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Diffstat (limited to 'spartan/theories/Spartan.thy')
| -rw-r--r-- | spartan/theories/Spartan.thy | 481 |
1 files changed, 0 insertions, 481 deletions
diff --git a/spartan/theories/Spartan.thy b/spartan/theories/Spartan.thy deleted file mode 100644 index d235041..0000000 --- a/spartan/theories/Spartan.thy +++ /dev/null @@ -1,481 +0,0 @@ -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" "prems" "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>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30) - "_lam2" :: \<open>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> -translations - "\<Prod>x: A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)" - "\<lambda>x xs: A. b" \<rightharpoonup> "CONST lam A (\<lambda>x. _lam2 xs A b)" - "_lam2 x A b" \<rightharpoonup> "\<lambda>x: A. 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> - -(*Potential to be retired*) -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 "?f"] = PiE and - [elims "?p"] = 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 old_elim = - \<open>Scan.option Args.term >> (fn tm => fn ctxt => - SIMPLE_METHOD' (SIDE_CONDS (old_elims_tac tm ctxt) ctxt))\<close> - -method_setup elim = - \<open>Scan.repeat Args.term >> (fn tms => fn ctxt => - CONTEXT_METHOD (K (elim_context_tac tms ctxt 1)))\<close> - -method elims = elim+ - -(*This could possibly use additional simplification hints via a simp: modifier*) -method_setup typechk = - \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' ( - SIDE_CONDS (typechk_tac ctxt) ctxt)) - (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close> - -method_setup rule = - \<open>Attrib.thms >> (fn ths => fn ctxt => - SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) 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 (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close> - -subsection \<open>Reflexivity\<close> - -named_theorems refl -method refl = (rule refl) - -subsection \<open>Trivial proofs modulo typechecking\<close> - -method_setup this = - \<open>Scan.succeed (fn ctxt => METHOD ( - EVERY o map (HEADGOAL o - (fn ths => SIDE_CONDS (resolve_tac ctxt ths) ctxt) o - single) - ))\<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> 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)+ - - -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> - -abbreviation id where "id A \<equiv> \<lambda>x: A. x" - -lemma - id_type[typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and - id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close> - 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" - 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" - 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 |