From 25225e0c637a43319fef6889dabc222df05bfd3c Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Wed, 8 Aug 2018 10:35:43 +0200
Subject: Rewriting projection functions

---
 Proj.thy | 212 ++-------------------------------------------------------------
 1 file changed, 4 insertions(+), 208 deletions(-)

(limited to 'Proj.thy')

diff --git a/Proj.thy b/Proj.thy
index d070ed0..717df65 100644
--- a/Proj.thy
+++ b/Proj.thy
@@ -13,217 +13,13 @@ theory Proj
 begin
 
 
-abbreviation fst :: Term where "fst \<equiv> \<^bold>\<lambda>p:\<Sum>x:?A. ?B(x). ind\<^sub>\<Sum>(\<lambda>x y. x)(p)"
-abbreviation snd :: "Term \<Rightarrow> Term" where ""snd \<equiv> \<^bold>\<lambda>p: \<Sum>x:A. B(x). ind\<^sub>\<Sum>(\<lambda>x y. y) p"
-
-
-section \<open>Overloaded syntax for dependent and nondependent pairs\<close>
-
-overloading
-  fst_dep \<equiv> fst
-  fst_nondep \<equiv> fst
-begin
-  definition fst_dep :: "[Term, Typefam] \<Rightarrow> Term" where
-    "fst_dep A B \<equiv> \<^bold>\<lambda>p:\<Sum>x:A. B(x). ind\<^sub>\<Sum>(\<lambda>x y. x)(p)"
-  
-  definition fst_nondep :: "[Term, Term] \<Rightarrow> Term" where
-    "fst_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. ind\<^sub>\<Sum>[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) p"
-end
-
-overloading
-  snd_dep \<equiv> snd
-  snd_nondep \<equiv> snd
-begin
-  definition snd_dep :: "[Term, Typefam] \<Rightarrow> Term" where
-    "snd_dep A B \<equiv> \<^bold>\<lambda>p: (\<Sum>x:A. B x). ind\<^sub>\<Sum>[A,B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) p"
-  
-  definition snd_nondep :: "[Term, Term] \<Rightarrow> Term" where
-    "snd_nondep A B \<equiv> \<^bold>\<lambda>p: A \<times> B. ind\<^sub>\<Sum>[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) p"
-end
-
-
-section \<open>Properties\<close>
+abbreviation fst :: "Term \<Rightarrow> Term" where "fst \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. x)(p)"
+abbreviation snd :: "Term \<Rightarrow> Term" where "snd \<equiv> \<lambda>p. ind\<^sub>\<Sum>(\<lambda>x y. y)(p)"
 
 text "Typing judgments and computation rules for the dependent and non-dependent projection functions."
 
-lemma fst_dep_type: assumes "\<Sum>x:A. B x : U(i)" and "p : \<Sum>x:A. B x" shows "fst[A,B]`p : A"
-unfolding fst_dep_def
-proof
-  show "lambda (Sum A B) (ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x)) : (\<Sum>x:A. B x) \<rightarrow> A"
-  proof
-    show "A : U(i)" using assms(1) ..
-    show "Sum A B : U(i)" by (rule assms)
-    show "\<And>p. p : Sum A B \<Longrightarrow> ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) p : A"
-    proof
-      show "A : U(i)" using assms(1) ..
-    qed
-  qed
-qed (rule assms)
-
-
-lemma fst_dep_comp:
-  assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "a : A" and "b : B a"
-  shows "fst[A,B]`(a,b) \<equiv> a"
-unfolding fst_dep_def
-proof (subst comp)
-  show "Sum A B : U(i)" using assms(1-2) ..
-  show "\<And>x. x : Sum A B \<Longrightarrow> ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) x : A" by (standard, rule assms(1), assumption+)
-  show "(a,b) : Sum A B" using assms ..
-  show "ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) (a, b) \<equiv> a" by (standard, (rule assms|assumption)+)
-qed (rule assms)
-
-\<comment> \<open> (* Old proof *)
-proof -
-  have "fst[A,B]`(a,b) \<equiv> indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)"
-    by (derive lems: assms unfolds: fst_dep_def)
-  
-  also have "indSum[A,B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a"
-    by (derive lems: assms)
-  
-  finally show "fst[A,B]`(a,b) \<equiv> a" .
-qed
-\<close>
-
-\<comment> \<open> (* Old lemma *)
-text "In proving results about the second dependent projection function we often use the following lemma."
-
-lemma lem:
-  assumes "B: A \<rightarrow> U" and "x : A" and "y : B x"
-  shows "y : B (fst[A,B]`(x,y))"
-
-proof -
-  have "fst[A,B]`(x,y) \<equiv> x" using assms by (rule fst_dep_comp)
-  then show "y : B (fst[A,B]`(x,y))" using assms by simp
-qed
-\<close>
-
-
-lemma snd_dep_type:
-  assumes "\<Sum>x:A. B x : U(i)" and "p : \<Sum>x:A. B x"
-  shows "snd[A,B]`p : B (fst[A,B]`p)"
-unfolding fst_dep_def snd_dep_def
-proof (subst (3) comp)
-  show "Sum A B : U(i)" by (rule assms)
-  show "\<And>x. x : Sum A B \<Longrightarrow> ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) x : A"
-  proof
-    show "A : U(i)" using assms(1) ..
-  qed
-  show "A : U(i)" using assms(1) ..
-  show "p : Sum A B" by (rule assms(2))
-  show "lambda
-          (Sum A B)
-          (ind\<^sub>\<Sum>[A, B] (\<lambda>q. B (lambda (Sum A B) (ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x))`q)) (\<lambda>x y. y))
-        ` p : B (ind\<^sub>\<Sum>[A, B] (\<lambda>_. A) (\<lambda>x y. x) p)"
-  proof (subst (2) comp)
-    show "Sum A B : U(i)" by (rule assms)
-    show "
-  
-
-\<comment> \<open> (* Old proof *)
-proof (derive lems: assms unfolds: snd_dep_def)
-  show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
-  
-  fix x y assume asm: "x : A" "y : B x"
-  have "B: A \<rightarrow> U" by (wellformed jdgmt: assms)
-  then show "y : B (fst[A, B]`(x,y))" using asm by (rule lem)
-qed (assumption | rule assms)+
-\<close>
-
-
-lemma snd_dep_comp:
-  assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "a : A" and "b : B a"
-  shows "snd[A,B]`(a,b) \<equiv> b"
-  unfolding snd_dep_def fst_dep_def
-  by (simplify lems: assms)
-
-\<comment> \<open> (* Old proof *)
-proof -
-  have "snd[A,B]`(a,b) \<equiv> indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b)"
-  proof (derive lems: assms unfolds: snd_dep_def)
-    show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
-
-    fix x y assume asm: "x : A" "y : B x"
-    with assms(1) show "y : B (fst[A, B]`(x,y))" by (rule lem)
-  qed (assumption | derive lems: assms)+
-
-  also have "indSum[A, B] (\<lambda>q. B (fst[A,B]`q)) (\<lambda>x y. y) (a,b) \<equiv> b"
-  proof (simple lems: assms)
-    show "fst[A, B] : (\<Sum>x:A. B x) \<rightarrow> A" by (derive lems: assms unfolds: fst_dep_def)
-
-    fix x y assume "x : A" and "y : B x"
-    with assms(1) show "y : B (fst[A,B]`(x,y))" by (rule lem)
-  qed (assumption | rule assms)+
-
-  finally show "snd[A,B]`(a,b) \<equiv> b" .
-qed
-\<close>
-
-
-text "Nondependent projections:"
-
-lemma fst_nondep_type: assumes "A \<times> B : U(i)" and "p : A \<times> B" shows "fst[A,B]`p : A"
-  unfolding fst_nondep_def 
-  by (derive lems: assms)
-
-
-lemma fst_nondep_comp:
-  assumes "A : U(i)" and "B : U(i)" and "a : A" and "b : B"
-  shows "fst[A,B]`(a,b) \<equiv> a"
-  unfolding fst_nondep_def
-  by (simplify lems: assms)
-
-\<comment> \<open> (* Old proof *)
-proof -
-  have "fst[A,B]`(a,b) \<equiv> indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) (a,b)"
-    by (derive lems: assms unfolds: fst_nondep_def)
-
-  also have "indSum[A, \<lambda>_. B] (\<lambda>_. A) (\<lambda>x y. x) (a,b) \<equiv> a"
-    by (derive lems: assms)
-
-  finally show "fst[A,B]`(a,b) \<equiv> a" .
-qed
-\<close>
-
-
-lemma snd_nondep_type: assumes "A \<times> B : U(i)" and "p : A \<times> B" shows "snd[A,B]`p : B"
-  unfolding snd_nondep_def
-  by (derive lems: assms)
-
-\<comment> \<open> (* Old proof *)
-proof
-  show "snd[A,B] : A \<times> B \<rightarrow> B"
-  proof (derive unfolds: snd_nondep_def)
-    fix q assume asm: "q : A \<times> B"
-    show "indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) q : B" by (derive lems: asm)
-  qed (wellformed jdgmt: assms)+
-qed (rule assms)
-\<close>
-
-
-lemma snd_nondep_comp:
-  assumes "A : U(i)" and "B : U(i)" and "a : A" and "b : B"
-  shows "snd[A,B]`(a,b) \<equiv> b"
-  unfolding snd_nondep_def
-  by (simplify lems: assms)
-
-\<comment> \<open> (* Old proof *)
-proof -
-  have "snd[A,B]`(a,b) \<equiv> indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) (a,b)"
-    by (derive lems: assms unfolds: snd_nondep_def)
-
-  also have "indSum[A, \<lambda>_. B] (\<lambda>_. B) (\<lambda>x y. y) (a,b) \<equiv> b"
-    by (derive lems: assms)
-
-  finally show "snd[A,B]`(a,b) \<equiv> b" .
-qed
-\<close>
-
-
-lemmas Proj_rules [intro] =
-  fst_dep_type snd_dep_type fst_nondep_type snd_nondep_type
-  fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp
-
-lemmas Proj_comps [comp] = fst_dep_comp snd_dep_comp fst_nondep_comp snd_nondep_comp
+lemma assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A"
+ by (derive lems: assms
 
 
 end
\ No newline at end of file
-- 
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