diff options
Diffstat (limited to 'spartan/core/Spartan.thy')
| -rw-r--r-- | spartan/core/Spartan.thy | 85 |
1 files changed, 38 insertions, 47 deletions
diff --git a/spartan/core/Spartan.thy b/spartan/core/Spartan.thy index 6b2ed58..ffe3778 100644 --- a/spartan/core/Spartan.thy +++ b/spartan/core/Spartan.thy @@ -9,23 +9,18 @@ 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 - "congruence" "print_coercions" :: thy_decl and + "calc" "print_coercions" :: thy_decl and "rhs" "def" "vars" :: quasi_command begin - -section \<open>Prelude\<close> - -paragraph \<open>Set up the meta-logic just so.\<close> - -paragraph \<open>Notation.\<close> +section \<open>Notation\<close> declare [[eta_contract=false]] text \<open> - Rebind notation for meta-lambdas since we want to use `\<lambda>` for the object - lambdas. Meta-functions now use the binder `fn`. +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 @@ -41,34 +36,31 @@ in end \<close> -syntax "_dollar" :: \<open>logic \<Rightarrow> logic \<Rightarrow> logic\<close> (infixr "$" 3) +syntax "_app" :: \<open>logic \<Rightarrow> logic \<Rightarrow> logic\<close> (infixr "$" 3) translations "a $ b" \<rightharpoonup> "a (b)" abbreviation (input) K where "K x \<equiv> fn _. x" -paragraph \<open> - Deeply embed dependent type theory: meta-type of expressions, and typing - judgment. + +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) -text \<open>Type annotations for type-checking and inference.\<close> - -definition anno :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("'(_ : _')") - where "(a : A) \<equiv> a" if "a: A" - -lemma anno: "a: A \<Longrightarrow> (a : A): A" - by (unfold anno_def) - section \<open>Axioms\<close> subsection \<open>Universes\<close> -typedecl lvl \<comment> \<open>Universe levels\<close> +text \<open>\<omega>-many cumulative Russell universes.\<close> + +typedecl lvl axiomatization O :: \<open>lvl\<close> and @@ -81,16 +73,15 @@ axiomatization axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where Ui_in_Uj: "i < j \<Longrightarrow> U i: U j" and - in_Uj_if_in_Ui: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j" + 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 in_USi_if_in_Ui: +lemma U_lift: "A: U i \<Longrightarrow> A: U (S i)" - by (erule in_Uj_if_in_Ui, rule lt_S) - + by (erule U_cumul, rule lt_S) subsection \<open>\<Prod>-type\<close> @@ -134,7 +125,6 @@ axiomatization where 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 @@ -207,7 +197,7 @@ lemma sub: shows "a: A'" using assms by simp -\<comment> \<open>Basic substitution of definitional equalities\<close> +\<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> @@ -305,7 +295,7 @@ method_setup this = subsection \<open>Rewriting\<close> -consts rewrite_HOLE :: "'a::{}" ("\<hole>") +consts rewrite_hole :: "'a::{}" ("\<hole>") lemma eta_expand: fixes f :: "'a::{} \<Rightarrow> 'b::{}" @@ -331,28 +321,28 @@ lemma imp_cong_eq: ML_file \<open>~~/src/HOL/Library/cconv.ML\<close> ML_file \<open>rewrite.ML\<close> -\<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close> +\<comment> \<open>\<open>reduce\<close> simplifies terms via computational equalities\<close> method reduce uses add = changed \<open>repeat_new \<open>(simp add: comp add | subst comp); typechk?\<close>\<close> -subsection \<open>Congruence relations\<close> +subsection \<open>Calculational reasoning\<close> consts "rhs" :: \<open>'a\<close> ("..") -ML_file \<open>congruence.ML\<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 + \<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> ("{}") + iarg :: \<open>'a\<close> ("{}") + hole :: \<open>'b\<close> ("?") ML_file \<open>implicits.ML\<close> @@ -363,11 +353,12 @@ ML \<open>val _ = Context.>> (Syntax_Phases.term_check 1 "" Implicits.make_holes text \<open>Automatically insert inhabitation judgments where needed:\<close> syntax inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)") -translations "inhabited A" \<rightharpoonup> "CONST has_type {} A" +translations "inhabited A" \<rightharpoonup> "CONST has_type ? A" + -text \<open>Implicit lambdas\<close> +subsection \<open>Implicit lambdas\<close> -definition lam_i where [implicit]: "lam_i f \<equiv> lam ? f" +definition lam_i where [implicit]: "lam_i f \<equiv> lam {} f" syntax "_lam_i" :: \<open>idts \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_./ _)" 30) @@ -410,7 +401,7 @@ Lemma refine_codomain: Lemma lift_universe_codomain: assumes "A: U i" "f: A \<rightarrow> U j" shows "f: A \<rightarrow> U (S j)" - using in_USi_if_in_Ui[of "f {}"] + using U_lift by (rule refine_codomain) subsection \<open>Function composition\<close> @@ -460,10 +451,10 @@ Lemma funcomp_apply_comp [comp]: shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)" unfolding funcomp_def by reduce -text \<open>Notation:\<close> +subsection \<open>Notation\<close> definition funcomp_i (infixr "\<circ>" 120) - where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>?\<^esub> f" + where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>{}\<^esub> f" translations "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f" @@ -520,10 +511,10 @@ Lemma snd_comp [comp]: subsection \<open>Notation\<close> definition fst_i ("fst") - where [implicit]: "fst \<equiv> Spartan.fst ? ?" + where [implicit]: "fst \<equiv> Spartan.fst {} {}" definition snd_i ("snd") - where [implicit]: "snd \<equiv> Spartan.snd ? ?" + where [implicit]: "snd \<equiv> Spartan.snd {} {}" translations "fst" \<leftharpoondown> "CONST Spartan.fst A B" @@ -550,10 +541,10 @@ 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)" +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)" |