diff options
Diffstat (limited to 'hott')
| -rw-r--r-- | hott/Equivalence.thy | 23 | ||||
| -rw-r--r-- | hott/Identity.thy | 82 |
2 files changed, 55 insertions, 50 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index d976677..d844b59 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -33,13 +33,16 @@ Lemma (derive) hsym: "g: \<Prod>x: A. B x" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" - shows "H: f ~ g \<Longrightarrow> g ~ f" - unfolding homotopy_def - apply intro - apply (rule pathinv) - \<guillemotright> by (elim H) - \<guillemotright> by typechk - done + "H: f ~ g" + shows "g ~ f" +unfolding homotopy_def +proof intro + fix x assume "x: A" then have "H x: f x = g x" + using \<open>H:_\<close>[unfolded homotopy_def] + \<comment> \<open>this should become unnecessary when definitional unfolding is implemented\<close> + by typechk + thus "g x = f x" by (rule pathinv) fact +qed typechk Lemma (derive) htrans: assumes @@ -71,9 +74,9 @@ Lemma (derive) commute_homotopy: assumes "A: U i" "B: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" "f: A \<rightarrow> B" "g: A \<rightarrow> B" - "H: homotopy A (fn _. B) f g" + "H: f ~ g" shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)" \<comment> \<open>Need this assumption unfolded for typechecking\<close> supply assms(8)[unfolded homotopy_def] @@ -94,7 +97,7 @@ Corollary (derive) commute_homotopy': "A: U i" "x: A" "f: A \<rightarrow> A" - "H: homotopy A (fn _. A) f (id A)" + "H: f ~ (id A)" shows "H (f x) = f [H x]" oops diff --git a/hott/Identity.thy b/hott/Identity.thy index 29ce26a..728537c 100644 --- a/hott/Identity.thy +++ b/hott/Identity.thy @@ -135,14 +135,14 @@ lemmas section \<open>Basic propositional equalities\<close> Lemma (derive) refl_pathcomp: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "(refl x) \<bullet> p = p" apply (eq p) apply (reduce; intro) done Lemma (derive) pathcomp_refl: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "p \<bullet> (refl y) = p" apply (eq p) apply (reduce; intro) @@ -166,24 +166,24 @@ Lemma ru_refl [comp]: unfolding pathcomp_refl_def by reduce Lemma (derive) inv_pathcomp: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "p\<inverse> \<bullet> p = refl y" by (eq p) (reduce; intro) Lemma (derive) pathcomp_inv: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "p \<bullet> p\<inverse> = refl x" by (eq p) (reduce; intro) Lemma (derive) pathinv_pathinv: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "p\<inverse>\<inverse> = p" by (eq p) (reduce; intro) Lemma (derive) pathcomp_assoc: assumes "A: U i" "x: A" "y: A" "z: A" "w: A" - "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" "r: z =\<^bsub>A\<^esub> w" + "p: x = y" "q: y = z" "r: z = w" shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r" apply (eq p) focus prems vars x p @@ -203,7 +203,7 @@ Lemma (derive) ap: "A: U i" "B: U i" "x: A" "y: A" "f: A \<rightarrow> B" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "f x = f y" by (eq p) intro @@ -222,7 +222,7 @@ Lemma (derive) ap_pathcomp: "A: U i" "B: U i" "x: A" "y: A" "z: A" "f: A \<rightarrow> B" - "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" + "p: x = y" "q: y = z" shows "f[p \<bullet> q] = f[p] \<bullet> f[q]" apply (eq p) @@ -237,7 +237,7 @@ Lemma (derive) ap_pathinv: "A: U i" "B: U i" "x: A" "y: A" "f: A \<rightarrow> B" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "f[p\<inverse>] = f[p]\<inverse>" by (eq p) (reduce; intro) @@ -248,7 +248,7 @@ Lemma (derive) ap_funcomp: "A: U i" "B: U i" "C: U i" "x: A" "y: A" "f: A \<rightarrow> B" "g: B \<rightarrow> C" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "(g \<circ> f)[p] = g[f[p]]" apply (eq p) \<guillemotright> by reduce @@ -256,7 +256,7 @@ Lemma (derive) ap_funcomp: done Lemma (derive) ap_id: - assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + assumes "A: U i" "x: A" "y: A" "p: x = y" shows "(id A)[p] = p" apply (eq p) \<guillemotright> by reduce @@ -271,7 +271,7 @@ Lemma (derive) transport: "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "P x \<rightarrow> P y" by (eq p) intro @@ -308,7 +308,7 @@ Lemma (derive) transport_left_inv: "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)" by (eq p) (reduce; refl) @@ -317,7 +317,7 @@ Lemma (derive) transport_right_inv: "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)" by (eq p) (reduce; intro) @@ -327,7 +327,7 @@ Lemma (derive) transport_pathcomp: "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" "z: A" "u: P x" - "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" + "p: x = y" "q: y = z" shows "trans P q (trans P p u) = trans P (p \<bullet> q) u" apply (eq p) focus prems vars x p @@ -342,7 +342,7 @@ Lemma (derive) transport_compose_typefam: "\<And>x. x: B \<Longrightarrow> P x: U i" "f: A \<rightarrow> B" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" "u: P (f x)" shows "trans (fn x. P (f x)) p u = trans P f[p] u" by (eq p) (reduce; intro) @@ -355,7 +355,7 @@ Lemma (derive) transport_function_family: "f: \<Prod>x: A. P x \<rightarrow> Q x" "x: A" "y: A" "u: P x" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "trans Q p ((f x) u) = (f y) (trans P p u)" by (eq p) (reduce; intro) @@ -363,7 +363,7 @@ Lemma (derive) transport_const: assumes "A: U i" "B: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "\<Prod>b: B. trans (fn _. B) p b = b" by (intro, eq p) (reduce; intro) @@ -384,7 +384,7 @@ Lemma (derive) pathlift: "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" "u: P x" shows "<x, u> = <y, trans P p u>" by (eq p) (reduce; intro) @@ -409,7 +409,7 @@ Lemma (derive) pathlift_fst: "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" "u: P x" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "fst[lift P p u] = p" apply (eq p) \<guillemotright> by reduce @@ -425,7 +425,7 @@ Lemma (derive) apd: "\<And>x. x: A \<Longrightarrow> P x: U i" "f: \<Prod>x: A. P x" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "trans P p (f x) = f y" by (eq p) (reduce; intro) @@ -448,7 +448,7 @@ Lemma (derive) apd_ap: "A: U i" "B: U i" "f: A \<rightarrow> B" "x: A" "y: A" - "p: x =\<^bsub>A\<^esub> y" + "p: x = y" shows "apd f p = trans_const B p (f x) \<bullet> f[p]" by (eq p) (reduce; intro) @@ -457,10 +457,9 @@ section \<open>Whiskering\<close> Lemma (derive) right_whisker: assumes "A: U i" "a: A" "b: A" "c: A" - shows "\<lbrakk>p: a = b; q: a = b; r: b = c; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> p \<bullet> r = q \<bullet> r" - \<comment> \<open>TODO: In the above we need to annotate the type of \<alpha> with the type `a = b` - in order for the `eq` method to work correctly. This should be fixed with a - pre-proof elaborator.\<close> + and "p: a = b" "q: a = b" "r: b = c" + and "\<alpha>: p = q" + shows "p \<bullet> r = q \<bullet> r" apply (eq r) focus prems vars x s t proof - @@ -473,7 +472,9 @@ Lemma (derive) right_whisker: Lemma (derive) left_whisker: assumes "A: U i" "a: A" "b: A" "c: A" - shows "\<lbrakk>p: b = c; q: b = c; r: a = b; \<alpha>: p =\<^bsub>b = c\<^esub> q\<rbrakk> \<Longrightarrow> r \<bullet> p = r \<bullet> q" + and "p: b = c" "q: b = c" "r: a = b" + and "\<alpha>: p = q" + shows "r \<bullet> p = r \<bullet> q" apply (eq r) focus prems prms vars x s t proof - @@ -495,15 +496,13 @@ translations "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>" Lemma whisker_refl [comp]: - assumes "A: U i" "a: A" "b: A" - shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p =\<^bsub>a = b\<^esub> q\<rbrakk> \<Longrightarrow> - \<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>" + assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q" + shows "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> (refl b) \<equiv> ru p \<bullet> \<alpha> \<bullet> (ru q)\<inverse>" unfolding right_whisker_def by reduce Lemma refl_whisker [comp]: - assumes "A: U i" "a: A" "b: A" - shows "\<lbrakk>p: a = b; q: a = b; \<alpha>: p = q\<rbrakk> \<Longrightarrow> - (refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>" + assumes "A: U i" "a: A" "b: A" "p: a = b" "q: a = b" "\<alpha>: p = q" + shows "(refl a) \<bullet>\<^sub>l\<^bsub>b\<^esub> \<alpha> \<equiv> (lu p) \<bullet> \<alpha> \<bullet> (lu q)\<inverse>" unfolding left_whisker_def by reduce method left_whisker = (rule left_whisker) @@ -524,20 +523,22 @@ begin Lemma (derive) horiz_pathcomp: notes assums - shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s" + assumes "\<alpha>: p = q" "\<beta>: r = s" + shows "p \<bullet> r = q \<bullet> s" proof (rule pathcomp) - show "\<alpha>: p = q \<Longrightarrow> p \<bullet> r = q \<bullet> r" by right_whisker - show "\<beta>: r = s \<Longrightarrow> .. = q \<bullet> s" by left_whisker + show "p \<bullet> r = q \<bullet> r" by right_whisker fact + show ".. = q \<bullet> s" by left_whisker fact qed typechk text \<open>A second horizontal composition:\<close> Lemma (derive) horiz_pathcomp': notes assums - shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> ?prf \<alpha> \<beta>: p \<bullet> r = q \<bullet> s" + assumes "\<alpha>: p = q" "\<beta>: r = s" + shows "p \<bullet> r = q \<bullet> s" proof (rule pathcomp) - show "\<beta>: r = s \<Longrightarrow> p \<bullet> r = p \<bullet> s" by left_whisker - show "\<alpha>: p = q \<Longrightarrow> .. = q \<bullet> s" by right_whisker + show "p \<bullet> r = p \<bullet> s" by left_whisker fact + show ".. = q \<bullet> s" by right_whisker fact qed typechk notation horiz_pathcomp (infix "\<star>" 121) @@ -545,7 +546,8 @@ notation horiz_pathcomp' (infix "\<star>''" 121) Lemma (derive) horiz_pathcomp_eq_horiz_pathcomp': notes assums - shows "\<lbrakk>\<alpha>: p = q; \<beta>: r = s\<rbrakk> \<Longrightarrow> \<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>" + assumes "\<alpha>: p = q" "\<beta>: r = s" + shows "\<alpha> \<star> \<beta> = \<alpha> \<star>' \<beta>" unfolding horiz_pathcomp_def horiz_pathcomp'_def apply (eq \<alpha>, eq \<beta>) focus vars p apply (eq p) |