases.term_check 101 "hole_expansion" (K prep_holes)) + #> Proof_Context.set_mode Proof_Context.mode_pattern + end + + +(* pattern conversions *) + +type patconv = Proof.context -> Type.tyenv * (string * term) list -> cterm -> thm + +fun (cv1 then_pconv cv2) ctxt tytenv ct = (cv1 ctxt tytenv then_conv cv2 ctxt tytenv) ct + +fun (cv1 else_pconv cv2) ctxt tytenv ct = (cv1 ctxt tytenv else_conv cv2 ctxt tytenv) ct + +fun raw_abs_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + Abs _ => CConv.abs_cconv (fn (x, ctxt') => cv x ctxt' tytenv) ctxt ct + | t => raise TERM ("raw_abs_pconv", [t]) + +fun raw_fun_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + _ $ _ => CConv.fun_cconv (cv ctxt tytenv) ct + | t => raise TERM ("raw_fun_pconv", [t]) + +fun raw_arg_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + _ $ _ => CConv.arg_cconv (cv ctxt tytenv) ct + | t => raise TERM ("raw_arg_pconv", [t]) + +fun abs_pconv cv (s,T) ctxt (tyenv, ts) ct = + let val u = Thm.term_of ct + in + case try (fastype_of #> dest_funT) u of + NONE => raise TERM ("abs_pconv: no function type", [u]) + | SOME (U, _) => + let + val tyenv' = + if T = dummyT then tyenv + else Sign.typ_match (Proof_Context.theory_of ctxt) (T, U) tyenv + val eta_expand_cconv = + case u of + Abs _=> Thm.reflexive + | _ => CConv.rewr_cconv @{thm eta_expand} + fun add_ident NONE _ l = l + | add_ident (SOME name) ct l = (name, Thm.term_of ct) :: l + val abs_cv = CConv.abs_cconv (fn (ct, ctxt) => cv ctxt (tyenv', add_ident s ct ts)) ctxt + in (eta_expand_cconv then_conv abs_cv) ct end + handle Pattern.MATCH => raise TYPE ("abs_pconv: types don't match", [T,U], [u]) + end + +fun fun_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + _ $ _ => CConv.fun_cconv (cv ctxt tytenv) ct + | Abs (_, T, _ $ Bound 0) => abs_pconv (fun_pconv cv) (NONE, T) ctxt tytenv ct + | t => raise TERM ("fun_pconv", [t]) + +local + +fun arg_pconv_gen cv0 cv ctxt tytenv ct = + case Thm.term_of ct of + _ $ _ => cv0 (cv ctxt tytenv) ct + | Abs (_, T, _ $ Bound 0) => abs_pconv (arg_pconv_gen cv0 cv) (NONE, T) ctxt tytenv ct + | t => raise TERM ("arg_pconv_gen", [t]) + +in + +fun arg_pconv ctxt = arg_pconv_gen CConv.arg_cconv ctxt +fun imp_pconv ctxt = arg_pconv_gen (CConv.concl_cconv 1) ctxt + +end + +(* Move to B in !!x_1 ... x_n. B. Do not eta-expand *) +fun params_pconv cv ctxt tytenv ct = + let val pconv = + case Thm.term_of ct of + Const (\<^const_name>\Pure.all\, _) $ Abs _ => (raw_arg_pconv o raw_abs_pconv) (fn _ => params_pconv cv) + | Const (\<^const_name>\Pure.all\, _) => raw_arg_pconv (params_pconv cv) + | _ => cv + in pconv ctxt tytenv ct end + +fun forall_pconv cv ident ctxt tytenv ct = + case Thm.term_of ct of + Const (\<^const_name>\Pure.all\, T) $ _ => + let + val def_U = T |> dest_funT |> fst |> dest_funT |> fst + val ident' = apsnd (the_default (def_U)) ident + in arg_pconv (abs_pconv cv ident') ctxt tytenv ct end + | t => raise TERM ("forall_pconv", [t]) + +fun all_pconv _ _ = Thm.reflexive + +fun for_pconv cv idents ctxt tytenv ct = + let + fun f rev_idents (Const (\<^const_name>\Pure.all\, _) $ t) = + let val (rev_idents', cv') = f rev_idents (case t of Abs (_,_,u) => u | _ => t) + in + case rev_idents' of + [] => ([], forall_pconv cv' (NONE, NONE)) + | (x :: xs) => (xs, forall_pconv cv' x) + end + | f rev_idents _ = (rev_idents, cv) + in + case f (rev idents) (Thm.term_of ct) of + ([], cv') => cv' ctxt tytenv ct + | _ => raise CTERM ("for_pconv", [ct]) + end + +fun concl_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + (Const (\<^const_name>\Pure.imp\, _) $ _) $ _ => imp_pconv (concl_pconv cv) ctxt tytenv ct + | _ => cv ctxt tytenv ct + +fun asm_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + (Const (\<^const_name>\Pure.imp\, _) $ _) $ _ => CConv.with_prems_cconv ~1 (cv ctxt tytenv) ct + | t => raise TERM ("asm_pconv", [t]) + +fun asms_pconv cv ctxt tytenv ct = + case Thm.term_of ct of + (Const (\<^const_name>\Pure.imp\, _) $ _) $ _ => + ((CConv.with_prems_cconv ~1 oo cv) else_pconv imp_pconv (asms_pconv cv)) ctxt tytenv ct + | t => raise TERM ("asms_pconv", [t]) + +fun judgment_pconv cv ctxt tytenv ct = + if Object_Logic.is_judgment ctxt (Thm.term_of ct) + then arg_pconv cv ctxt tytenv ct + else cv ctxt tytenv ct + +fun in_pconv cv ctxt tytenv ct = + (cv else_pconv + raw_fun_pconv (in_pconv cv) else_pconv + raw_arg_pconv (in_pconv cv) else_pconv + raw_abs_pconv (fn _ => in_pconv cv)) + ctxt tytenv ct + +fun replace_idents idents t = + let + fun subst ((n1, s)::ss) (t as Free (n2, _)) = if n1 = n2 then s else subst ss t + | subst _ t = t + in Term.map_aterms (subst idents) t end + +fun match_pconv cv (t,fixes) ctxt (tyenv, env_ts) ct = + let + val t' = replace_idents env_ts t + val thy = Proof_Context.theory_of ctxt + val u = Thm.term_of ct + + fun descend_hole fixes (Abs (_, _, t)) = + (case descend_hole fixes t of + NONE => NONE + | SOME (fix :: fixes', pos) => SOME (fixes', abs_pconv pos fix) + | SOME ([], _) => raise Match (* less fixes than abstractions on path to hole *)) + | descend_hole fixes (t as l $ r) = + let val (f, _) = strip_comb t + in + if is_hole f + then SOME (fixes, cv) + else + (case descend_hole fixes l of + SOME (fixes', pos) => SOME (fixes', fun_pconv pos) + | NONE => + (case descend_hole fixes r of + SOME (fixes', pos) => SOME (fixes', arg_pconv pos) + | NONE => NONE)) + end + | descend_hole fixes t = + if is_hole t then SOME (fixes, cv) else NONE + + val to_hole = descend_hole (rev fixes) #> the_default ([], cv) #> snd + in + case try (Pattern.match thy (apply2 Logic.mk_term (t',u))) (tyenv, Vartab.empty) of + NONE => raise TERM ("match_pconv: Does not match pattern", [t, t',u]) + | SOME (tyenv', _) => to_hole t ctxt (tyenv', env_ts) ct + end + +fun comps_pconv to thms ctxt (tyenv, env_ts) = + let + fun instantiate_normalize_env ctxt env thm = + let + val prop = Thm.prop_of thm + val norm_type = Envir.norm_type o Envir.type_env + val insts = Term.add_vars prop [] + |> map (fn x as (s, T) => + ((s, norm_type env T), Thm.cterm_of ctxt (Envir.norm_term env (Var x)))) + val tyinsts = Term.add_tvars prop [] + |> map (fn x => (x, Thm.ctyp_of ctxt (norm_type env (TVar x)))) + in Drule.instantiate_normalize (tyinsts, insts) thm end + + fun unify_with_rhs context to env thm = + let + val (_, rhs) = thm |> Thm.concl_of |> Logic.dest_equals + val env' = Pattern.unify context (Logic.mk_term to, Logic.mk_term rhs) env + handle Pattern.Unif => raise NO_TO_MATCH + in env' end + + fun inst_thm_to _ (NONE, _) thm = thm + | inst_thm_to (ctxt : Proof.context) (SOME to, env) thm = + instantiate_normalize_env ctxt (unify_with_rhs (Context.Proof ctxt) to env thm) thm + + fun inst_thm ctxt idents (to, tyenv) thm = + let + (* Replace any identifiers with their corresponding bound variables. *) + val maxidx = Term.maxidx_typs (map (snd o snd) (Vartab.dest tyenv)) 0 + val env = Envir.Envir {maxidx = maxidx, tenv = Vartab.empty, tyenv = tyenv} + val maxidx = Envir.maxidx_of env |> fold Term.maxidx_term (the_list to) + val thm' = Thm.incr_indexes (maxidx + 1) thm + in SOME (inst_thm_to ctxt (Option.map (replace_idents idents) to, env) thm') end + handle NO_TO_MATCH => NONE + + in CConv.rewrs_cconv (map_filter (inst_thm ctxt env_ts (to, tyenv)) thms) end + +fun compute_conv ctxt (pattern, to) thms ct = + let + fun apply_pat At = judgment_pconv + | apply_pat In = in_pconv + | apply_pat Asm = params_pconv o asms_pconv + | apply_pat Concl = params_pconv o concl_pconv + | apply_pat (For idents) = (fn cv => for_pconv cv (map (apfst SOME) idents)) + | apply_pat (Term x) = (fn cv => match_pconv cv (apsnd (map (apfst SOME)) x)) + + val cv = fold_rev apply_pat pattern + + fun distinct_prems th = + case Seq.pull (distinct_subgoals_tac th) of + NONE => th + | SOME (th', _) => th' + + val compute = comps_pconv to (maps (prep_meta_eq ctxt) thms) + in cv compute ctxt (Vartab.empty, []) ct |> distinct_prems end + +fun compute_export_tac ctxt (pat, pat_ctxt) thms = + let + val export = case pat_ctxt of + NONE => I + | SOME inner => singleton (Proof_Context.export inner ctxt) + in CCONVERSION (export o compute_conv ctxt pat thms) end + +val _ = + Theory.setup + let + fun mk_fix s = (Binding.name s, NONE, NoSyn) + + val raw_pattern : (string, binding * string option * mixfix) pattern list parser = + let + val sep = (Args.$$$ "at" >> K At) || (Args.$$$ "in" >> K In) + val atom = (Args.$$$ "asm" >> K Asm) || + (Args.$$$ "concl" >> K Concl) || + (Args.$$$ "for" |-- Args.parens (Scan.optional Parse.vars []) >> For) || + (Parse.term >> Term) + val sep_atom = sep -- atom >> (fn (s,a) => [s,a]) + + fun append_default [] = [Concl, In] + | append_default (ps as Term _ :: _) = Concl :: In :: ps + | append_default [For x, In] = [For x, Concl, In] + | append_default (For x :: (ps as In :: Term _:: _)) = For x :: Concl :: ps + | append_default ps = ps + + in Scan.repeats sep_atom >> (rev #> append_default) end + + fun context_lift (scan : 'a parser) f = fn (context : Context.generic, toks) => + let + val (r, toks') = scan toks + val (r', context') = Context.map_proof_result (fn ctxt => f ctxt r) context + in (r', (context', toks' : Token.T list)) end + + fun read_fixes fixes ctxt = + let fun read_typ (b, rawT, mx) = (b, Option.map (Syntax.read_typ ctxt) rawT, mx) + in Proof_Context.add_fixes (map read_typ fixes) ctxt end + + fun prep_pats ctxt (ps : (string, binding * string option * mixfix) pattern list) = + let + fun add_constrs ctxt n (Abs (x, T, t)) = + let + val (x', ctxt') = yield_singleton Proof_Context.add_fixes (mk_fix x) ctxt + in + (case add_constrs ctxt' (n+1) t of + NONE => NONE + | SOME ((ctxt'', n', xs), t') => + let + val U = Type_Infer.mk_param n [] + val u = Type.constraint (U --> dummyT) (Abs (x, T, t')) + in SOME ((ctxt'', n', (x', U) :: xs), u) end) + end + | add_constrs ctxt n (l $ r) = + (case add_constrs ctxt n l of + SOME (c, l') => SOME (c, l' $ r) + | NONE => + (case add_constrs ctxt n r of + SOME (c, r') => SOME (c, l $ r') + | NONE => NONE)) + | add_constrs ctxt n t = + if is_hole_const t then SOME ((ctxt, n, []), t) else NONE + + fun prep (Term s) (n, ctxt) = + let + val t = Syntax.parse_term ctxt s + val ((ctxt', n', bs), t') = + the_default ((ctxt, n, []), t) (add_constrs ctxt (n+1) t) + in (Term (t', bs), (n', ctxt')) end + | prep (For ss) (n, ctxt) = + let val (ns, ctxt') = read_fixes ss ctxt + in (For ns, (n, ctxt')) end + | prep At (n,ctxt) = (At, (n, ctxt)) + | prep In (n,ctxt) = (In, (n, ctxt)) + | prep Concl (n,ctxt) = (Concl, (n, ctxt)) + | prep Asm (n,ctxt) = (Asm, (n, ctxt)) + + val (xs, (_, ctxt')) = fold_map prep ps (0, ctxt) + + in (xs, ctxt') end + + fun prep_args ctxt (((raw_pats, raw_to), raw_ths)) = + let + + fun check_terms ctxt ps to = + let + fun safe_chop (0: int) xs = ([], xs) + | safe_chop n (x :: xs) = chop (n - 1) xs |>> cons x + | safe_chop _ _ = raise Match + + fun reinsert_pat _ (Term (_, cs)) (t :: ts) = + let val (cs', ts') = safe_chop (length cs) ts + in (Term (t, map dest_Free cs'), ts') end + | reinsert_pat _ (Term _) [] = raise Match + | reinsert_pat ctxt (For ss) ts = + let val fixes = map (fn s => (s, Variable.default_type ctxt s)) ss + in (For fixes, ts) end + | reinsert_pat _ At ts = (At, ts) + | reinsert_pat _ In ts = (In, ts) + | reinsert_pat _ Concl ts = (Concl, ts) + | reinsert_pat _ Asm ts = (Asm, ts) + + fun free_constr (s,T) = Type.constraint T (Free (s, dummyT)) + fun mk_free_constrs (Term (t, cs)) = t :: map free_constr cs + | mk_free_constrs _ = [] + + val ts = maps mk_free_constrs ps @ the_list to + |> Syntax.check_terms (hole_syntax ctxt) + val ctxt' = fold Variable.declare_term ts ctxt + val (ps', (to', ts')) = fold_map (reinsert_pat ctxt') ps ts + ||> (fn xs => case to of NONE => (NONE, xs) | SOME _ => (SOME (hd xs), tl xs)) + val _ = case ts' of (_ :: _) => raise Match | [] => () + in ((ps', to'), ctxt') end + + val (pats, ctxt') = prep_pats ctxt raw_pats + + val ths = Attrib.eval_thms ctxt' raw_ths + val to = Option.map (Syntax.parse_term ctxt') raw_to + + val ((pats', to'), ctxt'') = check_terms ctxt' pats to + + in ((pats', ths, (to', ctxt)), ctxt'') end + + val to_parser = Scan.option ((Args.$$$ "to") |-- Parse.term) + + val subst_parser = + let val scan = raw_pattern -- to_parser -- Parse.thms1 + in context_lift scan prep_args end + + fun compute_export_ctac inputs inthms = + CONTEXT_TACTIC' (fn ctxt => compute_export_tac ctxt inputs inthms) + in + Method.setup \<^binding>\cmp\ (subst_parser >> + (fn (pattern, inthms, (to, pat_ctxt)) => fn orig_ctxt => SIMPLE_METHOD' + (compute_export_tac orig_ctxt ((pattern, to), SOME pat_ctxt) inthms))) + "single-step rewriting, allowing subterm selection via patterns" #> + Method.setup \<^binding>\comp\ (subst_parser >> + (fn (pattern, inthms, (to, pat_ctxt)) => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 + (compute_export_ctac ((pattern, to), SOME pat_ctxt) inthms))))) + "single-step rewriting with auto-typechecking" + end +end diff --git a/mltt/core/context_facts.ML b/mltt/core/context_facts.ML new file mode 100644 index 0000000..5aa7c70 --- /dev/null +++ b/mltt/core/context_facts.ML @@ -0,0 +1,101 @@ +structure Context_Facts: sig + +val Known: Proof.context -> thm Item_Net.T +val known: Proof.context -> thm list +val known_of: Proof.context -> term -> thm list +val register_known: thm -> Context.generic -> Context.generic +val register_knowns: thm list -> Context.generic -> Context.generic + +val Cond: Proof.context -> thm Item_Net.T +val cond: Proof.context -> thm list +val cond_of: Proof.context -> term -> thm list +val register_cond: thm -> Context.generic -> Context.generic +val register_conds: thm list -> Context.generic -> Context.generic + +val Eq: Proof.context -> thm Item_Net.T +val eq: Proof.context -> thm list +val eq_of: Proof.context -> term -> thm list +val register_eq: thm -> Context.generic -> Context.generic +val register_eqs: thm list -> Context.generic -> Context.generic + +val register_facts: thm list -> Proof.context -> Proof.context + +end = struct + +(* Known types *) + +structure Known = Generic_Data ( + type T = thm Item_Net.T + val empty = Item_Net.init Thm.eq_thm + (single o Lib.term_of_typing o Thm.prop_of) + val extend = I + val merge = Item_Net.merge +) + +val Known = Known.get o Context.Proof +val known = Item_Net.content o Known +fun known_of ctxt tm = Item_Net.retrieve (Known ctxt) tm + +fun register_known typing = + if Lib.is_typing (Thm.prop_of typing) then Known.map (Item_Net.update typing) + else error "Not a type judgment" + +fun register_knowns typings = foldr1 (op o) (map register_known typings) + + +(* Conditional type rules *) + +(*Two important cases: 1. general type inference rules and 2. type family + judgments*) + +structure Cond = Generic_Data ( + type T = thm Item_Net.T + val empty = Item_Net.init Thm.eq_thm + (single o Lib.term_of_typing o Thm.concl_of) + val extend = I + val merge = Item_Net.merge +) + +val Cond = Cond.get o Context.Proof +val cond = Item_Net.content o Cond +fun cond_of ctxt tm = Item_Net.retrieve (Cond ctxt) tm + +fun register_cond rule = + if Lib.is_typing (Thm.concl_of rule) then Cond.map (Item_Net.update rule) + else error "Not a conditional type judgment" + +fun register_conds rules = foldr1 (op o) (map register_cond rules) + + +(* Equality statements *) + +structure Eq = Generic_Data ( + type T = thm Item_Net.T + val empty = Item_Net.init Thm.eq_thm + (single o (#1 o Lib.dest_eq) o Thm.concl_of) + val extend = I + val merge = Item_Net.merge +) + +val Eq = Eq.get o Context.Proof +val eq = Item_Net.content o Eq +fun eq_of ctxt tm = Item_Net.retrieve (Eq ctxt) tm + +fun register_eq rule = + if Lib.is_eq (Thm.concl_of rule) then Eq.map (Item_Net.update rule) + else error "Not a definitional equality judgment" + +fun register_eqs rules = foldr1 (op o) (map register_eq rules) + + +(* Context assumptions *) + +fun register_facts ths = + let + val (facts, conds, eqs) = Lib.partition_judgments ths + val f = register_knowns facts handle Empty => I + val c = register_conds conds handle Empty => I + val e = register_eqs eqs handle Empty => I + in Context.proof_map (e o c o f) end + +end diff --git a/mltt/core/context_tactical.ML b/mltt/core/context_tactical.ML new file mode 100644 index 0000000..d0fed61 --- /dev/null +++ b/mltt/core/context_tactical.ML @@ -0,0 +1,256 @@ +(* Title: context_tactical.ML + Author: Joshua Chen + +More context tactics, and context tactic combinators. + +Contains code modified from + ~~/Pure/search.ML + ~~/Pure/tactical.ML +*) + +infix 1 CTHEN CTHEN' CTHEN_ALL_NEW CTHEN_ALL_NEW_FWD +infix 0 CORELSE CAPPEND CORELSE' CAPPEND' + +structure Context_Tactical: +sig + +type context_tactic' = int -> context_tactic +val CONTEXT_TACTIC': (Proof.context -> int -> tactic) -> context_tactic' +val all_ctac: context_tactic +val no_ctac: context_tactic +val print_ctac: (Proof.context -> string) -> context_tactic +val CTHEN: context_tactic * context_tactic -> context_tactic +val CORELSE: context_tactic * context_tactic -> context_tactic +val CAPPEND: context_tactic * context_tactic -> context_tactic +val CTHEN': context_tactic' * context_tactic' -> context_tactic' +val CORELSE': context_tactic' * context_tactic' -> context_tactic' +val CAPPEND': context_tactic' * context_tactic' -> context_tactic' +val CTRY: context_tactic -> context_tactic +val CREPEAT: context_tactic -> context_tactic +val CREPEAT1: context_tactic -> context_tactic +val CREPEAT_N: int -> context_tactic -> context_tactic +val CFILTER: (context_state -> bool) -> context_tactic -> context_tactic +val CCHANGED: context_tactic -> context_tactic +val CTHEN_ALL_NEW: context_tactic' * context_tactic' -> context_tactic' +val CREPEAT_IN_RANGE: int -> int -> context_tactic' -> context_tactic +val CREPEAT_ALL_NEW: context_tactic' -> context_tactic' +val CTHEN_ALL_NEW_FWD: context_tactic' * context_tactic' -> context_tactic' +val CREPEAT_ALL_NEW_FWD: context_tactic' -> context_tactic' +val CHEADGOAL: context_tactic' -> context_tactic +val CALLGOALS: context_tactic' -> context_tactic +val CSOMEGOAL: context_tactic' -> context_tactic +val CRANGE: context_tactic' list -> context_tactic' +val CFIRST: context_tactic list -> context_tactic +val CFIRST': context_tactic' list -> context_tactic' +val CTHEN_BEST_FIRST: context_tactic -> (context_state -> bool) -> + (context_state -> int) -> context_tactic -> context_tactic +val CBEST_FIRST: (context_state -> bool) -> (context_state -> int) -> + context_tactic -> context_tactic +val CTHEN_ASTAR: context_tactic -> (context_state -> bool) -> + (int -> context_state -> int) -> context_tactic -> context_tactic +val CASTAR: (context_state -> bool) -> (int -> context_state -> int) -> + context_tactic -> context_tactic + +end = struct + +type context_tactic' = int -> context_tactic + +fun CONTEXT_TACTIC' tac i (ctxt, st) = TACTIC_CONTEXT ctxt ((tac ctxt i) st) + +val all_ctac = Seq.make_results o Seq.single +val no_ctac = K Seq.empty +fun print_ctac f (ctxt, st) = CONTEXT_TACTIC (print_tac ctxt (f ctxt)) (ctxt, st) + +fun (ctac1 CTHEN ctac2) cst = Seq.maps_results ctac2 (ctac1 cst) + +fun (ctac1 CORELSE ctac2) cst = + (case Seq.pull (ctac1 cst) of + NONE => ctac2 cst + | some => Seq.make (fn () => some)) + +fun (ctac1 CAPPEND ctac2) cst = + Seq.append (ctac1 cst) (Seq.make (fn () => Seq.pull (ctac2 cst))) + +fun (ctac1 CTHEN' ctac2) x = ctac1 x CTHEN ctac2 x +fun (ctac1 CORELSE' ctac2) x = ctac1 x CORELSE ctac2 x +fun (ctac1 CAPPEND' ctac2) x = ctac1 x CAPPEND ctac2 x + +fun CTRY ctac = ctac CORELSE all_ctac + +fun CREPEAT ctac = + let + fun rep qs cst = + (case Seq.pull (Seq.filter_results (ctac cst)) of + NONE => SOME (cst, Seq.make (fn () => repq qs)) + | SOME (cst', q) => rep (q :: qs) cst') + and repq [] = NONE + | repq (q :: qs) = + (case Seq.pull q of + NONE => repq qs + | SOME (cst, q) => rep (q :: qs) cst); + in fn cst => Seq.make_results (Seq.make (fn () => rep [] cst)) end + +fun CREPEAT1 ctac = ctac CTHEN CREPEAT ctac + +fun CREPEAT_N 0 _ = no_ctac + | CREPEAT_N n ctac = ctac CTHEN CREPEAT_N (n - 1) ctac + +fun CFILTER pred ctac cst = + ctac cst + |> Seq.filter_results + |> Seq.filter pred + |> Seq.make_results + +(*Only accept next states where the subgoals have changed*) +fun CCHANGED ctac (cst as (_, st)) = + CFILTER (fn (_, st') => not (Thm.eq_thm (st, st'))) ctac cst + +local + fun op THEN (f, g) x = Seq.maps_results g (f x) + + fun INTERVAL f i j x = + if i > j then Seq.make_results (Seq.single x) + else op THEN (f j, INTERVAL f i (j - 1)) x + + (*By Peter Lammich: apply tactic to subgoals in interval in a forward manner, + skipping over emerging subgoals*) + fun INTERVAL_FWD ctac l u (cst as (_, st)) = cst |> + (if l > u then all_ctac + else (ctac l CTHEN (fn cst' as (_, st') => + let val ofs = Thm.nprems_of st' - Thm.nprems_of st in + if ofs < ~1 + then raise THM ( + "INTERVAL_FWD: tactic solved more than one goal", ~1, [st, st']) + else INTERVAL_FWD ctac (l + 1 + ofs) (u + ofs) cst' + end))) +in + +fun (ctac1 CTHEN_ALL_NEW ctac2) i (cst as (_, st)) = + cst |> (ctac1 i CTHEN (fn cst' as (_, st') => + INTERVAL ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst')) + +(*By Peter Lammich: apply ctac2 to all subgoals emerging from ctac1, in forward + manner*) +fun (ctac1 CTHEN_ALL_NEW_FWD ctac2) i (cst as (_, st)) = + cst |> (ctac1 i CTHEN (fn cst' as (_, st') => + INTERVAL_FWD ctac2 i (i + Thm.nprems_of st' - Thm.nprems_of st) cst')) + +(*Repeatedly apply ctac to the i-th until the k-th-from-last subgoals + (i.e. leave the last k subgoals alone), until no more changes appear in the + goal state.*) +fun CREPEAT_IN_RANGE i k ctac = + let fun interval_ctac (cst as (_, st)) = + INTERVAL_FWD ctac i (Thm.nprems_of st - k) cst + in CREPEAT (CCHANGED interval_ctac) end + +end + +fun CREPEAT_ALL_NEW ctac = + ctac CTHEN_ALL_NEW (CTRY o (fn i => CREPEAT_ALL_NEW ctac i)) + +fun CREPEAT_ALL_NEW_FWD ctac = + ctac CTHEN_ALL_NEW_FWD (CTRY o (fn i => CREPEAT_ALL_NEW_FWD ctac i)) + +fun CHEADGOAL ctac = ctac 1 + +fun CALLGOALS ctac (cst as (_, st)) = + let + fun doall 0 = all_ctac + | doall n = ctac n CTHEN doall (n - 1); + in doall (Thm.nprems_of st) cst end + +fun CSOMEGOAL ctac (cst as (_, st)) = + let + fun find 0 = no_ctac + | find n = ctac n CORELSE find (n - 1); + in find (Thm.nprems_of st) cst end + +fun CRANGE [] _ = all_ctac + | CRANGE (ctac :: ctacs) i = CRANGE ctacs (i + 1) CTHEN ctac i + +fun CFIRST ctacs = fold_rev (curry op CORELSE) ctacs no_ctac + +(*FIRST' [tac1,...,tacn] i equals tac1 i ORELSE ... ORELSE tacn i*) +fun CFIRST' ctacs = fold_rev (curry op CORELSE') ctacs (K no_ctac) + + +(** Search tacticals **) + +(* Best-first search *) + +structure Thm_Heap = Heap ( + type elem = int * thm; + val ord = prod_ord int_ord (Term_Ord.term_ord o apply2 Thm.prop_of) +) + +structure Context_State_Heap = Heap ( + type elem = int * context_state; + val ord = prod_ord int_ord (Term_Ord.term_ord o apply2 (Thm.prop_of o #2)) +) + +fun some_of_list [] = NONE + | some_of_list (x :: l) = SOME (x, Seq.make (fn () => some_of_list l)) + +(*Check for and delete duplicate proof states*) +fun delete_all_min (cst as (_, st)) heap = + if Context_State_Heap.is_empty heap then heap + else if Thm.eq_thm (st, #2 (#2 (Context_State_Heap.min heap))) + then delete_all_min cst (Context_State_Heap.delete_min heap) + else heap + +(*Best-first search for a state that satisfies satp (incl initial state) + Function sizef estimates size of problem remaining (smaller means better). + tactic tac0 sets up the initial priority queue, while tac1 searches it. *) +fun CTHEN_BEST_FIRST ctac0 satp sizef ctac = + let + fun pairsize cst = (sizef cst, cst); + fun bfs (news, nst_heap) = + (case List.partition satp news of + ([], nonsats) => next (fold_rev Context_State_Heap.insert (map pairsize nonsats) nst_heap) + | (sats, _) => some_of_list sats) + and next nst_heap = + if Context_State_Heap.is_empty nst_heap then NONE + else + let + val (n, cst) = Context_State_Heap.min nst_heap; + in + bfs (Seq.list_of (Seq.filter_results (ctac cst)), delete_all_min cst (Context_State_Heap.delete_min nst_heap)) + end; + fun btac cst = bfs (Seq.list_of (Seq.filter_results (ctac0 cst)), Context_State_Heap.empty) + in fn cst => Seq.make_results (Seq.make (fn () => btac cst)) end + +(*Ordinary best-first search, with no initial tactic*) +val CBEST_FIRST = CTHEN_BEST_FIRST all_ctac + + +(* A*-like search *) + +(*Insertion into priority queue of states, marked with level*) +fun insert_with_level (lnth: int * int * context_state) [] = [lnth] + | insert_with_level (l, m, cst) ((l', n, cst') :: csts) = + if n < m then (l', n, cst') :: insert_with_level (l, m, cst) csts + else if n = m andalso Thm.eq_thm (#2 cst, #2 cst') then (l', n, cst') :: csts + else (l, m, cst) :: (l', n, cst') :: csts; + +fun CTHEN_ASTAR ctac0 satp costf ctac = + let + fun bfs (news, nst, level) = + let fun cost cst = (level, costf level cst, cst) in + (case List.partition satp news of + ([], nonsats) => next (fold_rev (insert_with_level o cost) nonsats nst) + | (sats, _) => some_of_list sats) + end + and next [] = NONE + | next ((level, n, cst) :: nst) = + bfs (Seq.list_of (Seq.filter_results (ctac cst)), nst, level + 1) + in fn cst => Seq.make_results + (Seq.make (fn () => bfs (Seq.list_of (Seq.filter_results (ctac0 cst)), [], 0))) + end + +(*Ordinary ASTAR, with no initial tactic*) +val CASTAR = CTHEN_ASTAR all_ctac; + + +end + +open Context_Tactical diff --git a/mltt/core/elaborated_statement.ML b/mltt/core/elaborated_statement.ML new file mode 100644 index 0000000..33f88cf --- /dev/null +++ b/mltt/core/elaborated_statement.ML @@ -0,0 +1,470 @@ +(* Title: elaborated_statement.ML + Author: Joshua Chen + +Term elaboration for goal statements and proof commands. + +Contains code from parts of + ~~/Pure/Isar/element.ML and + ~~/Pure/Isar/expression.ML +in both verbatim and modified forms. +*) + +structure Elaborated_Statement: sig + +val read_goal_statement: + (string, string, Facts.ref) Element.ctxt list -> + (string, string) Element.stmt -> + Proof.context -> + (Attrib.binding * (term * term list) list) list * Proof.context + +end = struct + + +(* Elaborated goal statements *) + +local + +fun mk_type T = (Logic.mk_type T, []) +fun mk_term t = (t, []) +fun mk_propp (p, pats) = (Type.constraint propT p, pats) + +fun dest_type (T, []) = Logic.dest_type T +fun dest_term (t, []) = t +fun dest_propp (p, pats) = (p, pats) + +fun extract_inst (_, (_, ts)) = map mk_term ts +fun restore_inst ((l, (p, _)), cs) = (l, (p, map dest_term cs)) + +fun extract_eqns es = map (mk_term o snd) es +fun restore_eqns (es, cs) = map2 (fn (b, _) => fn c => (b, dest_term c)) es cs + +fun extract_elem (Element.Fixes fixes) = map (#2 #> the_list #> map mk_type) fixes + | extract_elem (Element.Constrains csts) = map (#2 #> single #> map mk_type) csts + | extract_elem (Element.Assumes asms) = map (#2 #> map mk_propp) asms + | extract_elem (Element.Defines defs) = map (fn (_, (t, ps)) => [mk_propp (t, ps)]) defs + | extract_elem (Element.Notes _) = [] + | extract_elem (Element.Lazy_Notes _) = [] + +fun restore_elem (Element.Fixes fixes, css) = + (fixes ~~ css) |> map (fn ((x, _, mx), cs) => + (x, cs |> map dest_type |> try hd, mx)) |> Element.Fixes + | restore_elem (Element.Constrains csts, css) = + (csts ~~ css) |> map (fn ((x, _), cs) => + (x, cs |> map dest_type |> hd)) |> Element.Constrains + | restore_elem (Element.Assumes asms, css) = + (asms ~~ css) |> map (fn ((b, _), cs) => (b, map dest_propp cs)) |> Element.Assumes + | restore_elem (Element.Defines defs, css) = + (defs ~~ css) |> map (fn ((b, _), [c]) => (b, dest_propp c)) |> Element.Defines + | restore_elem (elem as Element.Notes _, _) = elem + | restore_elem (elem as Element.Lazy_Notes _, _) = elem + +fun prep (_, pats) (ctxt, t :: ts) = + let val ctxt' = Proof_Context.augment t ctxt + in + ((t, Syntax.check_props + (Proof_Context.set_mode Proof_Context.mode_pattern ctxt') pats), + (ctxt', ts)) + end + +fun check cs ctxt = + let + val (cs', (ctxt', _)) = fold_map prep cs + (ctxt, Syntax.check_terms + (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) (map fst cs)) + in (cs', ctxt') end + +fun inst_morphism params ((prfx, mandatory), insts') ctxt = + let + (*parameters*) + val parm_types = map #2 params; + val type_parms = fold Term.add_tfreesT parm_types []; + + (*type inference*) + val parm_types' = map (Type_Infer.paramify_vars o Logic.varifyT_global) parm_types; + val type_parms' = fold Term.add_tvarsT parm_types' []; + val checked = + (map (Logic.mk_type o TVar) type_parms' @ map2 Type.constraint parm_types' insts') + |> Syntax.check_terms (Config.put Type_Infer.object_logic false ctxt) + val (type_parms'', insts'') = chop (length type_parms') checked; + + (*context*) + val ctxt' = fold Proof_Context.augment checked ctxt; + val certT = Thm.trim_context_ctyp o Thm.ctyp_of ctxt'; + val cert = Thm.trim_context_cterm o Thm.cterm_of ctxt'; + + (*instantiation*) + val instT = + (type_parms ~~ map Logic.dest_type type_parms'') + |> map_filter (fn (v, T) => if TFree v = T then NONE else SOME (v, T)); + val cert_inst = + ((map #1 params ~~ map (Term_Subst.instantiateT_frees instT) parm_types) ~~ insts'') + |> map_filter (fn (v, t) => if Free v = t then NONE else SOME (v, cert t)); + in + (Element.instantiate_normalize_morphism (map (apsnd certT) instT, cert_inst) $> + Morphism.binding_morphism "Expression.inst" (Binding.prefix mandatory prfx), ctxt') + end; + +fun abs_def ctxt = + Thm.cterm_of ctxt #> Assumption.assume ctxt #> Local_Defs.abs_def_rule ctxt #> Thm.prop_of; + +fun declare_elem prep_var (Element.Fixes fixes) ctxt = + let val (vars, _) = fold_map prep_var fixes ctxt + in ctxt |> Proof_Context.add_fixes vars |> snd end + | declare_elem prep_var (Element.Constrains csts) ctxt = + ctxt |> fold_map (fn (x, T) => prep_var (Binding.name x, SOME T, NoSyn)) csts |> snd + | declare_elem _ (Element.Assumes _) ctxt = ctxt + | declare_elem _ (Element.Defines _) ctxt = ctxt + | declare_elem _ (Element.Notes _) ctxt = ctxt + | declare_elem _ (Element.Lazy_Notes _) ctxt = ctxt; + +fun parameters_of thy strict (expr, fixed) = + let + val ctxt = Proof_Context.init_global thy; + + fun reject_dups message xs = + (case duplicates (op =) xs of + [] => () + | dups => error (message ^ commas dups)); + + fun parm_eq ((p1, mx1), (p2, mx2)) = + p1 = p2 andalso + (Mixfix.equal (mx1, mx2) orelse + error ("Conflicting syntax for parameter " ^ quote p1 ^ " in expression" ^ + Position.here_list [Mixfix.pos_of mx1, Mixfix.pos_of mx2])); + + fun params_loc loc = Locale.params_of thy loc |> map (apfst #1); + fun params_inst (loc, (prfx, (Expression.Positional insts, eqns))) = + let + val ps = params_loc loc; + val d = length ps - length insts; + val insts' = + if d < 0 then + error ("More arguments than parameters in instantiation of locale " ^ + quote (Locale.markup_name ctxt loc)) + else insts @ replicate d NONE; + val ps' = (ps ~~ insts') |> + map_filter (fn (p, NONE) => SOME p | (_, SOME _) => NONE); + in (ps', (loc, (prfx, (Expression.Positional insts', eqns)))) end + | params_inst (loc, (prfx, (Expression.Named insts, eqns))) = + let + val _ = + reject_dups "Duplicate instantiation of the following parameter(s): " + (map fst insts); + val ps' = (insts, params_loc loc) |-> fold (fn (p, _) => fn ps => + if AList.defined (op =) ps p then AList.delete (op =) p ps + else error (quote p ^ " not a parameter of instantiated expression")); + in (ps', (loc, (prfx, (Expression.Named insts, eqns)))) end; + fun params_expr is = + let + val (is', ps') = fold_map (fn i => fn ps => + let + val (ps', i') = params_inst i; + val ps'' = distinct parm_eq (ps @ ps'); + in (i', ps'') end) is [] + in (ps', is') end; + + val (implicit, expr') = params_expr expr; + + val implicit' = map #1 implicit; + val fixed' = map (Variable.check_name o #1) fixed; + val _ = reject_dups "Duplicate fixed parameter(s): " fixed'; + val implicit'' = + if strict then [] + else + let + val _ = + reject_dups + "Parameter(s) declared simultaneously in expression and for clause: " + (implicit' @ fixed'); + in map (fn (x, mx) => (Binding.name x, NONE, mx)) implicit end; + in (expr', implicit'' @ fixed) end; + +fun parse_elem prep_typ prep_term ctxt = + Element.map_ctxt + {binding = I, + typ = prep_typ ctxt, + term = prep_term (Proof_Context.set_mode Proof_Context.mode_schematic ctxt), + pattern = prep_term (Proof_Context.set_mode Proof_Context.mode_pattern ctxt), + fact = I, + attrib = I}; + +fun prepare_stmt prep_prop prep_obtains ctxt stmt = + (case stmt of + Element.Shows raw_shows => + raw_shows |> (map o apsnd o map) (fn (t, ps) => + (prep_prop (Proof_Context.set_mode Proof_Context.mode_schematic ctxt) t, + map (prep_prop (Proof_Context.set_mode Proof_Context.mode_pattern ctxt)) ps)) + | Element.Obtains raw_obtains => + let + val ((_, thesis), thesis_ctxt) = Obtain.obtain_thesis ctxt; + val obtains = prep_obtains thesis_ctxt thesis raw_obtains; + in map (fn (b, t) => ((b, []), [(t, [])])) obtains end); + +fun finish_fixes (parms: (string * typ) list) = map (fn (binding, _, mx) => + let val x = Binding.name_of binding + in (binding, AList.lookup (op =) parms x, mx) end) + +fun finish_inst ctxt (loc, (prfx, inst)) = + let + val thy = Proof_Context.theory_of ctxt; + val (morph, _) = inst_morphism (map #1 (Locale.params_of thy loc)) (prfx, inst) ctxt; + in (loc, morph) end + +fun closeup _ _ false elem = elem + | closeup (outer_ctxt, ctxt) parms true elem = + let + (*FIXME consider closing in syntactic phase -- before type checking*) + fun close_frees t = + let + val rev_frees = + Term.fold_aterms (fn Free (x, T) => + if Variable.is_fixed outer_ctxt x orelse AList.defined (op =) parms x then I + else insert (op =) (x, T) | _ => I) t []; + in fold (Logic.all o Free) rev_frees t end; + + fun no_binds [] = [] + | no_binds _ = error "Illegal term bindings in context element"; + in + (case elem of + Element.Assumes asms => Element.Assumes (asms |> map (fn (a, propps) => + (a, map (fn (t, ps) => (close_frees t, no_binds ps)) propps))) + | Element.Defines defs => Element.Defines (defs |> map (fn ((name, atts), (t, ps)) => + let val ((c, _), t') = Local_Defs.cert_def ctxt (K []) (close_frees t) + in ((Thm.def_binding_optional (Binding.name c) name, atts), (t', no_binds ps)) end)) + | e => e) + end + +fun finish_elem _ parms _ (Element.Fixes fixes) = Element.Fixes (finish_fixes parms fixes) + | finish_elem _ _ _ (Element.Constrains _) = Element.Constrains [] + | finish_elem ctxts parms do_close (Element.Assumes asms) = closeup ctxts parms do_close (Element.Assumes asms) + | finish_elem ctxts parms do_close (Element.Defines defs) = closeup ctxts parms do_close (Element.Defines defs) + | finish_elem _ _ _ (elem as Element.Notes _) = elem + | finish_elem _ _ _ (elem as Element.Lazy_Notes _) = elem + +fun check_autofix insts eqnss elems concl ctxt = + let + val inst_cs = map extract_inst insts; + val eqns_cs = map extract_eqns eqnss; + val elem_css = map extract_elem elems; + val concl_cs = (map o map) mk_propp (map snd concl); + (*Type inference*) + val (inst_cs' :: eqns_cs' :: css', ctxt') = + (fold_burrow o fold_burrow) check (inst_cs :: eqns_cs :: elem_css @ [concl_cs]) ctxt; + val (elem_css', [concl_cs']) = chop (length elem_css) css'; + in + ((map restore_inst (insts ~~ inst_cs'), + map restore_eqns (eqnss ~~ eqns_cs'), + map restore_elem (elems ~~ elem_css'), + map fst concl ~~ concl_cs'), ctxt') + end + +fun prep_full_context_statement + parse_typ parse_prop + prep_obtains prep_var_elem prep_inst prep_eqns prep_attr prep_var_inst prep_expr + {strict, do_close, fixed_frees} raw_import init_body raw_elems raw_stmt + ctxt1 + = + let + val thy = Proof_Context.theory_of ctxt1 + val (raw_insts, fixed) = parameters_of thy strict (apfst (prep_expr thy) raw_import) + fun prep_insts_cumulative (loc, (prfx, (inst, eqns))) (i, insts, eqnss, ctxt) = + let + val params = map #1 (Locale.params_of thy loc) + val inst' = prep_inst ctxt (map #1 params) inst + val parm_types' = + params |> map (#2 #> Logic.varifyT_global #> + Term.map_type_tvar (fn ((x, _), S) => TVar ((x, i), S)) #> + Type_Infer.paramify_vars) + val inst'' = map2 Type.constraint parm_types' inst' + val insts' = insts @ [(loc, (prfx, inst''))] + val ((insts'', _, _, _), ctxt2) = check_autofix insts' [] [] [] ctxt + val inst''' = insts'' |> List.last |> snd |> snd + val (inst_morph, _) = inst_morphism params (prfx, inst''') ctxt + val ctxt' = Locale.activate_declarations (loc, inst_morph) ctxt2 + handle ERROR msg => if null eqns then error msg else + (Locale.tracing ctxt1 + (msg ^ "\nFalling back to reading rewrites clause before activation."); + ctxt2) + val attrss = map (apsnd (map (prep_attr ctxt)) o fst) eqns + val eqns' = (prep_eqns ctxt' o map snd) eqns + val eqnss' = [attrss ~~ eqns'] + val ((_, [eqns''], _, _), _) = check_autofix insts'' eqnss' [] [] ctxt' + val rewrite_morph = eqns' + |> map (abs_def ctxt') + |> Variable.export_terms ctxt' ctxt + |> Element.eq_term_morphism (Proof_Context.theory_of ctxt) + |> the_default Morphism.identity + val ctxt'' = Locale.activate_declarations (loc, inst_morph $> rewrite_morph) ctxt + val eqnss' = eqnss @ [attrss ~~ Variable.export_terms ctxt' ctxt eqns'] + in (i + 1, insts', eqnss', ctxt'') end + + fun prep_elem raw_elem ctxt = + let + val ctxt' = ctxt + |> Context_Position.set_visible false + |> declare_elem prep_var_elem raw_elem + |> Context_Position.restore_visible ctxt + val elems' = parse_elem parse_typ parse_prop ctxt' raw_elem + in (elems', ctxt') end + + val fors = fold_map prep_var_inst fixed ctxt1 |> fst + val ctxt2 = ctxt1 |> Proof_Context.add_fixes fors |> snd + val (_, insts', eqnss', ctxt3) = fold prep_insts_cumulative raw_insts (0, [], [], ctxt2) + + fun prep_stmt elems ctxt = + check_autofix insts' [] elems (prepare_stmt parse_prop prep_obtains ctxt raw_stmt) ctxt + + val _ = + if fixed_frees then () + else + (case fold (fold (Variable.add_frees ctxt3) o snd o snd) insts' [] of + [] => () + | frees => error ("Illegal free variables in expression: " ^ + commas_quote (map (Syntax.string_of_term ctxt3 o Free) (rev frees)))) + + val ((insts, _, elems', concl), ctxt4) = ctxt3 + |> init_body + |> fold_map prep_elem raw_elems + |-> prep_stmt + + (*parameters from expression and elements*) + val xs = maps (fn Element.Fixes fixes => map (Variable.check_name o #1) fixes | _ => []) + (Element.Fixes fors :: elems') + val (parms, ctxt5) = fold_map Proof_Context.inferred_param xs ctxt4 + val fors' = finish_fixes parms fors + val fixed = map (fn (b, SOME T, mx) => ((Binding.name_of b, T), mx)) fors' + val deps = map (finish_inst ctxt5) insts + val elems'' = map (finish_elem (ctxt1, ctxt5) parms do_close) elems' + in ((fixed, deps, eqnss', elems'', concl), (parms, ctxt5)) end + +fun prep_inst prep_term ctxt parms (Expression.Positional insts) = + (insts ~~ parms) |> map + (fn (NONE, p) => Free (p, dummyT) + | (SOME t, _) => prep_term ctxt t) + | prep_inst prep_term ctxt parms (Expression.Named insts) = + parms |> map (fn p => + (case AList.lookup (op =) insts p of + SOME t => prep_term ctxt t | + NONE => Free (p, dummyT))) +fun parse_inst x = prep_inst Syntax.parse_term x +fun check_expr thy instances = map (apfst (Locale.check thy)) instances + +val read_full_context_statement = prep_full_context_statement + Syntax.parse_typ Syntax.parse_prop Obtain.parse_obtains + Proof_Context.read_var parse_inst Syntax.read_props Attrib.check_src + Proof_Context.read_var check_expr + +fun filter_assumes ((x as Element.Assumes _) :: xs) = x :: filter_assumes xs + | filter_assumes (_ :: xs) = filter_assumes xs + | filter_assumes [] = [] + +fun prep_statement prep activate raw_elems raw_stmt ctxt = + let + val ((_, _, _, elems, concl), _) = + prep {strict = true, do_close = false, fixed_frees = true} + ([], []) I raw_elems raw_stmt ctxt + + val (elems', ctxt') = ctxt + |> Proof_Context.set_stmt true + |> fold_map activate elems + |> apsnd (Proof_Context.restore_stmt ctxt) + + val assumes = filter_assumes elems' + val assms = flat (flat (map + (fn (Element.Assumes asms) => + map (fn (_, facts) => map (Thm.cterm_of ctxt' o #1) facts) asms) + assumes)) + val concl' = Elab.elaborate ctxt' assms concl handle error => concl + in (concl', ctxt') end + +fun activate_i elem ctxt = + let + val elem' = + (case (Element.map_ctxt_attrib o map) Token.init_assignable elem of + Element.Defines defs => + Element.Defines (defs |> map (fn ((a, atts), (t, ps)) => + ((Thm.def_binding_optional + (Binding.name (#1 (#1 (Local_Defs.cert_def ctxt (K []) t)))) a, atts), + (t, ps)))) + | Element.Assumes assms => Element.Assumes (Elab.elaborate ctxt [] assms) + | e => e); + val ctxt' = Context.proof_map (Element.init elem') ctxt; + in ((Element.map_ctxt_attrib o map) Token.closure elem', ctxt') end + +fun activate raw_elem ctxt = + let val elem = raw_elem |> Element.map_ctxt + {binding = I, + typ = I, + term = I, + pattern = I, + fact = Proof_Context.get_fact ctxt, + attrib = Attrib.check_src ctxt} + in activate_i elem ctxt end + +in + +val read_goal_statement = prep_statement read_full_context_statement activate + +end + + +(* Proof assumption command *) + +local + +val structured_statement = + Parse_Spec.statement -- Parse_Spec.if_statement' -- Parse.for_fixes + >> (fn ((shows, assumes), fixes) => (fixes, assumes, shows)) + +fun these_factss more_facts (named_factss, state) = + (named_factss, state |> Proof.set_facts (maps snd named_factss @ more_facts)) + +fun gen_assume prep_statement prep_att export raw_fixes raw_prems raw_concls state = + let + val ctxt = Proof.context_of state; + + val bindings = map (apsnd (map (prep_att ctxt)) o fst) raw_concls; + val {fixes = params, assumes = prems_propss, shows = concl_propss, result_binds, text, ...} = + #1 (prep_statement raw_fixes raw_prems (map snd raw_concls) ctxt); + val propss = (map o map) (Logic.close_prop params (flat prems_propss)) concl_propss; + in + state + |> Proof.assert_forward + |> Proof.map_context_result (fn ctxt => + ctxt + |> Proof_Context.augment text + |> fold Variable.maybe_bind_term result_binds + |> fold_burrow (Assumption.add_assms export o map (Thm.cterm_of ctxt)) propss + |-> (fn premss => fn ctxt => + (premss, Context_Facts.register_facts (flat premss) ctxt)) + |-> (fn premss => + Proof_Context.note_thmss "" (bindings ~~ (map o map) (fn th => ([th], [])) premss))) + |> these_factss [] |> #2 + end + +val assume = + gen_assume Proof_Context.cert_statement (K I) Assumption.assume_export + +in + +val _ = Outer_Syntax.command \<^command_keyword>\assuming\ "elaborated assumption" + (structured_statement >> (fn (a, b, c) => Toplevel.proof (fn state => + let + val ctxt = Proof.context_of state + + fun read_option_typ NONE = NONE + | read_option_typ (SOME s) = SOME (Syntax.read_typ ctxt s) + fun read_terms (s, ss) = + let val f = Syntax.read_term ctxt in (f s, map f ss) end + + val a' = map (fn (b, s, m) => (b, read_option_typ s, m)) a + val b' = map (map read_terms) b + val c' = c |> map (fn ((b, atts), ss) => + ((b, map (Attrib.attribute_cmd ctxt) atts), map read_terms ss)) + val c'' = Elab.elaborate ctxt [] c' + in assume a' b' c'' state end))) + +end + + +end \ No newline at end of file diff --git a/mltt/core/elaboration.ML b/mltt/core/elaboration.ML new file mode 100644 index 0000000..9e5e0bd --- /dev/null +++ b/mltt/core/elaboration.ML @@ -0,0 +1,91 @@ +(* Title: elaboration.ML + Author: Joshua Chen + +Basic term elaboration. +*) + +structure Elab: sig + +val elab: Proof.context -> cterm list -> term -> Envir.env +val elab_stmt: Proof.context -> cterm list -> term -> Envir.env * term +val elaborate: Proof.context -> cterm list -> ('a * (term * term list) list) list -> ('a * (term * term list) list) list + +end = struct + +(*Elaborate `tm` by solving the inference problem `tm: {}`, knowing `assums`, + which are fully elaborated, in `ctxt`. Return a substitution.*) +fun elab ctxt assums tm = + if Lib.no_vars tm + then Envir.init + else + let + val inf = Goal.init (Thm.cterm_of ctxt (Lib.typing_of_term tm)) + val res = Types.check_infer (map Thm.assume assums) 1 (ctxt, inf) + val tm' = + Thm.prop_of (#2 (Seq.hd (Seq.filter_results res))) + |> Lib.dest_prop |> Lib.term_of_typing + handle TERM ("dest_typing", [t]) => + let val typ = Logic.unprotect (Logic.strip_assums_concl t) + |> Lib.term_of_typing + in + error ("Elaboration of " ^ Syntax.string_of_term ctxt typ ^ " failed") + end + in + Seq.hd (Unify.matchers (Context.Proof ctxt) [(tm, tm')]) + end + handle Option => error + ("Elaboration of " ^ Syntax.string_of_term ctxt tm ^ " failed") + +(*Recursively elaborate a statement \x ... y. \...\ \ P x ... y by elaborating + only the types of typing judgments (in particular, does not look at judgmental + equality statements). Could also elaborate the terms of typing judgments, but + for now we assume that these are always free variables in all the cases we're + interested in.*) +fun elab_stmt ctxt assums stmt = + let + val stmt = Lib.dest_prop stmt + fun subst_term env = Envir.subst_term (Envir.type_env env, Envir.term_env env) + in + if Lib.no_vars stmt orelse Lib.is_eq stmt then + (Envir.init, stmt) + else if Lib.is_typing stmt then + let + val typ = Lib.type_of_typing stmt + val subst = elab ctxt assums typ + in (subst, subst_term subst stmt) end + else + let + fun elab' assums (x :: xs) = + let + val (env, x') = elab_stmt ctxt assums x + val assums' = + if Lib.no_vars x' then Thm.cterm_of ctxt x' :: assums else assums + in env :: elab' assums' xs end + | elab' _ [] = [] + val (prems, concl) = Lib.decompose_goal ctxt stmt + val subst = fold (curry Envir.merge) (elab' assums prems) Envir.init + val prems' = map (Thm.cterm_of ctxt o subst_term subst) prems + val subst' = + if Lib.is_typing concl then + let val typ = Lib.type_of_typing concl + in Envir.merge (subst, elab ctxt (assums @ prems') typ) end + else subst + in (subst', subst_term subst' stmt) end + end + +(*Apply elaboration to the list format that assumptions and goal statements are + given in*) +fun elaborate ctxt known assms = + let + fun subst_term env = Envir.subst_term (Envir.type_env env, Envir.term_env env) + fun elab_fact (fact, xs) assums = + let val (subst, fact') = elab_stmt ctxt assums fact in + ((fact', map (subst_term subst) xs), Thm.cterm_of ctxt fact' :: assums) + end + fun elab (b, facts) assums = + let val (facts', assums') = fold_map elab_fact facts assums + in ((b, facts'), assums') end + in #1 (fold_map elab assms known) end + + +end diff --git a/mltt/core/elimination.ML b/mltt/core/elimination.ML new file mode 100644 index 0000000..cf9d21e --- /dev/null +++ b/mltt/core/elimination.ML @@ -0,0 +1,48 @@ +(* Title: elimination.ML + Author: Joshua Chen + +Type elimination setup. +*) + +structure Elim: sig + +val Rules: Proof.context -> (thm * indexname list) Termtab.table +val rules: Proof.context -> (thm * indexname list) list +val lookup_rule: Proof.context -> Termtab.key -> (thm * indexname list) option +val register_rule: term list -> thm -> Context.generic -> Context.generic + +end = struct + +(** Context data **) + +(* Elimination rule data *) + +(*Stores elimination rules together with a list of the indexnames of the + variables each rule eliminates. Keyed by head of the type being eliminated.*) +structure Rules = Generic_Data ( + type T = (thm * indexname list) Termtab.table + val empty = Termtab.empty + val extend = I + val merge = Termtab.merge (eq_fst Thm.eq_thm_prop) +) + +val Rules = Rules.get o Context.Proof +fun rules ctxt = map (op #2) (Termtab.dest (Rules ctxt)) +fun lookup_rule ctxt = Termtab.lookup (Rules ctxt) +fun register_rule tms rl = + let val hd = Term.head_of (Lib.type_of_typing (Thm.major_prem_of rl)) + in Rules.map (Termtab.update (hd, (rl, map (#1 o dest_Var) tms))) end + + +(* [elim] attribute *) +val _ = Theory.setup ( + Attrib.setup \<^binding>\elim\ + (Scan.repeat Args.term_pattern >> + (Thm.declaration_attribute o register_rule)) + "" + #> Global_Theory.add_thms_dynamic (\<^binding>\elim\, + fn context => map #1 (rules (Context.proof_of context))) +) + + +end diff --git a/mltt/core/eqsubst.ML b/mltt/core/eqsubst.ML new file mode 100644 index 0000000..5ae8c73 --- /dev/null +++ b/mltt/core/eqsubst.ML @@ -0,0 +1,442 @@ +(* Title: eqsubst.ML + Author: Lucas Dixon, University of Edinburgh + Modified: Joshua Chen, University of Innsbruck + +Perform a substitution using an equation. + +This code is slightly modified from the original at Tools/eqsubst..ML, +to incorporate auto-typechecking for type theory. +*) + +signature EQSUBST = +sig + type match = + ((indexname * (sort * typ)) list (* type instantiations *) + * (indexname * (typ * term)) list) (* term instantiations *) + * (string * typ) list (* fake named type abs env *) + * (string * typ) list (* type abs env *) + * term (* outer term *) + + type searchinfo = + Proof.context + * int (* maxidx *) + * Zipper.T (* focusterm to search under *) + + datatype 'a skipseq = SkipMore of int | SkipSeq of 'a Seq.seq Seq.seq + + val skip_first_asm_occs_search: ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> int -> 'b -> 'c skipseq + val skip_first_occs_search: int -> ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> 'b -> 'c Seq.seq + val skipto_skipseq: int -> 'a Seq.seq Seq.seq -> 'a skipseq + + (* tactics *) + val eqsubst_asm_tac: Proof.context -> int list -> thm list -> int -> tactic + val eqsubst_asm_tac': Proof.context -> + (searchinfo -> int -> term -> match skipseq) -> int -> thm -> int -> tactic + val eqsubst_tac: Proof.context -> + int list -> (* list of occurrences to rewrite, use [0] for any *) + thm list -> int -> tactic + val eqsubst_tac': Proof.context -> + (searchinfo -> term -> match Seq.seq) (* search function *) + -> thm (* equation theorem to rewrite with *) + -> int (* subgoal number in goal theorem *) + -> thm (* goal theorem *) + -> thm Seq.seq (* rewritten goal theorem *) + + (* search for substitutions *) + val valid_match_start: Zipper.T -> bool + val search_lr_all: Zipper.T -> Zipper.T Seq.seq + val search_lr_valid: (Zipper.T -> bool) -> Zipper.T -> Zipper.T Seq.seq + val searchf_lr_unify_all: searchinfo -> term -> match Seq.seq Seq.seq + val searchf_lr_unify_valid: searchinfo -> term -> match Seq.seq Seq.seq + val searchf_bt_unify_valid: searchinfo -> term -> match Seq.seq Seq.seq +end; + +structure EqSubst: EQSUBST = +struct + +(* changes object "=" to meta "==" which prepares a given rewrite rule *) +fun prep_meta_eq ctxt = + Simplifier.mksimps ctxt #> map Drule.zero_var_indexes; + +(* make free vars into schematic vars with index zero *) +fun unfix_frees frees = + fold (K (Thm.forall_elim_var 0)) frees o Drule.forall_intr_list frees; + + +type match = + ((indexname * (sort * typ)) list (* type instantiations *) + * (indexname * (typ * term)) list) (* term instantiations *) + * (string * typ) list (* fake named type abs env *) + * (string * typ) list (* type abs env *) + * term; (* outer term *) + +type searchinfo = + Proof.context + * int (* maxidx *) + * Zipper.T; (* focusterm to search under *) + + +(* skipping non-empty sub-sequences but when we reach the end + of the seq, remembering how much we have left to skip. *) +datatype 'a skipseq = + SkipMore of int | + SkipSeq of 'a Seq.seq Seq.seq; + +(* given a seqseq, skip the first m non-empty seq's, note deficit *) +fun skipto_skipseq m s = + let + fun skip_occs n sq = + (case Seq.pull sq of + NONE => SkipMore n + | SOME (h, t) => + (case Seq.pull h of + NONE => skip_occs n t + | SOME _ => if n <= 1 then SkipSeq (Seq.cons h t) else skip_occs (n - 1) t)) + in skip_occs m s end; + +(* note: outerterm is the taget with the match replaced by a bound + variable : ie: "P lhs" beocmes "%x. P x" + insts is the types of instantiations of vars in lhs + and typinsts is the type instantiations of types in the lhs + Note: Final rule is the rule lifted into the ontext of the + taget thm. *) +fun mk_foo_match mkuptermfunc Ts t = + let + val ty = Term.type_of t + val bigtype = rev (map snd Ts) ---> ty + fun mk_foo 0 t = t + | mk_foo i t = mk_foo (i - 1) (t $ (Bound (i - 1))) + val num_of_bnds = length Ts + (* foo_term = "fooabs y0 ... yn" where y's are local bounds *) + val foo_term = mk_foo num_of_bnds (Bound num_of_bnds) + in Abs ("fooabs", bigtype, mkuptermfunc foo_term) end; + +(* T is outer bound vars, n is number of locally bound vars *) +(* THINK: is order of Ts correct...? or reversed? *) +fun mk_fake_bound_name n = ":b_" ^ n; +fun fakefree_badbounds Ts t = + let val (FakeTs, Ts, newnames) = + fold_rev (fn (n, ty) => fn (FakeTs, Ts, usednames) => + let + val newname = singleton (Name.variant_list usednames) n + in + ((mk_fake_bound_name newname, ty) :: FakeTs, + (newname, ty) :: Ts, + newname :: usednames) + end) Ts ([], [], []) + in (FakeTs, Ts, Term.subst_bounds (map Free FakeTs, t)) end; + +(* before matching we need to fake the bound vars that are missing an + abstraction. In this function we additionally construct the + abstraction environment, and an outer context term (with the focus + abstracted out) for use in rewriting with RW_Inst.rw *) +fun prep_zipper_match z = + let + val t = Zipper.trm z + val c = Zipper.ctxt z + val Ts = Zipper.C.nty_ctxt c + val (FakeTs', Ts', t') = fakefree_badbounds Ts t + val absterm = mk_foo_match (Zipper.C.apply c) Ts' t' + in + (t', (FakeTs', Ts', absterm)) + end; + +(* Unification with exception handled *) +(* given context, max var index, pat, tgt; returns Seq of instantiations *) +fun clean_unify ctxt ix (a as (pat, tgt)) = + let + (* type info will be re-derived, maybe this can be cached + for efficiency? *) + val pat_ty = Term.type_of pat; + val tgt_ty = Term.type_of tgt; + (* FIXME is it OK to ignore the type instantiation info? + or should I be using it? *) + val typs_unify = + SOME (Sign.typ_unify (Proof_Context.theory_of ctxt) (pat_ty, tgt_ty) (Vartab.empty, ix)) + handle Type.TUNIFY => NONE; + in + (case typs_unify of + SOME (typinsttab, ix2) => + let + (* FIXME is it right to throw away the flexes? + or should I be using them somehow? *) + fun mk_insts env = + (Vartab.dest (Envir.type_env env), + Vartab.dest (Envir.term_env env)); + val initenv = + Envir.Envir {maxidx = ix2, tenv = Vartab.empty, tyenv = typinsttab}; + val useq = Unify.smash_unifiers (Context.Proof ctxt) [a] initenv + handle ListPair.UnequalLengths => Seq.empty + | Term.TERM _ => Seq.empty; + fun clean_unify' useq () = + (case (Seq.pull useq) of + NONE => NONE + | SOME (h, t) => SOME (mk_insts h, Seq.make (clean_unify' t))) + handle ListPair.UnequalLengths => NONE + | Term.TERM _ => NONE; + in + (Seq.make (clean_unify' useq)) + end + | NONE => Seq.empty) + end; + +(* Unification for zippers *) +(* Note: Ts is a modified version of the original names of the outer + bound variables. New names have been introduced to make sure they are + unique w.r.t all names in the term and each other. usednames' is + oldnames + new names. *) +fun clean_unify_z ctxt maxidx pat z = + let val (t, (FakeTs, Ts, absterm)) = prep_zipper_match z in + Seq.map (fn insts => (insts, FakeTs, Ts, absterm)) + (clean_unify ctxt maxidx (t, pat)) + end; + + +fun bot_left_leaf_of (l $ _) = bot_left_leaf_of l + | bot_left_leaf_of (Abs (_, _, t)) = bot_left_leaf_of t + | bot_left_leaf_of x = x; + +(* Avoid considering replacing terms which have a var at the head as + they always succeed trivially, and uninterestingly. *) +fun valid_match_start z = + (case bot_left_leaf_of (Zipper.trm z) of + Var _ => false + | _ => true); + +(* search from top, left to right, then down *) +val search_lr_all = ZipperSearch.all_bl_ur; + +(* search from top, left to right, then down *) +fun search_lr_valid validf = + let + fun sf_valid_td_lr z = + let val here = if validf z then [Zipper.Here z] else [] in + (case Zipper.trm z of + _ $ _ => + [Zipper.LookIn (Zipper.move_down_left z)] @ here @ + [Zipper.LookIn (Zipper.move_down_right z)] + | Abs _ => here @ [Zipper.LookIn (Zipper.move_down_abs z)] + | _ => here) + end; + in Zipper.lzy_search sf_valid_td_lr end; + +(* search from bottom to top, left to right *) +fun search_bt_valid validf = + let + fun sf_valid_td_lr z = + let val here = if validf z then [Zipper.Here z] else [] in + (case Zipper.trm z of + _ $ _ => + [Zipper.LookIn (Zipper.move_down_left z), + Zipper.LookIn (Zipper.move_down_right z)] @ here + | Abs _ => [Zipper.LookIn (Zipper.move_down_abs z)] @ here + | _ => here) + end; + in Zipper.lzy_search sf_valid_td_lr end; + +fun searchf_unify_gen f (ctxt, maxidx, z) lhs = + Seq.map (clean_unify_z ctxt maxidx lhs) (Zipper.limit_apply f z); + +(* search all unifications *) +val searchf_lr_unify_all = searchf_unify_gen search_lr_all; + +(* search only for 'valid' unifiers (non abs subterms and non vars) *) +val searchf_lr_unify_valid = searchf_unify_gen (search_lr_valid valid_match_start); + +val searchf_bt_unify_valid = searchf_unify_gen (search_bt_valid valid_match_start); + +(* apply a substitution in the conclusion of the theorem *) +(* cfvs are certified free var placeholders for goal params *) +(* conclthm is a theorem of for just the conclusion *) +(* m is instantiation/match information *) +(* rule is the equation for substitution *) +fun apply_subst_in_concl ctxt i st (cfvs, conclthm) rule m = + RW_Inst.rw ctxt m rule conclthm + |> unfix_frees cfvs + |> Conv.fconv_rule Drule.beta_eta_conversion + |> (fn r => resolve_tac ctxt [r] i st); + +(* substitute within the conclusion of goal i of gth, using a meta +equation rule. Note that we assume rule has var indicies zero'd *) +fun prep_concl_subst ctxt i gth = + let + val th = Thm.incr_indexes 1 gth; + val tgt_term = Thm.prop_of th; + + val (fixedbody, fvs) = IsaND.fix_alls_term ctxt i tgt_term; + val cfvs = rev (map (Thm.cterm_of ctxt) fvs); + + val conclterm = Logic.strip_imp_concl fixedbody; + val conclthm = Thm.trivial (Thm.cterm_of ctxt conclterm); + val maxidx = Thm.maxidx_of th; + val ft = + (Zipper.move_down_right (* ==> *) + o Zipper.move_down_left (* Trueprop *) + o Zipper.mktop + o Thm.prop_of) conclthm + in + ((cfvs, conclthm), (ctxt, maxidx, ft)) + end; + +(* substitute using an object or meta level equality *) +fun eqsubst_tac' ctxt searchf instepthm i st = + let + val (cvfsconclthm, searchinfo) = prep_concl_subst ctxt i st; + val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm); + fun rewrite_with_thm r = + let val (lhs,_) = Logic.dest_equals (Thm.concl_of r) in + searchf searchinfo lhs + |> Seq.maps (apply_subst_in_concl ctxt i st cvfsconclthm r) + end; + in stepthms |> Seq.maps rewrite_with_thm end; + + +(* General substitution of multiple occurrences using one of + the given theorems *) + +fun skip_first_occs_search occ srchf sinfo lhs = + (case skipto_skipseq occ (srchf sinfo lhs) of + SkipMore _ => Seq.empty + | SkipSeq ss => Seq.flat ss); + +(* The "occs" argument is a list of integers indicating which occurrence +w.r.t. the search order, to rewrite. Backtracking will also find later +occurrences, but all earlier ones are skipped. Thus you can use [0] to +just find all rewrites. *) + +fun eqsubst_tac ctxt occs thms i st = + let val nprems = Thm.nprems_of st in + if nprems < i then Seq.empty else + let + val thmseq = Seq.of_list thms; + fun apply_occ occ st = + thmseq |> Seq.maps (fn r => + eqsubst_tac' ctxt + (skip_first_occs_search occ searchf_lr_unify_valid) r + (i + (Thm.nprems_of st - nprems)) st); + val sorted_occs = Library.sort (rev_order o int_ord) occs; + in + Seq.maps distinct_subgoals_tac (Seq.EVERY (map apply_occ sorted_occs) st) + end + end; + + +(* apply a substitution inside assumption j, keeps asm in the same place *) +fun apply_subst_in_asm ctxt i st rule ((cfvs, j, _, pth),m) = + let + val st2 = Thm.rotate_rule (j - 1) i st; (* put premice first *) + val preelimrule = + RW_Inst.rw ctxt m rule pth + |> (Seq.hd o prune_params_tac ctxt) + |> Thm.permute_prems 0 ~1 (* put old asm first *) + |> unfix_frees cfvs (* unfix any global params *) + |> Conv.fconv_rule Drule.beta_eta_conversion; (* normal form *) + in + (* ~j because new asm starts at back, thus we subtract 1 *) + Seq.map (Thm.rotate_rule (~ j) (Thm.nprems_of rule + i)) + (dresolve_tac ctxt [preelimrule] i st2) + end; + + +(* prepare to substitute within the j'th premise of subgoal i of gth, +using a meta-level equation. Note that we assume rule has var indicies +zero'd. Note that we also assume that premt is the j'th premice of +subgoal i of gth. Note the repetition of work done for each +assumption, i.e. this can be made more efficient for search over +multiple assumptions. *) +fun prep_subst_in_asm ctxt i gth j = + let + val th = Thm.incr_indexes 1 gth; + val tgt_term = Thm.prop_of th; + + val (fixedbody, fvs) = IsaND.fix_alls_term ctxt i tgt_term; + val cfvs = rev (map (Thm.cterm_of ctxt) fvs); + + val asmt = nth (Logic.strip_imp_prems fixedbody) (j - 1); + val asm_nprems = length (Logic.strip_imp_prems asmt); + + val pth = Thm.trivial ((Thm.cterm_of ctxt) asmt); + val maxidx = Thm.maxidx_of th; + + val ft = + (Zipper.move_down_right (* trueprop *) + o Zipper.mktop + o Thm.prop_of) pth + in ((cfvs, j, asm_nprems, pth), (ctxt, maxidx, ft)) end; + +(* prepare subst in every possible assumption *) +fun prep_subst_in_asms ctxt i gth = + map (prep_subst_in_asm ctxt i gth) + ((fn l => Library.upto (1, length l)) + (Logic.prems_of_goal (Thm.prop_of gth) i)); + + +(* substitute in an assumption using an object or meta level equality *) +fun eqsubst_asm_tac' ctxt searchf skipocc instepthm i st = + let + val asmpreps = prep_subst_in_asms ctxt i st; + val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm); + fun rewrite_with_thm r = + let + val (lhs,_) = Logic.dest_equals (Thm.concl_of r); + fun occ_search occ [] = Seq.empty + | occ_search occ ((asminfo, searchinfo)::moreasms) = + (case searchf searchinfo occ lhs of + SkipMore i => occ_search i moreasms + | SkipSeq ss => + Seq.append (Seq.map (Library.pair asminfo) (Seq.flat ss)) + (occ_search 1 moreasms)) (* find later substs also *) + in + occ_search skipocc asmpreps |> Seq.maps (apply_subst_in_asm ctxt i st r) + end; + in stepthms |> Seq.maps rewrite_with_thm end; + + +fun skip_first_asm_occs_search searchf sinfo occ lhs = + skipto_skipseq occ (searchf sinfo lhs); + +fun eqsubst_asm_tac ctxt occs thms i st = + let val nprems = Thm.nprems_of st in + if nprems < i then Seq.empty + else + let + val thmseq = Seq.of_list thms; + fun apply_occ occ st = + thmseq |> Seq.maps (fn r => + eqsubst_asm_tac' ctxt + (skip_first_asm_occs_search searchf_lr_unify_valid) occ r + (i + (Thm.nprems_of st - nprems)) st); + val sorted_occs = Library.sort (rev_order o int_ord) occs; + in + Seq.maps distinct_subgoals_tac (Seq.EVERY (map apply_occ sorted_occs) st) + end + end; + +(* combination method that takes a flag (true indicates that subst + should be done to an assumption, false = apply to the conclusion of + the goal) as well as the theorems to use *) +val _ = + let + val parser = + Scan.lift (Args.mode "asm" + -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) + -- Attrib.thms + fun eqsubst_asm_ctac occs inthms = + CONTEXT_TACTIC' (fn ctxt => eqsubst_asm_tac ctxt occs inthms) + fun eqsubst_ctac occs inthms = + CONTEXT_TACTIC' (fn ctxt => eqsubst_tac ctxt occs inthms) + in + Theory.setup ( + Method.setup \<^binding>\sub\ + (parser >> (fn ((asm, occs), inthms) => fn ctxt => SIMPLE_METHOD' ( + (if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms))) + "single-step substitution" (* #> + Method.setup \<^binding>\subst\ + (parser >> (fn ((asm, occs), inthms) => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS 0 + ((if asm then eqsubst_asm_ctac else eqsubst_ctac) occs inthms))))) + "single-step substitution with automatic discharge of side conditions" *) + ) + end + +end; diff --git a/mltt/core/focus.ML b/mltt/core/focus.ML new file mode 100644 index 0000000..b963cfe --- /dev/null +++ b/mltt/core/focus.ML @@ -0,0 +1,158 @@ +(* Title: focus.ML + Author: Joshua Chen + +Focus on head subgoal, with optional variable renaming. + +Modified from code contained in ~~/Pure/Isar/subgoal.ML. +*) + +local + +fun reverse_prems imp = + let val (prems, concl) = (Drule.strip_imp_prems imp, Drule.strip_imp_concl imp) + in fold (curry mk_implies) prems concl end + +fun gen_focus ctxt i bindings raw_st = + let + val st = raw_st + |> Thm.solve_constraints + |> Thm.transfer' ctxt + |> Raw_Simplifier.norm_hhf_protect ctxt + + val ((schematic_types, [st']), ctxt1) = Variable.importT [st] ctxt + + val ((params, goal), ctxt2) = + Variable.focus_cterm bindings (Thm.cprem_of st' i) ctxt1 + + val (asms, concl) = + (Drule.strip_imp_prems goal, Drule.strip_imp_concl goal) + + fun intern_var_assms asm (asms, concl) = + if Lib.no_vars (Thm.term_of asm) + then (asm :: asms, concl) + else (asms, Drule.mk_implies (asm, concl)) + + val (asms', concl') = fold intern_var_assms asms ([], concl) + |> apfst rev |> apsnd reverse_prems + + val (inst, ctxt3) = Variable.import_inst true (map Thm.term_of (asms')) ctxt2 + val schematic_terms = map (apsnd (Thm.cterm_of ctxt3)) (#2 inst) + val schematics = (schematic_types, schematic_terms) + val asms' = map (Thm.instantiate_cterm schematics) asms' + val concl' = Thm.instantiate_cterm schematics concl' + val (prems, context) = Assumption.add_assumes asms' ctxt3 + in + ({context = context, params = params, prems = prems, + asms = asms', concl = concl', schematics = schematics}, Goal.init concl') + end + +fun param_bindings ctxt (param_suffix, raw_param_specs) st = + let + val _ = if Thm.no_prems st then error "No subgoals!" else () + val subgoal = #1 (Logic.dest_implies (Thm.prop_of st)) + val subgoal_params = + map (apfst (Name.internal o Name.clean)) (Term.strip_all_vars subgoal) + |> Term.variant_frees subgoal |> map #1 + + val n = length subgoal_params + val m = length raw_param_specs + val _ = + m <= n orelse + error ("Excessive subgoal parameter specification" ^ + Position.here_list (map snd (drop n raw_param_specs))) + + val param_specs = raw_param_specs + |> map + (fn (NONE, _) => NONE + | (SOME x, pos) => + let + val b = #1 (#1 (Proof_Context.cert_var (Binding.make (x, pos), NONE, NoSyn) ctxt)) + val _ = Variable.check_name b + in SOME b end) + |> param_suffix ? append (replicate (n - m) NONE) + + fun bindings (SOME x :: xs) (_ :: ys) = x :: bindings xs ys + | bindings (NONE :: xs) (y :: ys) = Binding.name y :: bindings xs ys + | bindings _ ys = map Binding.name ys + in bindings param_specs subgoal_params end + +fun gen_schematic_subgoal prep_atts raw_result_binding param_specs state = + let + val _ = Proof.assert_backward state + + val state1 = state + |> Proof.map_context (Proof_Context.set_mode Proof_Context.mode_schematic) + |> Proof.refine_insert [] + + val {context = ctxt, facts, goal = st} = Proof.raw_goal state1 + val result_binding = apsnd (map (prep_atts ctxt)) raw_result_binding + + val subgoal_focus = #1 + (gen_focus ctxt 1 (SOME (param_bindings ctxt param_specs st)) st) + + val prems = #prems subgoal_focus + + fun after_qed (ctxt'', [[result]]) = + Proof.end_block #> (fn state' => + let + val ctxt' = Proof.context_of state' + val results' = + Proof_Context.export ctxt'' ctxt' (Conjunction.elim_conjunctions result) + in + state' + |> Proof.refine_primitive (fn _ => fn _ => + Subgoal.retrofit ctxt'' ctxt' (#params subgoal_focus) (#asms subgoal_focus) 1 + (Goal.protect 0 result) st + |> Seq.hd) + |> Proof.map_context + (#2 o Proof_Context.note_thmss "" [(result_binding, [(results', [])])]) + end) + #> Proof.reset_facts + #> Proof.enter_backward + in + state1 + |> Proof.enter_forward + |> Proof.using_facts [] + |> Proof.begin_block + |> Proof.map_context (fn _ => + #context subgoal_focus + |> Proof_Context.note_thmss "" [((Binding.name "prems", []), [(prems, [])])] + |> snd + |> Context_Facts.register_facts prems) + |> Proof.internal_goal (K (K ())) (Proof_Context.get_mode ctxt) true "subgoal" + NONE after_qed [] [] [(Binding.empty_atts, [(Thm.term_of (#concl subgoal_focus), [])])] + |> #2 + |> Proof.using_facts (facts @ prems) + |> pair subgoal_focus + end + +val opt_fact_binding = + Scan.optional ((Parse.binding -- Parse.opt_attribs || Parse.attribs >> pair Binding.empty) --| Args.colon) + Binding.empty_atts + +val for_params = Scan.optional + (\<^keyword>\vars\ |-- + Parse.!!! ((Scan.option Parse.dots >> is_some) -- + (Scan.repeat1 (Parse.maybe_position Parse.name_position)))) + (false, []) + +val schematic_subgoal_cmd = gen_schematic_subgoal Attrib.attribute_cmd + +val parser = opt_fact_binding -- for_params >> (fn (fact, params) => + Toplevel.proofs (Seq.make_results o Seq.single o #2 o schematic_subgoal_cmd fact params)) + +in + +(** Outer syntax commands **) + +val _ = Outer_Syntax.command \<^command_keyword>\focus\ + "focus on first subgoal within backward refinement, without instantiating schematic vars" + parser + +val _ = Outer_Syntax.command \<^command_keyword>\\<^item>\ "focus bullet" parser +val _ = Outer_Syntax.command \<^command_keyword>\\<^enum>\ "focus bullet" parser +val _ = Outer_Syntax.command \<^command_keyword>\\\ "focus bullet" parser +val _ = Outer_Syntax.command \<^command_keyword>\\\ "focus bullet" parser +val _ = Outer_Syntax.command \<^command_keyword>\~\ "focus bullet" parser + +end diff --git a/mltt/core/goals.ML b/mltt/core/goals.ML new file mode 100644 index 0000000..7d52495 --- /dev/null +++ b/mltt/core/goals.ML @@ -0,0 +1,213 @@ +(* Title: goals.ML + Author: Joshua Chen + +Goal statements and proof term export. + +Modified from code contained in ~~/Pure/Isar/specification.ML. +*) + +local + +val long_keyword = + Parse_Spec.includes >> K "" || + Parse_Spec.long_statement_keyword + +val long_statement = + Scan.optional + (Parse_Spec.opt_thm_name ":" --| Scan.ahead long_keyword) + Binding.empty_atts + -- Scan.optional Parse_Spec.includes [] + -- Parse_Spec.long_statement >> + (fn ((binding, includes), (elems, concl)) => + (true, binding, includes, elems, concl)) + +val short_statement = + Parse_Spec.statement -- Parse_Spec.if_statement -- Parse.for_fixes >> + (fn ((shows, assumes), fixes) => + (false, Binding.empty_atts, [], + [Element.Fixes fixes, Element.Assumes assumes], Element.Shows shows) + ) + +fun prep_statement prep_att prep_stmt raw_elems raw_stmt ctxt = + let + val (stmt, elems_ctxt) = prep_stmt raw_elems raw_stmt ctxt + val prems = Assumption.local_prems_of elems_ctxt ctxt + val stmt_ctxt = + fold (fold (Proof_Context.augment o fst) o snd) stmt elems_ctxt + in case raw_stmt + of Element.Shows _ => + let val stmt' = Attrib.map_specs (map prep_att) stmt + in (([], prems, stmt', NONE), stmt_ctxt) end + | Element.Obtains raw_obtains => + let + val asms_ctxt = stmt_ctxt + |> fold (fn ((name, _), asm) => + snd o Proof_Context.add_assms Assumption.assume_export + [((name, [Context_Rules.intro_query NONE]), asm)]) stmt + val that = Assumption.local_prems_of asms_ctxt stmt_ctxt + val ([(_, that')], that_ctxt) = asms_ctxt + |> Proof_Context.set_stmt true + |> Proof_Context.note_thmss "" + [((Binding.name Auto_Bind.thatN, []), [(that, [])])] + ||> Proof_Context.restore_stmt asms_ctxt + val stmt' = + [(Binding.empty_atts, [(#2 (#1 (Obtain.obtain_thesis ctxt)), [])])] + in + ((Obtain.obtains_attribs raw_obtains, prems, stmt', SOME that'), + that_ctxt) + end + end + +fun define_proof_term name (local_name, [th]) lthy = + let + fun make_name_binding suffix local_name = + let val base_local_name = Long_Name.base_name local_name + in Binding.qualified_name + ((case base_local_name of "" => name | _ => base_local_name) ^ + (case suffix + of SOME "prf" => "_prf" + | SOME "def" => "_def" + | _ => "")) + end + + val (prems, concl) = + (Logic.strip_assums_hyp (Thm.prop_of th), + Logic.strip_assums_concl (Thm.prop_of th)) + in + if not (Lib.is_typing concl) then ([], lthy) + else let + val prems_vars = distinct Term.aconv (flat + (map (Lib.collect_subterms is_Var) prems)) + + val concl_vars = Lib.collect_subterms is_Var + (Lib.term_of_typing concl) + + val params = inter Term.aconv concl_vars prems_vars + + val prf_tm = fold_rev lambda params (Lib.term_of_typing concl) + + val ((_, (_, raw_def)), lthy') = Local_Theory.define + ((make_name_binding NONE local_name, Mixfix.NoSyn), + ((make_name_binding (SOME "prf") local_name, []), prf_tm)) lthy + + val def = fold + (fn th1 => fn th2 => Thm.combination th2 th1) + (map (Thm.reflexive o Thm.cterm_of lthy) params) + raw_def + + val ((_, def'), lthy'') = Local_Theory.note + ((make_name_binding (SOME "def") local_name, []), [def]) + lthy' + in + (def', lthy'') + end + end + | define_proof_term _ _ _ = error + ("Unimplemented: proof terms for multiple facts in one statement") + +fun gen_schematic_theorem + bundle_includes prep_att prep_stmt + gen_prf_tm long kind + before_qed after_qed + (name, raw_atts) raw_includes raw_elems raw_concl + do_print lthy + = + let + val _ = Local_Theory.assert lthy + val elems = raw_elems |> map (Element.map_ctxt_attrib (prep_att lthy)) + val ((more_atts, prems, stmt, facts), goal_ctxt) = lthy + |> bundle_includes raw_includes + |> prep_statement (prep_att lthy) prep_stmt elems raw_concl + val atts = more_atts @ map (prep_att lthy) raw_atts + val pos = Position.thread_data () + val prems_name = if long then Auto_Bind.assmsN else Auto_Bind.thatN + + fun gen_and_after_qed results goal_ctxt' = + let + val results' = burrow + (map (Goal.norm_result lthy) o Proof_Context.export goal_ctxt' lthy) + results + + val ((res, lthy'), substmts) = + if forall (Binding.is_empty_atts o fst) stmt + then ((map (pair "") results', lthy), false) + else + (Local_Theory.notes_kind kind + (map2 (fn (b, _) => fn ths => (b, [(ths, [])])) stmt results') + lthy, + true) + + val (res', lthy'') = + if gen_prf_tm + then + let + val (prf_tm_defs, new_lthy) = fold + (fn result => fn (defs, lthy) => + apfst (fn new_defs => defs @ new_defs) + (define_proof_term (Binding.name_of name) result lthy)) + res + ([], lthy') + val res_folded = + map (apsnd (map (Local_Defs.fold new_lthy prf_tm_defs))) res + in + Local_Theory.notes_kind kind + [((name, @{attributes [type]} @ atts), + [(maps #2 res_folded, [])])] + new_lthy + end + else + Local_Theory.notes_kind kind + [((name, atts), [(maps #2 res, [])])] + lthy' + + val _ = Proof_Display.print_results do_print pos lthy'' + ((kind, Binding.name_of name), map (fn (_, ths) => ("", ths)) res') + + val _ = + if substmts then map + (fn (name, ths) => Proof_Display.print_results do_print pos lthy'' + (("and", name), [("", ths)])) + res + else [] + in + after_qed results' lthy'' + end + in + goal_ctxt + |> not (null prems) ? + (Proof_Context.note_thmss "" [((Binding.name prems_name, []), [(prems, [])])] + #> snd #> Context_Facts.register_facts prems) + |> Proof.theorem before_qed gen_and_after_qed (map snd stmt) + |> (case facts of NONE => I | SOME ths => Proof.refine_insert ths) + end + +val schematic_theorem_cmd = + gen_schematic_theorem + Bundle.includes_cmd + Attrib.check_src + Elaborated_Statement.read_goal_statement + +fun theorem spec descr = + Outer_Syntax.local_theory_to_proof' spec ("state " ^ descr) + (Scan.option (Args.parens (Args.$$$ "def")) + -- (long_statement || short_statement) >> + (fn (opt_derive, (long, binding, includes, elems, concl)) => + schematic_theorem_cmd + (case opt_derive of SOME "def" => true | _ => false) + long descr NONE (K I) binding includes elems concl)) + +fun definition spec descr = + Outer_Syntax.local_theory_to_proof' spec "definition via proof" + ((long_statement || short_statement) >> + (fn (long, binding, includes, elems, concl) => schematic_theorem_cmd + true long descr NONE (K I) binding includes elems concl)) + +in + +val _ = theorem \<^command_keyword>\Theorem\ "Theorem" +val _ = theorem \<^command_keyword>\Lemma\ "Lemma" +val _ = theorem \<^command_keyword>\Corollary\ "Corollary" +val _ = theorem \<^command_keyword>\Proposition\ "Proposition" +val _ = definition \<^command_keyword>\Definition\ "Definition" + +end diff --git a/mltt/core/implicits.ML b/mltt/core/implicits.ML new file mode 100644 index 0000000..2b63f49 --- /dev/null +++ b/mltt/core/implicits.ML @@ -0,0 +1,87 @@ +(* Title: implicits.ML + Author: Joshua Chen + +Implicit arguments. +*) + +structure Implicits : +sig + +val implicit_defs: Proof.context -> (term * term) Symtab.table +val implicit_defs_attr: attribute +val make_holes: Proof.context -> term list -> term list + +end = struct + +structure Defs = Generic_Data ( + type T = (term * term) Symtab.table + val empty = Symtab.empty + val extend = I + val merge = Symtab.merge (Term.aconv o apply2 #1) +) + +val implicit_defs = Defs.get o Context.Proof + +val implicit_defs_attr = Thm.declaration_attribute (fn th => + let + val (t, def) = Lib.dest_eq (Thm.prop_of th) + val (head, args) = Term.strip_comb t + val def' = fold_rev lambda args def + in + Defs.map (Symtab.update (Term.term_name head, (head, def'))) + end) + +fun make_holes_single ctxt tm name_ctxt = + let + fun iarg_to_hole (Const (\<^const_name>\iarg\, T)) = + Const (\<^const_name>\hole\, T) + | iarg_to_hole t = t + + fun expand head args = + let fun betapplys (head', args') = + Term.betapplys (map_aterms iarg_to_hole head', args') + in + case head of + Abs (x, T, t) => + list_comb (Abs (x, T, Lib.traverse_term expand t), args) + | _ => + case Symtab.lookup (implicit_defs ctxt) (Term.term_name head) of + SOME (t, def) => betapplys + (Envir.expand_atom + (Term.fastype_of head) + (Term.fastype_of t, def), + args) + | NONE => list_comb (head, args) + end + + fun holes_to_vars t = + let + val count = Lib.subterm_count (Const (\<^const_name>\hole\, dummyT)) + + fun subst (Const (\<^const_name>\hole\, T)) (Var (idx, _)::_) Ts = + let + val bounds = map Bound (0 upto (length Ts - 1)) + val T' = foldr1 (op -->) (Ts @ [T]) + in + foldl1 (op $) (Var (idx, T')::bounds) + end + | subst (Abs (x, T, t)) vs Ts = Abs (x, T, subst t vs (T::Ts)) + | subst (t $ u) vs Ts = + let val n = count t + in subst t (take n vs) Ts $ subst u (drop n vs) Ts end + | subst t _ _ = t + + val names = Name.invent name_ctxt "*" (count t) + val vars = map (fn n => Var ((n, 0), dummyT)) names + in + (subst t vars [], fold Name.declare names name_ctxt) + end + in + holes_to_vars (Lib.traverse_term expand tm) + end + +fun make_holes ctxt tms = #1 + (fold_map (make_holes_single ctxt) tms (Variable.names_of ctxt)) + + +end diff --git a/mltt/core/lib.ML b/mltt/core/lib.ML new file mode 100644 index 0000000..e43ad98 --- /dev/null +++ b/mltt/core/lib.ML @@ -0,0 +1,193 @@ +structure Lib : +sig + +(*Lists*) +val max: ('a * 'a -> bool) -> 'a list -> 'a +val maxint: int list -> int + +(*Terms*) +val no_vars: term -> bool +val is_rigid: term -> bool +val is_eq: term -> bool +val dest_prop: term -> term +val dest_eq: term -> term * term +val mk_Var: string -> int -> typ -> term +val lambda_var: term -> term -> term + +val is_typing: term -> bool +val mk_typing: term -> term -> term +val dest_typing: term -> term * term +val term_of_typing: term -> term +val type_of_typing: term -> term +val mk_Pi: term -> term -> term -> term + +val typing_of_term: term -> term + +(*Goals*) +val decompose_goal: Proof.context -> term -> term list * term +val rigid_typing_concl: term -> bool + +(*Theorems*) +val partition_judgments: thm list -> thm list * thm list * thm list + +(*Subterms*) +val has_subterm: term list -> term -> bool +val subterm_count: term -> term -> int +val subterm_count_distinct: term list -> term -> int +val traverse_term: (term -> term list -> term) -> term -> term +val collect_subterms: (term -> bool) -> term -> term list + +(*Orderings*) +val subterm_order: term -> term -> order +val cond_order: order -> order -> order + +end = struct + + +(** Lists **) + +fun max gt (x::xs) = fold (fn a => fn b => if gt (a, b) then a else b) xs x + | max _ [] = error "max of empty list" + +val maxint = max (op >) + + +(** Terms **) + +(* Meta *) + +val no_vars = not o exists_subterm is_Var + +val is_rigid = not o is_Var o head_of + +fun is_eq (Const (\<^const_name>\Pure.eq\, _) $ _ $ _) = true + | is_eq _ = false + +fun dest_prop (Const (\<^const_name>\Pure.prop\, _) $ P) = P + | dest_prop P = P + +fun dest_eq (Const (\<^const_name>\Pure.eq\, _) $ t $ def) = (t, def) + | dest_eq _ = error "dest_eq" + +fun mk_Var name idx T = Var ((name, idx), T) + +fun lambda_var x tm = + let + fun var_args (Var (idx, T)) = Var (idx, \<^typ>\o\ --> T) $ x + | var_args t = t + in + tm |> map_aterms var_args + |> lambda x + end + +(* Object *) + +fun is_typing (Const (\<^const_name>\has_type\, _) $ _ $ _) = true + | is_typing _ = false + +fun mk_typing t T = \<^const>\has_type\ $ t $ T + +fun dest_typing (Const (\<^const_name>\has_type\, _) $ t $ T) = (t, T) + | dest_typing t = raise TERM ("dest_typing", [t]) + +val term_of_typing = #1 o dest_typing +val type_of_typing = #2 o dest_typing + +fun mk_Pi v typ body = Const (\<^const_name>\Pi\, dummyT) $ typ $ lambda v body + +fun typing_of_term tm = \<^const>\has_type\ $ tm $ Var (("*?", 0), \<^typ>\o\) +(*The above is a bit hacky; basically we need to guarantee that the schematic + var is fresh. This works for now because no other code in the Isabelle system + or the current logic uses this identifier.*) + + +(** Goals **) + +(*Breaks a goal \x ... y. \P1; ... Pn\ \ Q into ([P1, ..., Pn], Q), fixing + \-quantified variables and keeping schematics.*) +fun decompose_goal ctxt goal = + let + val focus = + #1 (Subgoal.focus_prems ctxt 1 NONE (Thm.trivial (Thm.cterm_of ctxt goal))) + + val schematics = #2 (#schematics focus) + |> map (fn (v, ctm) => (Thm.term_of ctm, Var v)) + in + map Thm.prop_of (#prems focus) @ [Thm.term_of (#concl focus)] + |> map (subst_free schematics) + |> (fn xs => chop (length xs - 1) xs) |> apsnd the_single + end + handle List.Empty => error "Lib.decompose_goal" + +fun rigid_typing_concl goal = + let val concl = Logic.strip_assums_concl goal + in is_typing concl andalso is_rigid (term_of_typing concl) end + + +(** Theorems **) +fun partition_judgments ths = + let + fun part [] facts conds eqs = (facts, conds, eqs) + | part (th::ths) facts conds eqs = + if is_typing (Thm.prop_of th) then + part ths (th::facts) conds eqs + else if is_typing (Thm.concl_of th) then + part ths facts (th::conds) eqs + else part ths facts conds (th::eqs) + in part ths [] [] [] end + + +(** Subterms **) + +fun has_subterm tms = + Term.exists_subterm + (foldl1 (op orf) (map (fn t => fn s => Term.aconv_untyped (s, t)) tms)) + +fun subterm_count s t = + let + fun count (t1 $ t2) i = i + count t1 0 + count t2 0 + | count (Abs (_, _, t)) i = i + count t 0 + | count t i = if Term.aconv_untyped (s, t) then i + 1 else i + in + count t 0 + end + +(*Number of distinct subterms in `tms` that appear in `tm`*) +fun subterm_count_distinct tms tm = + length (filter I (map (fn t => has_subterm [t] tm) tms)) + +(* + "Folds" a function f over the term structure of t by traversing t from child + nodes upwards through parents. At each node n in the term syntax tree, f is + additionally passed a list of the results of f at all children of n. +*) +fun traverse_term f t = + let + fun map_aux (Abs (x, T, t)) = Abs (x, T, map_aux t) + | map_aux t = + let + val (head, args) = Term.strip_comb t + val args' = map map_aux args + in + f head args' + end + in + map_aux t + end + +fun collect_subterms f (t $ u) = collect_subterms f t @ collect_subterms f u + | collect_subterms f (Abs (_, _, t)) = collect_subterms f t + | collect_subterms f t = if f t then [t] else [] + + +(** Orderings **) + +fun subterm_order t1 t2 = + if has_subterm [t1] t2 then LESS + else if has_subterm [t2] t1 then GREATER + else EQUAL + +fun cond_order o1 o2 = case o1 of EQUAL => o2 | _ => o1 + + +end diff --git a/mltt/core/tactics.ML b/mltt/core/tactics.ML new file mode 100644 index 0000000..6876d5c --- /dev/null +++ b/mltt/core/tactics.ML @@ -0,0 +1,180 @@ +(* Title: tactics.ML + Author: Joshua Chen + +General tactics for dependent type theory. +*) + +structure Tactics: +sig + +val solve_side_conds: int Config.T +val SIDE_CONDS: int -> context_tactic' -> thm list -> context_tactic' +val rule_ctac: thm list -> context_tactic' +val dest_ctac: int option -> thm list -> context_tactic' +val intro_ctac: context_tactic' +val elim_ctac: term list -> context_tactic' +val cases_ctac: term -> context_tactic' + +end = struct + + +(* Side conditions *) +val solve_side_conds = Attrib.setup_config_int \<^binding>\solve_side_conds\ (K 2) + +fun SIDE_CONDS j ctac facts i (cst as (ctxt, st)) = cst |> + (case Config.get ctxt solve_side_conds of + 1 => (ctac CTHEN_ALL_NEW (CTRY o Types.known_ctac facts)) i + | 2 => ctac i CTHEN CREPEAT_IN_RANGE (i + j) (Thm.nprems_of st - i) + (CTRY o CREPEAT_ALL_NEW_FWD (Types.check_infer facts)) + | _ => ctac i) + + +(* rule, dest, intro *) + +local + fun mk_rules _ ths [] = ths + | mk_rules n ths ths' = + let val ths'' = foldr1 (op @) + (map + (fn th => [rotate_prems n (th RS @{thm PiE})] handle THM _ => []) + ths') + in + mk_rules n (ths @ ths') ths'' + end +in + +(*Resolves with given rules*) +fun rule_ctac ths i (ctxt, st) = + TACTIC_CONTEXT ctxt (resolve_tac ctxt (mk_rules 0 [] ths) i st) + +(*Attempts destruct-resolution with the n-th premise of the given rules*) +fun dest_ctac opt_n ths i (ctxt, st) = + TACTIC_CONTEXT ctxt (dresolve_tac ctxt + (mk_rules (case opt_n of NONE => 0 | SOME 0 => 0 | SOME n => n-1) [] ths) + i st) + +end + +(*Applies an appropriate introduction rule*) +val intro_ctac = CONTEXT_TACTIC' (fn ctxt => SUBGOAL (fn (goal, i) => + let val concl = Logic.strip_assums_concl goal in + if Lib.is_typing concl andalso Lib.is_rigid (Lib.type_of_typing concl) + then resolve_tac ctxt (Named_Theorems.get ctxt \<^named_theorems>\intro\) i + else no_tac + end)) + + +(* Induction/elimination *) + +(*Pushes a known typing t:T into a \-type. + This tactic is well-behaved only when t is sufficiently well specified + (otherwise there might be multiple possible judgments t:T that unify, and + which is chosen is undefined).*) +fun internalize_fact_tac t = + Subgoal.FOCUS_PARAMS (fn {context = ctxt, concl = raw_concl, ...} => + let + val concl = Logic.strip_assums_concl (Thm.term_of raw_concl) + val C = Lib.type_of_typing concl + val B = Thm.cterm_of ctxt (Lib.lambda_var t C) + val a = Thm.cterm_of ctxt t + (*The resolvent is PiE[where ?B=B and ?a=a]*) + val resolvent = + Drule.infer_instantiate' ctxt [NONE, NONE, SOME B, SOME a] @{thm PiE} + in + HEADGOAL (resolve_tac ctxt [resolvent]) + (*Unify with the correct type T*) + THEN SOMEGOAL (NO_CONTEXT_TACTIC ctxt o Types.known_ctac []) + end) + +fun elim_core_tac tms types ctxt = + let + val rule_insts = map ((Elim.lookup_rule ctxt) o Term.head_of) types + val rules = flat (map + (fn rule_inst => case rule_inst of + NONE => [] + | SOME (rl, idxnames) => [Drule.infer_instantiate ctxt + (idxnames ~~ map (Thm.cterm_of ctxt) tms) rl]) + rule_insts) + in + resolve_tac ctxt rules + THEN' RANGE (replicate (length tms) (NO_CONTEXT_TACTIC ctxt o Types.check_infer [])) + end handle Option => K no_tac + +(*Premises that have already been pushed into the \-type*) +structure Inserts = Proof_Data ( + type T = term Item_Net.T + val init = K (Item_Net.init Term.aconv_untyped single) +) + +fun elim_ctac tms = + case tms of + [] => CONTEXT_TACTIC' (fn ctxt => eresolve_tac ctxt (map #1 (Elim.rules ctxt))) + | major :: _ => CONTEXT_SUBGOAL (fn (goal, _) => fn cst as (ctxt, st) => + let + val facts = map Thm.prop_of (Context_Facts.known ctxt) + val prems = Logic.strip_assums_hyp goal + val template = Lib.typing_of_term major + val types = filter (fn th => Term.could_unify (template, th)) (facts @ prems) + |> map Lib.type_of_typing + in case types of + [] => no_ctac cst + | _ => + let + val inserts = facts @ prems + |> filter Lib.is_typing + |> map Lib.dest_typing + |> filter_out (fn (t, _) => + Term.aconv (t, major) orelse Item_Net.member (Inserts.get ctxt) t) + |> map (fn (t, T) => ((t, T), Lib.subterm_count_distinct tms T)) + |> filter (fn (_, i) => i > 0) + (*`t1: T1` comes before `t2: T2` if T1 contains t2 as subterm. + If they are incomparable, then order by decreasing + `subterm_count_distinct tms T`*) + |> sort (fn (((t1, _), i), ((_, T2), j)) => + Lib.cond_order (Lib.subterm_order T2 t1) (int_ord (j, i))) + |> map (#1 o #1) + val record_inserts = Inserts.map (fold Item_Net.update inserts) + val tac = + (*Push premises having a subterm in `tms` into a \*) + fold (fn t => fn tac => + tac THEN HEADGOAL (internalize_fact_tac t ctxt)) + inserts all_tac + (*Apply elimination rule*) + THEN HEADGOAL ( + elim_core_tac tms types ctxt + (*Pull pushed premises back out*) + THEN_ALL_NEW (SUBGOAL (fn (_, i) => + REPEAT_DETERM_N (length inserts) + (resolve_tac ctxt @{thms PiI[rotated]} i)))) + in + TACTIC_CONTEXT (record_inserts ctxt) (tac st) + end + end) + +fun cases_ctac tm = + let fun tac ctxt = + SUBGOAL (fn (goal, i) => + let + val facts = Proof_Context.facts_of ctxt + val prems = Logic.strip_assums_hyp goal + val template = Lib.typing_of_term tm + val types = + map (Thm.prop_of o #1) (Facts.could_unify facts template) + @ filter (fn prem => Term.could_unify (template, prem)) prems + |> map Lib.type_of_typing + val res = (case types of + [typ] => Drule.infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt tm)] + (the (Case.lookup_rule ctxt (Term.head_of typ))) + | [] => raise Option + | _ => raise error (Syntax.string_of_term ctxt tm ^ "not uniquely typed")) + handle Option => + error ("No case rule known for " ^ (Syntax.string_of_term ctxt tm)) + in + resolve_tac ctxt [res] i + end) + in CONTEXT_TACTIC' tac end + + +end + +open Tactics diff --git a/mltt/core/types.ML b/mltt/core/types.ML new file mode 100644 index 0000000..5e0d484 --- /dev/null +++ b/mltt/core/types.ML @@ -0,0 +1,113 @@ +(* Title: types.ML + Author: Joshua Chen + +Type-checking infrastructure. +*) + +structure Types: sig + +val debug_typechk: bool Config.T + +val known_ctac: thm list -> int -> context_tactic +val check_infer: thm list -> int -> context_tactic + +end = struct + +open Context_Facts + +(** [type] attribute **) + +val _ = Theory.setup ( + Attrib.setup \<^binding>\type\ + (Scan.succeed (Thm.declaration_attribute (fn th => + if Thm.no_prems th then register_known th else register_cond th))) + "" + #> Global_Theory.add_thms_dynamic (\<^binding>\type\, + fn context => let val ctxt = Context.proof_of context in + known ctxt @ cond ctxt end) +) + + +(** Context tactics for type-checking and elaboration **) + +val debug_typechk = Attrib.setup_config_bool \<^binding>\debug_typechk\ (K false) + +fun debug_tac ctxt s = + if Config.get ctxt debug_typechk then print_tac ctxt s else all_tac + +(*Solves goals without metavariables and type inference problems by assumption + from inline premises or resolution with facts*) +fun known_ctac facts = CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) => + TACTIC_CONTEXT ctxt + let val concl = Logic.strip_assums_concl goal in + if Lib.no_vars concl orelse + (Lib.is_typing concl andalso Lib.no_vars (Lib.term_of_typing concl)) + then + let val ths = known ctxt @ facts + in st |> + (assume_tac ctxt ORELSE' resolve_tac ctxt ths THEN_ALL_NEW K no_tac) i + end + else Seq.empty + end) + +(*Simple bidirectional typing tactic with some backtracking search over input + facts.*) +fun check_infer_step facts i (ctxt, st) = + let + val refine_tac = SUBGOAL (fn (goal, i) => + if Lib.rigid_typing_concl goal + then + let + val net = Tactic.build_net ( + map (Simplifier.norm_hhf ctxt) facts + @(cond ctxt) + @(Named_Theorems.get ctxt \<^named_theorems>\form\) + @(Named_Theorems.get ctxt \<^named_theorems>\intr\) + @(map #1 (Elim.rules ctxt))) + in resolve_from_net_tac ctxt net i end + else no_tac) + + val sub_tac = SUBGOAL (fn (goal, i) => + let val concl = Logic.strip_assums_concl goal in + if Lib.is_typing concl + andalso Lib.is_rigid (Lib.term_of_typing concl) + andalso Lib.no_vars (Lib.type_of_typing concl) + then + (resolve_tac ctxt @{thms sub} + THEN' SUBGOAL (fn (_, i) => + NO_CONTEXT_TACTIC ctxt (check_infer facts i)) + THEN' compute_tac ctxt facts + THEN_ALL_NEW K no_tac) i + else no_tac end) + + val ctxt' = ctxt (*TODO: Use this to store already-derived typing judgments*) + in + TACTIC_CONTEXT ctxt' ( + (NO_CONTEXT_TACTIC ctxt' o known_ctac facts + ORELSE' refine_tac + ORELSE' sub_tac) i st) + end + +and check_infer facts i (cst as (_, st)) = + let + val ctac = check_infer_step facts + in + cst |> (ctac i CTHEN + CREPEAT_IN_RANGE i (Thm.nprems_of st - i) (CTRY o CREPEAT_ALL_NEW_FWD ctac)) + end + +and compute_tac ctxt facts = + let + val comps = Named_Theorems.get ctxt \<^named_theorems>\comp\ + val ss = simpset_of ctxt + val ss' = simpset_of (empty_simpset ctxt addsimps comps) + val ctxt' = put_simpset (merge_ss (ss, ss')) ctxt + in + SUBGOAL (fn (_, i) => + ((CHANGED o asm_simp_tac ctxt' ORELSE' EqSubst.eqsubst_tac ctxt [0] comps) + THEN_ALL_NEW SUBGOAL (fn (_, i) => + NO_CONTEXT_TACTIC ctxt (check_infer facts i))) i) + end + + +end diff --git a/mltt/lib/List.thy b/mltt/lib/List.thy new file mode 100644 index 0000000..4beb9b6 --- /dev/null +++ b/mltt/lib/List.thy @@ -0,0 +1,191 @@ +chapter \Lists\ + +theory List +imports Maybe + +begin + +(*TODO: Inductive type and recursive function definitions. The ad-hoc + axiomatization below should be subsumed once general inductive types are + properly implemented.*) + +axiomatization + List :: \o \ o\ and + nil :: \o \ o\ and + cons :: \o \ o \ o \ o\ and + ListInd :: \o \ (o \ o) \ o \ (o \ o \ o \ o) \ o \ o\ +where + ListF: "A: U i \ List A: U i" and + + List_nil: "A: U i \ nil A: List A" and + + List_cons: "\x: A; xs: List A\ \ cons A x xs: List A" and + + ListE: "\ + xs: List A; + c\<^sub>0: C (nil A); + \x xs rec. \x: A; xs: List A; rec: C xs\ \ f x xs rec: C (cons A x xs); + \xs. xs: List A \ C xs: U i + \ \ ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs: C xs" and + + List_comp_nil: "\ + c\<^sub>0: C (nil A); + \x xs rec. \x: A; xs: List A; rec: C xs\ \ f x xs rec: C (cons A x xs); + \xs. xs: List A \ C xs: U i + \ \ ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) (nil A) \ c\<^sub>0" and + + List_comp_cons: "\ + xs: List A; + c\<^sub>0: C (nil A); + \x xs rec. \x: A; xs: List A; rec: C xs\ \ f x xs rec: C (cons A x xs); + \xs. xs: List A \ C xs: U i + \ \ + ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) (cons A x xs) \ + f x xs (ListInd A (fn xs. C xs) c\<^sub>0 (fn x xs rec. f x xs rec) xs)" + +lemmas + [form] = ListF and + [intr, intro] = List_nil List_cons and + [elim "?xs"] = ListE and + [comp] = List_comp_nil List_comp_cons + +abbreviation "ListRec A C \ ListInd A (fn _. C)" + +Lemma list_cases [cases]: + assumes + "xs: List A" and + nil_case: "c\<^sub>0: C (nil A)" and + cons_case: "\x xs. \x: A; xs: List A\ \ f x xs: C (cons A x xs)" and + "\xs. xs: List A \ C xs: U i" + shows "C xs" + by (elim xs) (fact nil_case, rule cons_case) + + +section \Notation\ + +definition nil_i ("[]") + where [implicit]: "[] \ nil {}" + +definition cons_i (infixr "#" 120) + where [implicit]: "x # xs \ cons {} x xs" + +translations + "[]" \ "CONST List.nil A" + "x # xs" \ "CONST List.cons A x xs" +syntax + "_list" :: \args \ o\ ("[_]") +translations + "[x, xs]" \ "x # [xs]" + "[x]" \ "x # []" + + +section \Standard functions\ + +subsection \Head and tail\ + +Definition head: + assumes "A: U i" "xs: List A" + shows "Maybe A" +proof (cases xs) + show "none: Maybe A" by intro + show "\x. x: A \ some x: Maybe A" by intro +qed + +Definition tail: + assumes "A: U i" "xs: List A" + shows "List A" +proof (cases xs) + show "[]: List A" by intro + show "\xs. xs: List A \ xs: List A" . +qed + +definition head_i ("head") where [implicit]: "head xs \ List.head {} xs" +definition tail_i ("tail") where [implicit]: "tail xs \ List.tail {} xs" + +translations + "head" \ "CONST List.head A" + "tail" \ "CONST List.tail A" + +Lemma head_type [type]: + assumes "A: U i" "xs: List A" + shows "head xs: Maybe A" + unfolding head_def by typechk + +Lemma head_of_cons [comp]: + assumes "A: U i" "x: A" "xs: List A" + shows "head (x # xs) \ some x" + unfolding head_def by compute + +Lemma tail_type [type]: + assumes "A: U i" "xs: List A" + shows "tail xs: List A" + unfolding tail_def by typechk + +Lemma tail_of_cons [comp]: + assumes "A: U i" "x: A" "xs: List A" + shows "tail (x # xs) \ xs" + unfolding tail_def by compute + +subsection \Append\ + +Definition app: + assumes "A: U i" "xs: List A" "ys: List A" + shows "List A" + apply (elim xs) + \<^item> by (fact \ys:_\) + \<^item> vars x _ rec + proof - show "x # rec: List A" by typechk qed + done + +definition app_i ("app") where [implicit]: "app xs ys \ List.app {} xs ys" + +translations "app" \ "CONST List.app A" + +subsection \Map\ + +Definition map: + assumes "A: U i" "B: U i" "f: A \ B" "xs: List A" + shows "List B" +proof (elim xs) + show "[]: List B" by intro + next fix x ys + assuming "x: A" "ys: List B" + show "f x # ys: List B" by typechk +qed + +definition map_i ("map") where [implicit]: "map \ List.map {} {}" + +translations "map" \ "CONST List.map A B" + +Lemma map_type [type]: + assumes "A: U i" "B: U i" "f: A \ B" "xs: List A" + shows "map f xs: List B" + unfolding map_def by typechk + + +subsection \Reverse\ + +Definition rev: + assumes "A: U i" "xs: List A" + shows "List A" + apply (elim xs) + \<^item> by (rule List_nil) + \<^item> vars x _ rec proof - show "app rec [x]: List A" by typechk qed + done + +definition rev_i ("rev") where [implicit]: "rev \ List.rev {}" + +translations "rev" \ "CONST List.rev A" + +Lemma rev_type [type]: + assumes "A: U i" "xs: List A" + shows "rev xs: List A" + unfolding rev_def by typechk + +Lemma rev_nil [comp]: + assumes "A: U i" + shows "rev (nil A) \ nil A" + unfolding rev_def by compute + + +end diff --git a/mltt/lib/Maybe.thy b/mltt/lib/Maybe.thy new file mode 100644 index 0000000..452acc2 --- /dev/null +++ b/mltt/lib/Maybe.thy @@ -0,0 +1,75 @@ +chapter \Maybe type\ + +theory Maybe +imports Prelude + +begin + +text \Defined as a sum.\ + +definition "Maybe A \ A \ \" +definition "none A \ inr A \ tt" +definition "some A a \ inl A \ a" + +lemma + MaybeF: "A: U i \ Maybe A: U i" and + Maybe_none: "A: U i \ none A: Maybe A" and + Maybe_some: "a: A \ some A a: Maybe A" + unfolding Maybe_def none_def some_def by typechk+ + +Definition MaybeInd: + assumes + "A: U i" + "\m. m: Maybe A \ C m: U i" + "c\<^sub>0: C (none A)" + "\a. a: A \ f a: C (some A a)" + "m: Maybe A" + shows "C m" + using assms[unfolded Maybe_def none_def some_def, type] + apply (elim m) + apply fact + apply (elim, fact) + done + +Lemma Maybe_comp_none: + assumes + "A: U i" + "c\<^sub>0: C (none A)" + "\a. a: A \ f a: C (some A a)" + "\m. m: Maybe A \ C m: U i" + shows "MaybeInd A C c\<^sub>0 f (none A) \ c\<^sub>0" + using assms + unfolding Maybe_def MaybeInd_def none_def some_def + by compute + +Lemma Maybe_comp_some: + assumes + "A: U i" + "a: A" + "c\<^sub>0: C (none A)" + "\a. a: A \ f a: C (some A a)" + "\m. m: Maybe A \ C m: U i" + shows "MaybeInd A C c\<^sub>0 f (some A a) \ f a" + using assms + unfolding Maybe_def MaybeInd_def none_def some_def + by compute + +lemmas + [form] = MaybeF and + [intr, intro] = Maybe_none Maybe_some and + [comp] = Maybe_comp_none Maybe_comp_some and + MaybeE [elim "?m"] = MaybeInd[rotated 4] +lemmas + Maybe_cases [cases] = MaybeE + +abbreviation "MaybeRec A C \ MaybeInd A (K C)" + +definition none_i ("none") where [implicit]: "none \ Maybe.none {}" +definition some_i ("some") where [implicit]: "some a \ Maybe.some {} a" + +translations + "none" \ "CONST Maybe.none A" + "some a" \ "CONST Maybe.some A a" + + +end diff --git a/mltt/lib/Prelude.thy b/mltt/lib/Prelude.thy new file mode 100644 index 0000000..0393968 --- /dev/null +++ b/mltt/lib/Prelude.thy @@ -0,0 +1,153 @@ +theory Prelude +imports MLTT + +begin + +section \Sum type\ + +axiomatization + Sum :: \o \ o \ o\ and + inl :: \o \ o \ o \ o\ and + inr :: \o \ o \ o \ o\ and + SumInd :: \o \ o \ (o \ o) \ (o \ o) \ (o \ o) \ o \ o\ + +notation Sum (infixl "\" 50) + +axiomatization where + SumF: "\A: U i; B: U i\ \ A \ B: U i" and + + Sum_inl: "\B: U i; a: A\ \ inl A B a: A \ B" and + + Sum_inr: "\A: U i; b: B\ \ inr A B b: A \ B" and + + SumE: "\ + s: A \ B; + \s. s: A \ B \ C s: U i; + \a. a: A \ c a: C (inl A B a); + \b. b: B \ d b: C (inr A B b) + \ \ SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) s: C s" and + + Sum_comp_inl: "\ + a: A; + \s. s: A \ B \ C s: U i; + \a. a: A \ c a: C (inl A B a); + \b. b: B \ d b: C (inr A B b) + \ \ SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inl A B a) \ c a" and + + Sum_comp_inr: "\ + b: B; + \s. s: A \ B \ C s: U i; + \a. a: A \ c a: C (inl A B a); + \b. b: B \ d b: C (inr A B b) + \ \ SumInd A B (fn s. C s) (fn a. c a) (fn b. d b) (inr A B b) \ d b" + +lemmas + [form] = SumF and + [intr] = Sum_inl Sum_inr and + [intro] = Sum_inl[rotated] Sum_inr[rotated] and + [elim ?s] = SumE and + [comp] = Sum_comp_inl Sum_comp_inr + +method left = rule Sum_inl +method right = rule Sum_inr + + +section \Empty and unit types\ + +axiomatization + Top :: \o\ and + tt :: \o\ and + TopInd :: \(o \ o) \ o \ o \ o\ +and + Bot :: \o\ and + BotInd :: \(o \ o) \ o \ o\ + +notation Top ("\") and Bot ("\") + +axiomatization where + TopF: "\: U i" and + + TopI: "tt: \" and + + TopE: "\a: \; \x. x: \ \ C x: U i; c: C tt\ \ TopInd (fn x. C x) c a: C a" and + + Top_comp: "\\x. x: \ \ C x: U i; c: C tt\ \ TopInd (fn x. C x) c tt \ c" +and + BotF: "\: U i" and + + BotE: "\x: \; \x. x: \ \ C x: U i\ \ BotInd (fn x. C x) x: C x" + +lemmas + [form] = TopF BotF and + [intr, intro] = TopI and + [elim ?a] = TopE and + [elim ?x] = BotE and + [comp] = Top_comp + +abbreviation (input) Not ("\_" [1000] 1000) where "\A \ A \ \" + + +section \Booleans\ + +definition "Bool \ \ \ \" +definition "true \ inl \ \ tt" +definition "false \ inr \ \ tt" + +Lemma + BoolF: "Bool: U i" and + Bool_true: "true: Bool" and + Bool_false: "false: Bool" + unfolding Bool_def true_def false_def by typechk+ + +\ \Definitions like these should be handled by a future function package\ +Definition ifelse [rotated 1]: + assumes *[unfolded Bool_def true_def false_def]: + "\x. x: Bool \ C x: U i" + "x: Bool" + "a: C true" + "b: C false" + shows "C x" + using assms[unfolded Bool_def true_def false_def, type] + by (elim x) (elim, fact)+ + +Lemma if_true: + assumes + "\x. x: Bool \ C x: U i" + "a: C true" + "b: C false" + shows "ifelse C true a b \ a" + unfolding ifelse_def true_def + using assms unfolding Bool_def true_def false_def + by compute + +Lemma if_false: + assumes + "\x. x: Bool \ C x: U i" + "a: C true" + "b: C false" + shows "ifelse C false a b \ b" + unfolding ifelse_def false_def + using assms unfolding Bool_def true_def false_def + by compute + +lemmas + [form] = BoolF and + [intr, intro] = Bool_true Bool_false and + [comp] = if_true if_false and + [elim ?x] = ifelse +lemmas + BoolE = ifelse + +subsection \Notation\ + +definition ifelse_i ("if _ then _ else _") + where [implicit]: "if x then a else b \ ifelse {} x a b" + +translations "if x then a else b" \ "CONST ifelse C x a b" + +subsection \Logical connectives\ + +definition not ("!_") where "!x \ ifelse (K Bool) x false true" + + +end -- cgit v1.2.3