diff options
Diffstat (limited to 'EqualProps.thy')
| -rw-r--r-- | EqualProps.thy | 40 |
1 files changed, 20 insertions, 20 deletions
diff --git a/EqualProps.thy b/EqualProps.thy index 3d99456..2da7e2f 100644 --- a/EqualProps.thy +++ b/EqualProps.thy @@ -48,17 +48,17 @@ text " Raw composition function, of type \<open>\<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close> polymorphic over the type \<open>A\<close>. " -axiomatization rcompose :: Term where - rcompose_def: "rcompose \<equiv> \<^bold>\<lambda>x y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= (\<lambda>x. refl(x)) q) p" +axiomatization reqcompose :: Term where + reqcompose_def: "reqcompose \<equiv> \<^bold>\<lambda>x y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= (\<lambda>x. refl(x)) q) p" text " More complicated proofs---the nested path inductions require more explicit step-by-step rule applications: " -lemma rcompose_type: +lemma reqcompose_type: assumes "A: U(i)" - shows "rcompose: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" -unfolding rcompose_def + shows "reqcompose: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" +unfolding reqcompose_def proof fix x assume 1: "x: A" show "\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" @@ -85,17 +85,17 @@ qed fact corollary assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" - shows "rcompose`x`y`p`z`q: x =\<^sub>A z" - by (simple lem: assms rcompose_type) + shows "reqcompose`x`y`p`z`q: x =\<^sub>A z" + by (simple lem: assms reqcompose_type) text " The following proof is very long, chiefly because for every application of \<open>`\<close> we have to show the wellformedness of the type family appearing in the equality computation rule. " -lemma rcompose_comp: +lemma reqcompose_comp: assumes "A: U(i)" and "a: A" - shows "rcompose`a`a`refl(a)`a`refl(a) \<equiv> refl(a)" -unfolding rcompose_def + shows "reqcompose`a`a`refl(a)`a`refl(a) \<equiv> refl(a)" +unfolding reqcompose_def proof (subst comp) { fix x assume 1: "x: A" show "\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" @@ -197,28 +197,28 @@ qed fact text "The raw object lambda term is cumbersome to use, so we define a simpler constant instead." -axiomatization compose :: "[Term, Term] \<Rightarrow> Term" (infixl "\<bullet>" 60) where - compose_def: "\<lbrakk> +axiomatization eqcompose :: "[Term, Term] \<Rightarrow> Term" (infixl "\<bullet>" 60) where + eqcompose_def: "\<lbrakk> A: U(i); x: A; y: A; z: A; p: x =\<^sub>A y; q: y =\<^sub>A z - \<rbrakk> \<Longrightarrow> p \<bullet> q \<equiv> rcompose`x`y`p`z`q" + \<rbrakk> \<Longrightarrow> p \<bullet> q \<equiv> reqcompose`x`y`p`z`q" -lemma compose_type: +lemma eqcompose_type: assumes "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" shows "p \<bullet> q: x =\<^sub>A z" -proof (subst compose_def) +proof (subst eqcompose_def) show "A: U(i)" "x: A" "y: A" "z: A" "p: x =\<^sub>A y" "q: y =\<^sub>A z" by fact+ -qed (simple lem: assms rcompose_type) +qed (simple lem: assms reqcompose_type) -lemma compose_comp: +lemma eqcompose_comp: assumes "A : U(i)" and "a : A" shows "refl(a) \<bullet> refl(a) \<equiv> refl(a)" -by (subst compose_def) (simple lem: assms rcompose_comp) +by (subst eqcompose_def) (simple lem: assms reqcompose_comp) -lemmas EqualProps_rules [intro] = inv_type compose_type -lemmas EqualProps_comps [comp] = inv_comp compose_comp +lemmas EqualProps_rules [intro] = inv_type eqcompose_type +lemmas EqualProps_comps [comp] = inv_comp eqcompose_comp end
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