diff options
Diffstat (limited to 'spartan/core/Spartan.thy')
| -rw-r--r-- | spartan/core/Spartan.thy | 571 |
1 files changed, 0 insertions, 571 deletions
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy deleted file mode 100644 index 5046b6a..0000000 --- a/spartan/core/Spartan.thy +++ /dev/null @@ -1,571 +0,0 @@ -text \<open>Spartan type theory\<close> - -theory Spartan -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>" "\<circ>" "\<diamondop>" "~" :: prf_script_goal % "proof" and - "calc" "print_coercions" :: thy_decl and - "rhs" "def" "vars" :: quasi_command - -begin - -section \<open>Notation\<close> - -declare [[eta_contract=false]] - -text \<open> -Rebind notation for meta-lambdas since we want to use \<open>\<lambda>\<close> for the object -lambdas. Metafunctions now use the binder \<open>fn\<close>. -\<close> -setup \<open> -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 \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3%_./ _)", [0, 3], 3))] - #> Sign.del_syntax Syntax.mode_default - [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3\<lambda>_./ _)", [0, 3], 3))] - #> Sign.add_syntax Syntax.mode_default - [("_lambda", typ "pttrns \<Rightarrow> 'a \<Rightarrow> logic", mixfix ("(3fn _./ _)", [0, 3], 3))] -end -\<close> - -syntax "_app" :: \<open>logic \<Rightarrow> logic \<Rightarrow> logic\<close> (infixr "$" 3) -translations "a $ b" \<rightharpoonup> "a (b)" - -abbreviation (input) K where "K x \<equiv> fn _. x" - - -section \<open>Metalogic\<close> - -text \<open> -HOAS embedding of dependent type theory: metatype of expressions, and typing -judgment. -\<close> - -typedecl o - -consts has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999) - - -section \<open>Axioms\<close> - -subsection \<open>Universes\<close> - -text \<open>\<omega>-many cumulative Russell universes.\<close> - -typedecl lvl - -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 - Ui_in_Uj: "i < j \<Longrightarrow> U i: U j" and - U_cumul: "A: U i \<Longrightarrow> i < j \<Longrightarrow> 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 \<Longrightarrow> A: U (S i)" - by (erule U_cumul, rule lt_S) - -subsection \<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>idts \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Prod>_: _./ _)" 30) - "_Pi2" :: \<open>idts \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> - "_lam" :: \<open>idts \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30) - "_lam2" :: \<open>idts \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> -translations - "\<Prod>x xs: A. B" \<rightharpoonup> "CONST Pi A (fn x. _Pi2 xs A B)" - "_Pi2 x A B" \<rightharpoonup> "\<Prod>x: A. B" - "\<Prod>x: A. B" \<rightleftharpoons> "CONST Pi A (fn x. B)" - "\<lambda>x xs: A. b" \<rightharpoonup> "CONST lam A (fn x. _lam2 xs A b)" - "_lam2 x A b" \<rightharpoonup> "\<lambda>x: A. b" - "\<lambda>x: A. b" \<rightleftharpoons> "CONST lam A (fn x. b)" - -abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" - -axiomatization where - PiF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and - - PiI: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> b x: B x\<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> - \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x; - A: U i; - \<And>x. x: A \<Longrightarrow> B x: U j; - \<And>x. x: A \<Longrightarrow> B' x: U j - \<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" - -subsection \<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 (fn x. B)" - -abbreviation Prod (infixl "\<times>" 60) - where "A \<times> B \<equiv> \<Sum>_: A. B" - -abbreviation "and" (infixl "\<and>" 60) - where "A \<and> B \<equiv> A \<times> B" - -axiomatization where - SigF: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: 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 j; - \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U k; - \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y> - \<rbrakk> \<Longrightarrow> SigInd A (fn x. B x) (fn 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 (fn x. B x) (fn 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 j; - \<And>x. x : A \<Longrightarrow> B' x: U j - \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x" - - -section \<open>Type checking & inference\<close> - -ML_file \<open>lib.ML\<close> -ML_file \<open>context_facts.ML\<close> -ML_file \<open>context_tactical.ML\<close> - -\<comment> \<open>Rule attributes for the typechecker\<close> -named_theorems form and intr and comp - -\<comment> \<open>Elimination/induction automation and the `elim` attribute\<close> -ML_file \<open>elimination.ML\<close> - -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 - -\<comment> \<open>Subsumption rule\<close> -lemma sub: - assumes "a: A" "A \<equiv> A'" - shows "a: A'" - using assms by simp - -\<comment> \<open>Basic rewriting of computational equality\<close> -ML_file \<open>~~/src/Tools/misc_legacy.ML\<close> -ML_file \<open>~~/src/Tools/IsaPlanner/isand.ML\<close> -ML_file \<open>~~/src/Tools/IsaPlanner/rw_inst.ML\<close> -ML_file \<open>~~/src/Tools/IsaPlanner/zipper.ML\<close> -ML_file \<open>~~/src/Tools/eqsubst.ML\<close> - -\<comment> \<open>Term normalization, type checking & inference\<close> -ML_file \<open>types.ML\<close> - -method_setup typechk = - \<open>Scan.succeed (K (CONTEXT_METHOD ( - CHEADGOAL o Types.check_infer)))\<close> - -method_setup known = - \<open>Scan.succeed (K (CONTEXT_METHOD ( - CHEADGOAL o Types.known_ctac)))\<close> - -setup \<open> -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 -\<close> - - -section \<open>Statements and goals\<close> - -ML_file \<open>focus.ML\<close> -ML_file \<open>elaboration.ML\<close> -ML_file \<open>elaborated_statement.ML\<close> -ML_file \<open>goals.ML\<close> - - -section \<open>Proof methods\<close> - -named_theorems intro \<comment> \<open>Logical introduction rules\<close> - -lemmas [intro] = PiI[rotated] SigI - -\<comment> \<open>Case reasoning rules\<close> -ML_file \<open>cases.ML\<close> - -ML_file \<open>tactics.ML\<close> - -method_setup rule = - \<open>Attrib.thms >> (fn ths => K (CONTEXT_METHOD ( - CHEADGOAL o SIDE_CONDS 0 (rule_ctac ths))))\<close> - -method_setup dest = - \<open>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))))\<close> - -method_setup intro = - \<open>Scan.succeed (K (CONTEXT_METHOD ( - CHEADGOAL o SIDE_CONDS 0 intro_ctac)))\<close> - -method_setup intros = - \<open>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)))))))\<close> - -method_setup elim = - \<open>Scan.repeat Args.term >> (fn tms => K (CONTEXT_METHOD ( - CHEADGOAL o SIDE_CONDS 0 (elim_ctac tms))))\<close> - -method_setup cases = - \<open>Args.term >> (fn tm => K (CONTEXT_METHOD ( - CHEADGOAL o SIDE_CONDS 0 (cases_ctac tm))))\<close> - -method elims = elim+ -method facts = fact+ - - -subsection \<open>Reflexivity\<close> - -named_theorems refl -method refl = (rule refl) - - -subsection \<open>Trivial proofs (modulo automatic discharge of side conditions)\<close> - -method_setup this = - \<open>Scan.succeed (K (CONTEXT_METHOD (fn facts => - CHEADGOAL (SIDE_CONDS 0 - (CONTEXT_TACTIC' (fn ctxt => resolve_tac ctxt facts)) - facts))))\<close> - - -subsection \<open>Rewriting\<close> - -consts compute_hole :: "'a::{}" ("\<hole>") - -lemma eta_expand: - fixes f :: "'a::{} \<Rightarrow> 'b::{}" - shows "f \<equiv> fn 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 (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 \<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>comp.ML\<close> - -\<comment> \<open>\<open>compute\<close> simplifies terms via computational equalities\<close> -method compute uses add = - changed \<open>repeat_new \<open>(simp add: comp add | subst comp); typechk?\<close>\<close> - - -subsection \<open>Calculational reasoning\<close> - -consts "rhs" :: \<open>'a\<close> ("..") - -ML_file \<open>calc.ML\<close> - - -section \<open>Implicits\<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>implicits.ML\<close> - -attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close> - -ML \<open>val _ = Context.>> (Syntax_Phases.term_check 1 "" Implicits.make_holes)\<close> - -text \<open>Automatically insert inhabitation judgments where needed:\<close> - -syntax inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)") -translations "inhabited A" \<rightharpoonup> "CONST has_type ? A" - - -subsection \<open>Implicit lambdas\<close> - -definition lam_i where [implicit]: "lam_i f \<equiv> lam {} f" - -syntax - "_lam_i" :: \<open>idts \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_./ _)" 30) - "_lam_i2" :: \<open>idts \<Rightarrow> o \<Rightarrow> o\<close> -translations - "\<lambda>x xs. b" \<rightharpoonup> "CONST lam_i (fn x. _lam_i2 xs b)" - "_lam_i2 x b" \<rightharpoonup> "\<lambda>x. b" - "\<lambda>x. b" \<rightleftharpoons> "CONST lam_i (fn x. b)" - -translations "\<lambda>x. b" \<leftharpoondown> "\<lambda>x: A. b" - - -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> fn 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 refine_codomain: - assumes - "A: U i" - "f: \<Prod>x: A. B x" - "\<And>x. x: A \<Longrightarrow> f `x: C x" - shows "f: \<Prod>x: A. C x" - by (comp eta_exp) - -Lemma lift_universe_codomain: - assumes "A: U i" "f: A \<rightarrow> U j" - shows "f: A \<rightarrow> U (S j)" - using U_lift - by (rule refine_codomain) - -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 [type]: - 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 [comp]: - assumes - "A: U i" - "f: A \<rightarrow> B" - "g: B \<rightarrow> C" - "h: \<Prod>x: C. D x" - 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 compute - -Lemma funcomp_lambda_comp [comp]: - 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 compute - -Lemma funcomp_apply_comp [comp]: - assumes - "A: U i" "B: U i" "\<And>x y. x: B \<Longrightarrow> C x: U i" - "f: A \<rightarrow> B" "g: \<Prod>x: B. C x" - "x: A" - shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)" - unfolding funcomp_def by compute - -subsection \<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 [type]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and - id_comp [comp]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close> - by compute+ - -Lemma id_left [comp]: - assumes "A: U i" "B: U i" "f: A \<rightarrow> B" - shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f" - by (comp eta_exp[of f]) (compute, rule eta) - -Lemma id_right [comp]: - assumes "A: U i" "B: U i" "f: A \<rightarrow> B" - shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f" - by (comp eta_exp[of f]) (compute, rule eta) - -lemma id_U [type]: - "id (U i): U i \<rightarrow> U i" - using Ui_in_USi by typechk - - -section \<open>Pairs\<close> - -definition "fst A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (fn _. A) (fn x y. x) p" -definition "snd A B \<equiv> \<lambda>p: \<Sum>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" "\<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 [comp]: - assumes - "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" "a: A" "b: B a" - shows "fst A B <a, b> \<equiv> a" - unfolding fst_def by compute - -Lemma snd_type [type]: - 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 - -Lemma snd_comp [comp]: - assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" "a: A" "b: B a" - shows "snd A B <a, b> \<equiv> b" - unfolding snd_def by compute - -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 [type]: - assumes - "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" - "p: \<Sum>x: A. B x" - shows "fst p: A" - by typechk - -Lemma snd [type]: - assumes - "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" - "p: \<Sum>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 \<open>Double projections:\<close> - -definition [implicit]: "p\<^sub>1\<^sub>1 p \<equiv> Spartan.fst {} {} (Spartan.fst {} {} p)" -definition [implicit]: "p\<^sub>1\<^sub>2 p \<equiv> Spartan.snd {} {} (Spartan.fst {} {} p)" -definition [implicit]: "p\<^sub>2\<^sub>1 p \<equiv> Spartan.fst {} {} (Spartan.snd {} {} p)" -definition [implicit]: "p\<^sub>2\<^sub>2 p \<equiv> Spartan.snd {} {} (Spartan.snd {} {} p)" - -translations - "CONST p\<^sub>1\<^sub>1 p" \<leftharpoondown> "fst (fst p)" - "CONST p\<^sub>1\<^sub>2 p" \<leftharpoondown> "snd (fst p)" - "CONST p\<^sub>2\<^sub>1 p" \<leftharpoondown> "fst (snd p)" - "CONST p\<^sub>2\<^sub>2 p" \<leftharpoondown> "snd (snd p)" - -Lemma (def) distribute_Sig: - assumes - "A: U i" - "\<And>x. x: A \<Longrightarrow> B x: U i" - "\<And>x. x: A \<Longrightarrow> C x: U i" - "p: \<Sum>x: A. B x \<times> C x" - shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)" - proof intro - have "fst p: A" and "snd p: B (fst p) \<times> C (fst p)" - by typechk+ - thus "<fst p, fst (snd p)>: \<Sum>x: A. B x" - and "<fst p, snd (snd p)>: \<Sum>x: A. C x" - by typechk+ - qed - - -end |