Brand-spanking new version using Spartan infrastructure

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-(********
-Isabelle/HoTT: Dependent product (dependent function)
-Feb 2019
-
-********)
-
-theory Prod
-imports HoTT_Base HoTT_Methods
-
-begin
-
-
-section \<open>Basic type definitions\<close>
-
-axiomatization
-  Prod :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
-  lam  :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and
-  app  :: "[t, t] \<Rightarrow> t"  ("(2_`/_)" [120, 121] 120)
-  \<comment> \<open>Application should bind tighter than abstraction.\<close>
-
-syntax
-  "_Prod"  :: "[idt, t, t] \<Rightarrow> t"  ("(2\<Prod>'(_: _')./ _)" 30)
-  "_Prod'" :: "[idt, t, t] \<Rightarrow> t"  ("(2\<Prod>_: _./ _)" 30)
-  "_lam"   :: "[idt, t, t] \<Rightarrow> t"  ("(2\<lambda>'(_: _')./ _)" 30)
-  "_lam'"  :: "[idt, t, t] \<Rightarrow> t"  ("(2\<lambda>_: _./ _)" 30)
-translations
-  "\<Prod>x: A. B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
-  "\<Prod>(x: A). B" \<rightleftharpoons> "(CONST Prod) A (\<lambda>x. B)"
-  "\<lambda>(x: A). b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
-  "\<lambda>x: A. b" \<rightleftharpoons> "(CONST lam) A (\<lambda>x. b)"
-
-text \<open>
-The syntax translations above bind the variable @{term x} in the expressions @{term B} and @{term b}.
-\<close>
-
-text \<open>Non-dependent functions are a special case:\<close>
-
-abbreviation Fun :: "[t, t] \<Rightarrow> t"  (infixr "\<rightarrow>" 40)
-where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"
-
-axiomatization where
-\<comment> \<open>Type rules\<close>
-
-  Prod_form: "\<lbrakk>A: U i; B: A \<leadsto> U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and
-
-  Prod_intro: "\<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
-
-  Prod_elim: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f`a: B a" and
-
-  Prod_cmp: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; a: A\<rbrakk> \<Longrightarrow> (\<lambda>x: A. b x)`a \<equiv> b a" and
-
-  Prod_uniq: "f: \<Prod>x: A. B x \<Longrightarrow> \<lambda>x: A. f`x \<equiv> f" and
-
-\<comment> \<open>Congruence rules\<close>
-
-  Prod_form_eq: "\<lbrakk>A: U i; B: A \<leadsto> U i; C: A \<leadsto> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk>
-    \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. C x" and
-
-  Prod_intro_eq: "\<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"
-
-text \<open>
-The Pure rules for \<open>\<equiv>\<close> only let us judge strict syntactic equality of object lambda expressions.
-The actual definitional equality rule in the type theory is @{thm Prod_intro_eq}.
-Note that this is a separate rule from function extensionality.
-\<close>
-
-lemmas Prod_form [form]
-lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim
-lemmas Prod_comp [comp] = Prod_cmp Prod_uniq
-lemmas Prod_cong [cong] = Prod_form_eq Prod_intro_eq
-
-
-section \<open>Function composition\<close>
-
-definition compose :: "[t, t, t] \<Rightarrow> t"
-where "compose A g f \<equiv> \<lambda>x: A. g`(f`x)"
-
-syntax "_compose" :: "[t, t, t] \<Rightarrow> t"  ("(2_ o[_]/ _)" [111, 0, 110] 110)
-translations "g o[A] f" \<rightleftharpoons> "(CONST compose) A g f"
-
-text \<open>The composition @{term "g o[A] f"} is annotated with the domain @{term A} of @{term f}.\<close>
-
-syntax "_compose'" :: "[t, t] \<Rightarrow> t"  (infixr "o" 110)
-
-text \<open>Pretty-printing switch for composition; hides domain type information.\<close>
-
-ML \<open>val pretty_compose = Attrib.setup_config_bool @{binding "pretty_compose"} (K true)\<close>
-
-print_translation \<open>
-let fun compose_tr' ctxt [A, g, f] =
-  if Config.get ctxt pretty_compose
-  then Syntax.const @{syntax_const "_compose'"} $ g $ f
-  else @{const compose} $ A $ g $ f
-in
-  [(@{const_syntax compose}, compose_tr')]
-end
-\<close>
-
-lemma compose_type:
-  assumes
-    "A: U i" and "B: U i" and "C: B \<leadsto> U i" and
-    "f: A \<rightarrow> B" and "g: \<Prod>x: B. C x"
-  shows "g o[A] f: \<Prod>x: A. C (f`x)"
-unfolding compose_def by (derive lems: assms)
-
-lemma compose_comp:
-  assumes "A: U i" and "\<And>x. x: A \<Longrightarrow> b x: B" and "\<And>x. x: B \<Longrightarrow> c x: C x"
-  shows "(\<lambda>x: B. c x) o[A] (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)"
-unfolding compose_def by (derive lems: assms cong)
-
-declare
-  compose_type [intro]
-  compose_comp [comp]
-
-lemma compose_assoc:
-  assumes "A: U i" and "f: A \<rightarrow> B" and "g: B \<rightarrow> C" and "h: \<Prod>x: C. D x"
-  shows "(h o[B] g) o[A] f \<equiv> h o[A] g o[A] f"
-unfolding compose_def by (derive lems: assms cong)
-
-abbreviation id :: "t \<Rightarrow> t"  ("(id _)" [115] 114)  where "id A \<equiv> \<lambda>x: A. x"
-
-lemma id_type: "\<And>A. A: U i \<Longrightarrow> id A: A \<rightarrow> A" by derive
-
-lemma id_compl:
-  assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "id B o[A] f \<equiv> f"
-unfolding compose_def proof -
-  {
-  fix x assume [intro]: "x: A"
-  have "(id B)`(f`x) \<equiv> f`x" by derive
-  }
-  hence "\<lambda>x: A. (id B)`(f`x) \<equiv> \<lambda>x: A. f`x" by (derive lems: cong) derive
-  also have "\<lambda>x: A. f`x \<equiv> f" by derive
-  finally show "\<lambda>(x: A). (id B)`(f`x) \<equiv> f" by simp
-qed
-
-lemma id_compr:
-  assumes [intro]: "A: U i" "B: U i" "f: A \<rightarrow> B"
-  shows "f o[A] id A \<equiv> f"
-unfolding compose_def proof -
-  {
-  fix x assume [intro]: "x: A"
-  have "f`((id A)`x) \<equiv> f`x" by derive
-  }
-  hence "\<lambda>x: A. f`((id A)`x) \<equiv> \<lambda>x: A. f`x" by (derive lems: cong) derive
-  also have "\<lambda>x: A. f`x \<equiv> f" by derive
-  finally show "\<lambda>x: A. f`((id A)`x) \<equiv> f" by simp
-qed
-
-declare id_type [intro]
-lemmas id_comp [comp] = id_compl id_compr
-
-section \<open>Universal quantification\<close>
-
-text \<open>
-It will often be useful to convert a proof goal asserting the inhabitation of a dependent product to one that instead uses Pure universal quantification.
-
-Method @{theory_text quantify_all} converts the goal
-@{text "t: \<Prod>x1: A1. ... \<Prod>xn: An. B x1 ... xn"},
-where @{term B} is not a product, to
-@{text "\<And>x1 ... xn . \<lbrakk>x1: A1; ...; xn: An\<rbrakk> \<Longrightarrow> ?b x1 ... xn: B x1 ... xn"}.
-
-Method @{theory_text "quantify k"} does the same, but only for the first k unknowns.
-\<close>
-
-method quantify_all = (rule Prod_intro)+
-method_setup quantify = \<open>repeat (fn ctxt => Method.rule_tac ctxt [@{thm Prod_intro}] [] 1)\<close>
-
-end