From c2dfffffb7586662c67e44a2d255a1a97ab0398b Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Thu, 2 Apr 2020 17:57:48 +0200
Subject: Brand-spanking new version using Spartan infrastructure

---
 spartan/theories/Spartan.thy | 463 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 463 insertions(+)
 create mode 100644 spartan/theories/Spartan.thy

(limited to 'spartan/theories/Spartan.thy')

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
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