diff options
Diffstat (limited to 'Sum.thy')
-rw-r--r-- | Sum.thy | 56 |
1 files changed, 0 insertions, 56 deletions
diff --git a/Sum.thy b/Sum.thy deleted file mode 100644 index a1c0d34..0000000 --- a/Sum.thy +++ /dev/null @@ -1,56 +0,0 @@ -(******** -Isabelle/HoTT: Dependent sum (dependent pair) -Feb 2019 - -********) - -theory Sum -imports HoTT_Base - -begin - -axiomatization - Sum :: "[t, t \<Rightarrow> t] \<Rightarrow> t" and - pair :: "[t, t] \<Rightarrow> t" ("(2<_,/ _>)") and - indSum :: "[t \<Rightarrow> t, [t, t] \<Rightarrow> t, t] \<Rightarrow> t" - -syntax - "_Sum" :: "[idt, t, t] \<Rightarrow> t" ("(2\<Sum>'(_: _')./ _)" 20) - "_Sum'" :: "[idt, t, t] \<Rightarrow> t" ("(2\<Sum>_: _./ _)" 20) -translations - "\<Sum>(x: A). B" \<rightleftharpoons> "(CONST Sum) A (\<lambda>x. B)" - "\<Sum>x: A. B" \<rightleftharpoons> "(CONST Sum) A (\<lambda>x. B)" - -abbreviation Pair :: "[t, t] \<Rightarrow> t" (infixr "\<times>" 50) - where "A \<times> B \<equiv> \<Sum>_: A. B" - -axiomatization where -\<comment> \<open>Type rules\<close> - - Sum_form: "\<lbrakk>A: U i; \<And>x. x: A \<Longrightarrow> B x: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and - - Sum_intro: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and - - Sum_elim: "\<lbrakk> - p: \<Sum>x: A. B x; - C: \<Sum>x: A. B x \<leadsto> U i; - \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> indSum C f p: C p" and - - Sum_cmp: "\<lbrakk> - a: A; - b: B a; - B: A \<leadsto> U i; - C: \<Sum>x: A. B x \<leadsto> U i; - \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x,y> \<rbrakk> \<Longrightarrow> indSum C f <a, b> \<equiv> f a b" and - -\<comment> \<open>Congruence rules\<close> - - Sum_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> \<Sum>x: A. B x \<equiv> \<Sum>x: A. C x" - -lemmas Sum_form [form] -lemmas Sum_routine [intro] = Sum_form Sum_intro Sum_elim -lemmas Sum_comp [comp] = Sum_cmp -lemmas Sum_cong [cong] = Sum_form_eq - -end |