Renaming

1 files changed, 0 insertions, 43 deletions

diff --git a/Proj.thy b/Proj.thy
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index e76e8d3..0000000
--- a/Proj.thy
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-(*
-Title:  Proj.thy
-Author: Joshua Chen
-Date:   2018
-
-Projection functions \<open>fst\<close> and \<open>snd\<close> for the dependent sum type.
-*)
-
-theory Proj
-imports HoTT_Methods Prod Sum
-
-begin
-
-
-definition fst :: "t \<Rightarrow> t" where "fst p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. x) p"
-definition snd :: "t \<Rightarrow> t" where "snd p \<equiv> ind\<^sub>\<Sum> (\<lambda>x y. y) p"
-
-lemma fst_type:
-  assumes "A: U i" and "B: A \<longrightarrow> U i" and "p: \<Sum>x:A. B x" shows "fst p: A"
-unfolding fst_def by (derive lems: assms)
-
-lemma fst_comp:
-  assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "fst <a,b> \<equiv> a"
-unfolding fst_def by compute (derive lems: assms)
-
-lemma snd_type:
-  assumes "A: U i" and "B: A \<longrightarrow> U i" and "p: \<Sum>x:A. B x" shows "snd p: B (fst p)"
-unfolding snd_def proof (derive lems: assms)
-  show "\<And>p. p: \<Sum>x:A. B x \<Longrightarrow> fst p: A" using assms(1-2) by (rule fst_type)
-
-  fix x y assume asm: "x: A" "y: B x"
-  show "y: B (fst <x,y>)" by (derive lems: asm assms fst_comp)
-qed
-
-lemma snd_comp:
-  assumes "A: U i" and "B: A \<longrightarrow> U i" and "a: A" and "b: B a" shows "snd <a,b> \<equiv> b"
-unfolding snd_def by (derive lems: assms)
-
-lemmas Proj_types [intro] = fst_type snd_type
-lemmas Proj_comps [comp] = fst_comp snd_comp
-
-
-end