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Diffstat (limited to '')
| -rw-r--r-- | Prod.thy | 169 |
1 files changed, 0 insertions, 169 deletions
diff --git a/Prod.thy b/Prod.thy deleted file mode 100644 index a35138c..0000000 --- a/Prod.thy +++ /dev/null @@ -1,169 +0,0 @@ -(******** -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 |