diff options
Diffstat (limited to '')
-rw-r--r-- | spartan/theories/Spartan.thy | 463 |
1 files changed, 463 insertions, 0 deletions
diff --git a/spartan/theories/Spartan.thy b/spartan/theories/Spartan.thy new file mode 100644 index 0000000..fb901d5 --- /dev/null +++ b/spartan/theories/Spartan.thy @@ -0,0 +1,463 @@ +text \<open>Spartan type theory\<close> + +theory Spartan +imports + Pure + "HOL-Eisbach.Eisbach" + "HOL-Eisbach.Eisbach_Tools" +keywords + "Theorem" "Lemma" "Corollary" "Proposition" :: thy_goal_stmt and + "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and + "derive" "vars":: quasi_command and + "print_coercions" :: thy_decl + +begin + + +section \<open>Preamble\<close> + +declare [[eta_contract=false]] + + +section \<open>Metatype setup\<close> + +typedecl o + + +section \<open>Judgments\<close> + +judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999) + + +section \<open>Universes\<close> + +typedecl lvl \<comment> \<open>Universe levels\<close> + +axiomatization + O :: \<open>lvl\<close> and + S :: \<open>lvl \<Rightarrow> lvl\<close> and + lt :: \<open>lvl \<Rightarrow> lvl \<Rightarrow> prop\<close> (infix "<" 900) + where + O_min: "O < S i" and + lt_S: "i < S i" and + lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k" + +axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where + U_hierarchy: "i < j \<Longrightarrow> U i: U j" and + U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j" + +lemma U_in_U: + "U i: U (S i)" + by (rule U_hierarchy, rule lt_S) + +lemma lift_universe: + "A: U i \<Longrightarrow> A: U (S i)" + by (erule U_cumulative, rule lt_S) + + +section \<open>\<Prod>-type\<close> + +axiomatization + Pi :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + lam :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + app :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(1_ `_)" [120, 121] 120) + +syntax + "_Pi" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Prod>_: _./ _)" 30) + "_lam" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30) +translations + "\<Prod>x: A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)" + "\<lambda>x: A. b" \<rightleftharpoons> "CONST lam A (\<lambda>x. b)" + +abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" + +axiomatization where + PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and + + PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and + + PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and + + beta: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> (\<lambda>x: A. b x) `a \<equiv> b a" and + + eta: "f: \<Prod>x: A. B x \<Longrightarrow> \<lambda>x: A. f `x \<equiv> f" and + + Pi_cong: "\<lbrakk> + A: U i; + \<And>x. x: A \<Longrightarrow> B x: U i; + \<And>x. x: A \<Longrightarrow> B' x: U i; + \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x + \<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and + + lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x" + + +section \<open>\<Sum>-type\<close> + +axiomatization + Sig :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + pair :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(2<_,/ _>)") and + SigInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close> + +syntax "_Sum" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Sum>_: _./ _)" 20) + +translations "\<Sum>x: A. B" \<rightleftharpoons> "CONST Sig A (\<lambda>x. B)" + +abbreviation Prod (infixl "\<times>" 50) + where "A \<times> B \<equiv> \<Sum>_: A. B" + +axiomatization where + SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and + + SigI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and + + SigE: "\<lbrakk> + p: \<Sum>x: A. B x; + A: U i; + \<And>x. x : A \<Longrightarrow> B x: U i; + \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i; + \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y> + \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f p: C p" and + + Sig_comp: "\<lbrakk> + a: A; + b: B a; + \<And>x. x: A \<Longrightarrow> B x: U i; + \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i; + \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y> + \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f <a, b> \<equiv> f a b" and + + Sig_cong: "\<lbrakk> + \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x; + A: U i; + \<And>x. x : A \<Longrightarrow> B x: U i; + \<And>x. x : A \<Longrightarrow> B' x: U i + \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x" + + + + + +section \<open>Proof commands\<close> + +named_theorems typechk + +ML_file \<open>../lib/lib.ML\<close> +ML_file \<open>../lib/goals.ML\<close> +ML_file \<open>../lib/focus.ML\<close> + + +section \<open>Congruence automation\<close> + +ML_file \<open>../lib/congruence.ML\<close> + + +section \<open>Methods\<close> + +ML_file \<open>../lib/elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close> + +named_theorems intros and comps +lemmas + [intros] = PiF PiI SigF SigI and + [elims] = PiE SigE and + [comps] = beta Sig_comp and + [cong] = Pi_cong lam_cong Sig_cong + +ML_file \<open>../lib/tactics.ML\<close> + +method_setup assumptions = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (TRYALL (assumptions_tac ctxt))))\<close> + +method_setup known = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (TRYALL (known_tac ctxt))))\<close> + +method_setup intro = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close> + +method_setup intros = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close> + +method_setup elim = + \<open>Scan.option Args.term >> (fn tm => fn ctxt => + SIMPLE_METHOD' (SIDE_CONDS (elims_tac tm ctxt) ctxt))\<close> + +method elims = elim+ + +method_setup typechk = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (ALLGOALS (TRY o typechk_tac ctxt))))\<close> + +method_setup rule = + \<open>Attrib.thms >> (fn ths => fn ctxt => + SIMPLE_METHOD (HEADGOAL (rule_tac ths ctxt)))\<close> + +method_setup dest = + \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms + >> (fn (opt_n, ths) => fn ctxt => + SIMPLE_METHOD (HEADGOAL (dest_tac opt_n ths ctxt)))\<close> + +subsection \<open>Rewriting\<close> + +\<comment> \<open>\<open>subst\<close> method\<close> +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 "../lib/eqsubst.ML" + +\<comment> \<open>\<open>rewrite\<close> method\<close> +consts rewrite_HOLE :: "'a::{}" ("\<hole>") + +lemma eta_expand: + fixes f :: "'a::{} \<Rightarrow> 'b::{}" + shows "f \<equiv> \<lambda>x. f x" . + +lemma rewr_imp: + assumes "PROP A \<equiv> PROP B" + shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)" + apply (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 \<Longrightarrow> (PROP B \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP C')) \<equiv> + ((PROP B \<Longrightarrow> PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP A \<Longrightarrow> 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 \<open>~~/src/HOL/Library/cconv.ML\<close> +ML_file \<open>../lib/rewrite.ML\<close> + +\<comment> \<open>\<open>reduce\<close> method computes terms via judgmental equalities\<close> +setup \<open> + map_theory_simpset (fn ctxt => + ctxt addSolver (mk_solver "" typechk_tac)) +\<close> + +method reduce uses add = (simp add: comps add | subst comps, reduce add: add)+ + + +section \<open>Implicit notations\<close> + +text \<open> + \<open>?\<close> is used to mark implicit arguments in definitions, while \<open>{}\<close> is expanded + immediately for elaboration in statements. +\<close> + +consts + iarg :: \<open>'a\<close> ("?") + hole :: \<open>'b\<close> ("{}") + +ML_file \<open>../lib/implicits.ML\<close> + +attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close> + +ML \<open> +val _ = Context.>> + (Syntax_Phases.term_check 1 "" (fn ctxt => map (Implicits.make_holes ctxt))) +\<close> + +text \<open>Automatically insert inhabitation judgments where needed:\<close> + +consts inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)") +translations "CONST inhabited A" \<rightharpoonup> "CONST has_type {} A" + + +section \<open>Lambda coercion\<close> + +\<comment> \<open>Coerce object lambdas to meta-lambdas\<close> +abbreviation (input) lambda :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> + where "lambda f \<equiv> \<lambda>x. f `x" + +ML_file \<open>~~/src/Tools/subtyping.ML\<close> +declare [[coercion_enabled, coercion lambda]] + +translations "f x" \<leftharpoondown> "f `x" + + +section \<open>Functions\<close> + +lemma eta_exp: + assumes "f: \<Prod>x: A. B x" + shows "f \<equiv> \<lambda>x: A. f x" + by (rule eta[symmetric]) + +lemma lift_universe_codomain: + assumes "A: U i" "f: A \<rightarrow> U j" + shows "f: A \<rightarrow> U (S j)" + apply (sub eta_exp) + apply known + apply (Pure.rule intros; rule lift_universe) + done + +subsection \<open>Function composition\<close> + +definition "funcomp A g f \<equiv> \<lambda>x: A. g `(f `x)" + +syntax + "_funcomp" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ \<circ>\<^bsub>_\<^esub>/ _)" [111, 0, 110] 110) +translations + "g \<circ>\<^bsub>A\<^esub> f" \<rightleftharpoons> "CONST funcomp A g f" + +lemma funcompI [typechk]: + assumes + "A: U i" + "B: U i" + "\<And>x. x: B \<Longrightarrow> C x: U i" + "f: A \<rightarrow> B" + "g: \<Prod>x: B. C x" + shows + "g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)" + unfolding funcomp_def by typechk + +lemma funcomp_assoc [comps]: + assumes + "f: A \<rightarrow> B" + "g: B \<rightarrow> C" + "h: \<Prod>x: C. D x" + "A: U i" + shows + "(h \<circ>\<^bsub>B\<^esub> g) \<circ>\<^bsub>A\<^esub> f \<equiv> h \<circ>\<^bsub>A\<^esub> g \<circ>\<^bsub>A\<^esub> f" + unfolding funcomp_def by reduce + +lemma funcomp_lambda_comp [comps]: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> b x: B" + "\<And>x. x: B \<Longrightarrow> c x: C x" + shows + "(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)" + unfolding funcomp_def by reduce + +lemma funcomp_apply_comp [comps]: + assumes + "f: A \<rightarrow> B" "g: \<Prod>x: B. C x" + "x: A" + "A: U i" "B: U i" + "\<And>x y. x: B \<Longrightarrow> C x: U i" + shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)" + unfolding funcomp_def by reduce + +text \<open>Notation:\<close> + +definition funcomp_i (infixr "\<circ>" 120) + where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>?\<^esub> f" + +translations "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f" + +subsection \<open>Identity function\<close> + +definition id where "id A \<equiv> \<lambda>x: A. x" + +lemma + idI [typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and + id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" + unfolding id_def by reduce + +lemma id_left [comps]: + assumes "f: A \<rightarrow> B" "A: U i" "B: U i" + shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f" + unfolding id_def + by (subst eta_exp[of f]) (reduce, rule eta) + +lemma id_right [comps]: + assumes "f: A \<rightarrow> B" "A: U i" "B: U i" + shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f" + unfolding id_def + by (subst eta_exp[of f]) (reduce, rule eta) + +lemma id_U [typechk]: + "id (U i): U i \<rightarrow> U i" + by typechk (fact U_in_U) + + +section \<open>Pairs\<close> + +definition "fst A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>_. A) (\<lambda>x y. x) p" +definition "snd A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>p. B (fst A B p)) (\<lambda>x y. y) p" + +lemma fst_type [typechk]: + assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A" + unfolding fst_def by typechk + +lemma fst_comp [comps]: + assumes + "a: A" + "b: B a" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst A B <a, b> \<equiv> a" + unfolding fst_def by reduce + +lemma snd_type [typechk]: + assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)" + unfolding snd_def by typechk reduce + +lemma snd_comp [comps]: + assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd A B <a, b> \<equiv> b" + unfolding snd_def by reduce + +subsection \<open>Notation\<close> + +definition fst_i ("fst") + where [implicit]: "fst \<equiv> Spartan.fst ? ?" + +definition snd_i ("snd") + where [implicit]: "snd \<equiv> Spartan.snd ? ?" + +translations + "fst" \<leftharpoondown> "CONST Spartan.fst A B" + "snd" \<leftharpoondown> "CONST Spartan.snd A B" + +subsection \<open>Projections\<close> + +Lemma fst [typechk]: + assumes + "p: \<Sum>x: A. B x" + "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst p: A" + by typechk + +Lemma snd [typechk]: + assumes + "p: \<Sum>x: A. B x" + "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd p: B (fst p)" + by typechk + +method fst for p::o = rule fst[of p] +method snd for p::o = rule snd[of p] + +subsection \<open>Properties of \<Sigma>\<close> + +Lemma (derive) Sig_dist_exp: + assumes + "p: \<Sum>x: A. B x \<times> C x" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + "\<And>x. x: A \<Longrightarrow> C x: U i" + shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)" + apply (elim p) + focus vars x y + apply intro + \<guillemotright> apply intro + apply assumption + apply (fst y) + done + \<guillemotright> apply intro + apply assumption + apply (snd y) + done + done + done + + +end |