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 \ \<^bold>\p:\x:?A. ?B(x). ind\<^sub>\(\x y. x)(p)" -abbreviation snd :: "Term \ Term" where ""snd \ \<^bold>\p: \x:A. B(x). ind\<^sub>\(\x y. y) p" - - -section \Overloaded syntax for dependent and nondependent pairs\ - -overloading - fst_dep \ fst - fst_nondep \ fst -begin - definition fst_dep :: "[Term, Typefam] \ Term" where - "fst_dep A B \ \<^bold>\p:\x:A. B(x). ind\<^sub>\(\x y. x)(p)" - - definition fst_nondep :: "[Term, Term] \ Term" where - "fst_nondep A B \ \<^bold>\p: A \ B. ind\<^sub>\[A, \_. B] (\_. A) (\x y. x) p" -end - -overloading - snd_dep \ snd - snd_nondep \ snd -begin - definition snd_dep :: "[Term, Typefam] \ Term" where - "snd_dep A B \ \<^bold>\p: (\x:A. B x). ind\<^sub>\[A,B] (\q. B (fst[A,B]`q)) (\x y. y) p" - - definition snd_nondep :: "[Term, Term] \ Term" where - "snd_nondep A B \ \<^bold>\p: A \ B. ind\<^sub>\[A, \_. B] (\_. B) (\x y. y) p" -end - - -section \Properties\ +abbreviation fst :: "Term \ Term" where "fst \ \p. ind\<^sub>\(\x y. x)(p)" +abbreviation snd :: "Term \ Term" where "snd \ \p. ind\<^sub>\(\x y. y)(p)" text "Typing judgments and computation rules for the dependent and non-dependent projection functions." -lemma fst_dep_type: assumes "\x:A. B x : U(i)" and "p : \x:A. B x" shows "fst[A,B]`p : A" -unfolding fst_dep_def -proof - show "lambda (Sum A B) (ind\<^sub>\[A, B] (\_. A) (\x y. x)) : (\x:A. B x) \ A" - proof - show "A : U(i)" using assms(1) .. - show "Sum A B : U(i)" by (rule assms) - show "\p. p : Sum A B \ ind\<^sub>\[A, B] (\_. A) (\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 \ U(i)" and "a : A" and "b : B a" - shows "fst[A,B]`(a,b) \ a" -unfolding fst_dep_def -proof (subst comp) - show "Sum A B : U(i)" using assms(1-2) .. - show "\x. x : Sum A B \ ind\<^sub>\[A, B] (\_. A) (\x y. x) x : A" by (standard, rule assms(1), assumption+) - show "(a,b) : Sum A B" using assms .. - show "ind\<^sub>\[A, B] (\_. A) (\x y. x) (a, b) \ a" by (standard, (rule assms|assumption)+) -qed (rule assms) - -\ \ (* Old proof *) -proof - - have "fst[A,B]`(a,b) \ indSum[A,B] (\_. A) (\x y. x) (a,b)" - by (derive lems: assms unfolds: fst_dep_def) - - also have "indSum[A,B] (\_. A) (\x y. x) (a,b) \ a" - by (derive lems: assms) - - finally show "fst[A,B]`(a,b) \ a" . -qed -\ - -\ \ (* Old lemma *) -text "In proving results about the second dependent projection function we often use the following lemma." - -lemma lem: - assumes "B: A \ U" and "x : A" and "y : B x" - shows "y : B (fst[A,B]`(x,y))" - -proof - - have "fst[A,B]`(x,y) \ x" using assms by (rule fst_dep_comp) - then show "y : B (fst[A,B]`(x,y))" using assms by simp -qed -\ - - -lemma snd_dep_type: - assumes "\x:A. B x : U(i)" and "p : \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 "\x. x : Sum A B \ ind\<^sub>\[A, B] (\_. A) (\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>\[A, B] (\q. B (lambda (Sum A B) (ind\<^sub>\[A, B] (\_. A) (\x y. x))`q)) (\x y. y)) - ` p : B (ind\<^sub>\[A, B] (\_. A) (\x y. x) p)" - proof (subst (2) comp) - show "Sum A B : U(i)" by (rule assms) - show " - - -\ \ (* Old proof *) -proof (derive lems: assms unfolds: snd_dep_def) - show "fst[A, B] : (\x:A. B x) \ A" by (derive lems: assms unfolds: fst_dep_def) - - fix x y assume asm: "x : A" "y : B x" - have "B: A \ U" by (wellformed jdgmt: assms) - then show "y : B (fst[A, B]`(x,y))" using asm by (rule lem) -qed (assumption | rule assms)+ -\ - - -lemma snd_dep_comp: - assumes "A : U(i)" and "B: A \ U(i)" and "a : A" and "b : B a" - shows "snd[A,B]`(a,b) \ b" - unfolding snd_dep_def fst_dep_def - by (simplify lems: assms) - -\ \ (* Old proof *) -proof - - have "snd[A,B]`(a,b) \ indSum[A, B] (\q. B (fst[A,B]`q)) (\x y. y) (a,b)" - proof (derive lems: assms unfolds: snd_dep_def) - show "fst[A, B] : (\x:A. B x) \ 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] (\q. B (fst[A,B]`q)) (\x y. y) (a,b) \ b" - proof (simple lems: assms) - show "fst[A, B] : (\x:A. B x) \ 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) \ b" . -qed -\ - - -text "Nondependent projections:" - -lemma fst_nondep_type: assumes "A \ B : U(i)" and "p : A \ 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) \ a" - unfolding fst_nondep_def - by (simplify lems: assms) - -\ \ (* Old proof *) -proof - - have "fst[A,B]`(a,b) \ indSum[A, \_. B] (\_. A) (\x y. x) (a,b)" - by (derive lems: assms unfolds: fst_nondep_def) - - also have "indSum[A, \_. B] (\_. A) (\x y. x) (a,b) \ a" - by (derive lems: assms) - - finally show "fst[A,B]`(a,b) \ a" . -qed -\ - - -lemma snd_nondep_type: assumes "A \ B : U(i)" and "p : A \ B" shows "snd[A,B]`p : B" - unfolding snd_nondep_def - by (derive lems: assms) - -\ \ (* Old proof *) -proof - show "snd[A,B] : A \ B \ B" - proof (derive unfolds: snd_nondep_def) - fix q assume asm: "q : A \ B" - show "indSum[A, \_. B] (\_. B) (\x y. y) q : B" by (derive lems: asm) - qed (wellformed jdgmt: assms)+ -qed (rule assms) -\ - - -lemma snd_nondep_comp: - assumes "A : U(i)" and "B : U(i)" and "a : A" and "b : B" - shows "snd[A,B]`(a,b) \ b" - unfolding snd_nondep_def - by (simplify lems: assms) - -\ \ (* Old proof *) -proof - - have "snd[A,B]`(a,b) \ indSum[A, \_. B] (\_. B) (\x y. y) (a,b)" - by (derive lems: assms unfolds: snd_nondep_def) - - also have "indSum[A, \_. B] (\_. B) (\x y. y) (a,b) \ b" - by (derive lems: assms) - - finally show "snd[A,B]`(a,b) \ b" . -qed -\ - - -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 "\x:A. B(x): U(i)" and "p: \x:A. B(x)" shows "fst(p): A" + by (derive lems: assms end \ No newline at end of file -- cgit v1.2.3