diff options
Diffstat (limited to 'hott')
| -rw-r--r-- | hott/Equivalence.thy | 35 | ||||
| -rw-r--r-- | hott/Identity.thy | 92 | ||||
| -rw-r--r-- | hott/List_HoTT.thy (renamed from hott/List+.thy) | 4 | ||||
| -rw-r--r-- | hott/Nat.thy | 14 |
4 files changed, 76 insertions, 69 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index 99300a0..379dc81 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -8,7 +8,7 @@ section \<open>Homotopy\<close> definition "homotopy A B f g \<equiv> \<Prod>x: A. f `x =\<^bsub>B x\<^esub> g `x" definition homotopy_i (infix "~" 100) - where [implicit]: "f ~ g \<equiv> homotopy ? ? f g" + where [implicit]: "f ~ g \<equiv> homotopy {} {} f g" translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g" @@ -79,7 +79,7 @@ Lemma (def) htrans: section \<open>Rewriting homotopies\<close> -congruence "f ~ g" rhs g +calc "f ~ g" rhs g lemmas homotopy_sym [sym] = hsym[rotated 4] and @@ -129,7 +129,6 @@ Lemma funcomp_right_htpy: method lhtpy = rule funcomp_left_htpy[rotated 6] method rhtpy = rule funcomp_right_htpy[rotated 6] - Lemma (def) commute_homotopy: assumes "A: U i" "B: U i" @@ -152,8 +151,16 @@ Corollary (def) commute_homotopy': "H: f ~ (id A)" shows "H (f x) = f [H x]" proof - - (*FIXME: Bug; if the following type declaration isn't made then bad things - happen on the next lines.*) + (* Rewrite the below proof + have *: "H (f x) \<bullet> (id A)[H x] = f[H x] \<bullet> H x" + using \<open>H:_\<close> unfolding homotopy_def by (rule commute_homotopy) + + thus "H (f x) = f[H x]" + apply (pathcomp_cancelr) + ... + *) + (*FUTURE: Because we don't have very good normalization integrated into + things yet, we need to manually unfold the type of H.*) from \<open>H: f ~ id A\<close> have [type]: "H: \<Prod>x: A. f x = x" by (reduce add: homotopy_def) @@ -161,7 +168,7 @@ Corollary (def) commute_homotopy': by (rule left_whisker, transport eq: ap_id, refl) also have [simplified id_comp]: "H (f x) \<bullet> (id A)[H x] = f[H x] \<bullet> H x" by (rule commute_homotopy) - finally have "{}" by this + finally have "?" by this thus "H (f x) = f [H x]" by pathcomp_cancelr (fact, typechk+) qed @@ -181,7 +188,7 @@ Lemma is_qinv_type [type]: by typechk definition is_qinv_i ("is'_qinv") - where [implicit]: "is_qinv f \<equiv> Equivalence.is_qinv ? ? f" + where [implicit]: "is_qinv f \<equiv> Equivalence.is_qinv {} {} f" no_translations "is_qinv f" \<leftharpoondown> "CONST Equivalence.is_qinv A B f" @@ -241,7 +248,7 @@ Lemma (def) funcomp_is_qinv: have "(finv \<circ> ginv) \<circ> g \<circ> f ~ finv \<circ> (ginv \<circ> g) \<circ> f" by reduce refl also have ".. ~ finv \<circ> id B \<circ> f" by (rhtpy, lhtpy) fact also have ".. ~ id A" by reduce fact - finally show "{}" by this + finally show "?" by this qed show "(g \<circ> f) \<circ> finv \<circ> ginv ~ id C" @@ -249,7 +256,7 @@ Lemma (def) funcomp_is_qinv: have "(g \<circ> f) \<circ> finv \<circ> ginv ~ g \<circ> (f \<circ> finv) \<circ> ginv" by reduce refl also have ".. ~ g \<circ> id B \<circ> ginv" by (rhtpy, lhtpy) fact also have ".. ~ id C" by reduce fact - finally show "{}" by this + finally show "?" by this qed qed done @@ -267,7 +274,7 @@ Lemma is_biinv_type [type]: unfolding is_biinv_def by typechk definition is_biinv_i ("is'_biinv") - where [implicit]: "is_biinv f \<equiv> Equivalence.is_biinv ? ? f" + where [implicit]: "is_biinv f \<equiv> Equivalence.is_biinv {} {} f" translations "is_biinv f" \<leftharpoondown> "CONST Equivalence.is_biinv A B f" @@ -279,8 +286,8 @@ Lemma is_biinvI: shows "is_biinv f" unfolding is_biinv_def proof intro - show "<g, H1>: {}" by typechk - show "<h, H2>: {}" by typechk + show "<g, H1>: \<Sum>g: B \<rightarrow> A. g \<circ> f ~ id A" by typechk + show "<h, H2>: \<Sum>g: B \<rightarrow> A. f \<circ> g ~ id B" by typechk qed Lemma is_biinv_components [type]: @@ -408,7 +415,7 @@ text \<open> Lemma (def) equiv_if_equal: assumes "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B" - shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B" + shows "<transp (id (U i)) p, ?isequiv>: A \<simeq> B" unfolding equivalence_def apply intro defer \<^item> apply (eq p) @@ -422,7 +429,7 @@ Lemma (def) equiv_if_equal: using [[solve_side_conds=1]] apply (rewrite transport_comp) \<circ> by (rule Ui_in_USi) - \<circ> by reduce (rule in_USi_if_in_Ui) + \<circ> by reduce (rule U_lift) \<circ> by reduce (rule id_is_biinv) done done diff --git a/hott/Identity.thy b/hott/Identity.thy index dc27ee8..0a31dc7 100644 --- a/hott/Identity.thy +++ b/hott/Identity.thy @@ -20,7 +20,7 @@ syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b" axiomatization where - \<comment> \<open>Here `A: U i` comes last because A is often implicit\<close> + \<comment> \<open>Here \<open>A: U i\<close> comes last because A is often implicit\<close> IdF: "\<lbrakk>a: A; b: A; A: U i\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and @@ -49,8 +49,8 @@ lemmas section \<open>Path induction\<close> -\<comment> \<open>With `p: x = y` in the context the invokation `eq p` is essentially - `elim p x y`, with some extra bits before and after.\<close> +\<comment> \<open>With \<open>p: x = y\<close> in the context the invokation \<open>eq p\<close> is essentially + \<open>elim p x y\<close>, with some extra bits before and after.\<close> method_setup eq = \<open>Args.term >> (fn tm => K (CONTEXT_METHOD ( @@ -109,13 +109,13 @@ method pathcomp for p q :: o = rule pathcomp[where ?p=p and ?q=q] section \<open>Notation\<close> definition Id_i (infix "=" 110) - where [implicit]: "Id_i x y \<equiv> x =\<^bsub>?\<^esub> y" + where [implicit]: "Id_i x y \<equiv> x =\<^bsub>{}\<^esub> y" definition pathinv_i ("_\<inverse>" [1000]) - where [implicit]: "pathinv_i p \<equiv> pathinv ? ? ? p" + where [implicit]: "pathinv_i p \<equiv> pathinv {} {} {} p" definition pathcomp_i (infixl "\<bullet>" 121) - where [implicit]: "pathcomp_i p q \<equiv> pathcomp ? ? ? ? p q" + where [implicit]: "pathcomp_i p q \<equiv> pathcomp {} {} {} {} p q" translations "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y" @@ -125,7 +125,7 @@ translations section \<open>Calculational reasoning\<close> -congruence "x = y" rhs y +calc "x = y" rhs y lemmas [sym] = pathinv[rotated 3] and @@ -144,8 +144,8 @@ Lemma (def) pathcomp_refl: shows "p \<bullet> (refl y) = p" by (eq p) (reduce, refl) -definition [implicit]: "lu p \<equiv> refl_pathcomp ? ? ? p" -definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p" +definition [implicit]: "lu p \<equiv> refl_pathcomp {} {} {} p" +definition [implicit]: "ru p \<equiv> pathcomp_refl {} {} {} p" translations "CONST lu p" \<leftharpoondown> "CONST refl_pathcomp A x y p" @@ -204,7 +204,7 @@ Lemma (def) ap: by (eq p) intro definition ap_i ("_[_]" [1000, 0]) - where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p" + where [implicit]: "ap_i f p \<equiv> ap {} {} {} {} f p" translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p" @@ -265,17 +265,17 @@ Lemma (def) transport: shows "P x \<rightarrow> P y" by (eq p) intro -definition transport_i ("trans") - where [implicit]: "trans P p \<equiv> transport ? P ? ? p" +definition transport_i ("transp") + where [implicit]: "transp P p \<equiv> transport {} P {} {} p" -translations "trans P p" \<leftharpoondown> "CONST transport A P x y p" +translations "transp P p" \<leftharpoondown> "CONST transport A P x y p" Lemma transport_comp [comp]: assumes "a: A" "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" - shows "trans P (refl a) \<equiv> id (P a)" + shows "transp P (refl a) \<equiv> id (P a)" unfolding transport_def by reduce Lemma apply_transport: @@ -284,7 +284,7 @@ Lemma apply_transport: "x: A" "y: A" "p: y =\<^bsub>A\<^esub> x" "u: P x" - shows "trans P p\<inverse> u: P y" + shows "transp P p\<inverse> u: P y" by typechk method transport uses eq = (rule apply_transport[OF _ _ _ _ eq]) @@ -304,7 +304,7 @@ Lemma (def) pathcomp_cancel_left: by (transport eq: pathcomp_assoc, transport eq: inv_pathcomp, transport eq: refl_pathcomp) refl - finally show "{}" by this + finally show "?" by this qed Lemma (def) pathcomp_cancel_right: @@ -323,7 +323,7 @@ Lemma (def) pathcomp_cancel_right: transport eq: pathcomp_assoc[symmetric], transport eq: pathcomp_inv, transport eq: pathcomp_refl) refl - finally show "{}" by this + finally show "?" by this qed method pathcomp_cancell = rule pathcomp_cancel_left[rotated 7] @@ -335,7 +335,7 @@ Lemma (def) transport_left_inv: "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" "p: x = y" - shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)" + shows "(transp P p\<inverse>) \<circ> (transp P p) = id (P x)" by (eq p) (reduce, refl) Lemma (def) transport_right_inv: @@ -344,7 +344,7 @@ Lemma (def) transport_right_inv: "\<And>x. x: A \<Longrightarrow> P x: U i" "x: A" "y: A" "p: x = y" - shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)" + shows "(transp P p) \<circ> (transp P p\<inverse>) = id (P y)" by (eq p) (reduce, refl) Lemma (def) transport_pathcomp: @@ -354,11 +354,11 @@ Lemma (def) transport_pathcomp: "x: A" "y: A" "z: A" "u: P x" "p: x = y" "q: y = z" - shows "trans P q (trans P p u) = trans P (p \<bullet> q) u" + shows "transp P q (transp P p u) = transp P (p \<bullet> q) u" proof (eq p) fix x q u assuming "x: A" "q: x = z" "u: P x" - show "trans P q (trans P (refl x) u) = trans P ((refl x) \<bullet> q) u" + show "transp P q (transp P (refl x) u) = transp P ((refl x) \<bullet> q) u" by (eq q) (reduce, refl) qed @@ -370,7 +370,7 @@ Lemma (def) transport_compose_typefam: "x: A" "y: A" "p: x = y" "u: P (f x)" - shows "trans (fn x. P (f x)) p u = trans P f[p] u" + shows "transp (fn x. P (f x)) p u = transp P f[p] u" by (eq p) (reduce, refl) Lemma (def) transport_function_family: @@ -382,7 +382,7 @@ Lemma (def) transport_function_family: "x: A" "y: A" "u: P x" "p: x = y" - shows "trans Q p ((f x) u) = (f y) (trans P p u)" + shows "transp Q p ((f x) u) = (f y) (transp P p u)" by (eq p) (reduce, refl) Lemma (def) transport_const: @@ -390,19 +390,19 @@ Lemma (def) transport_const: "A: U i" "B: U i" "x: A" "y: A" "p: x = y" - shows "\<Prod>b: B. trans (fn _. B) p b = b" + shows "\<Prod>b: B. transp (fn _. B) p b = b" by intro (eq p, reduce, refl) -definition transport_const_i ("trans'_const") - where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p" +definition transport_const_i ("transp'_c") + where [implicit]: "transp_c B p \<equiv> transport_const {} B {} {} p" -translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p" +translations "transp_c B p" \<leftharpoondown> "CONST transport_const A B x y p" Lemma transport_const_comp [comp]: assumes "x: A" "b: B" "A: U i" "B: U i" - shows "trans_const B (refl x) b \<equiv> refl b" + shows "transp_c B (refl x) b \<equiv> refl b" unfolding transport_const_def by reduce Lemma (def) pathlift: @@ -412,11 +412,11 @@ Lemma (def) pathlift: "x: A" "y: A" "p: x = y" "u: P x" - shows "<x, u> = <y, trans P p u>" + shows "<x, u> = <y, transp P p u>" by (eq p) (reduce, refl) definition pathlift_i ("lift") - where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u" + where [implicit]: "lift P p u \<equiv> pathlift {} P {} {} p u" translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u" @@ -449,11 +449,11 @@ Lemma (def) apd: "f: \<Prod>x: A. P x" "x: A" "y: A" "p: x = y" - shows "trans P p (f x) = f y" + shows "transp P p (f x) = f y" by (eq p) (reduce, refl) definition apd_i ("apd") - where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p" + where [implicit]: "apd f p \<equiv> Identity.apd {} {} f {} {} p" translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p" @@ -472,7 +472,7 @@ Lemma (def) apd_ap: "f: A \<rightarrow> B" "x: A" "y: A" "p: x = y" - shows "apd f p = trans_const B p (f x) \<bullet> f[p]" + shows "apd f p = transp_c B p (f x) \<bullet> f[p]" by (eq p) (reduce, refl) @@ -488,7 +488,7 @@ Lemma (def) right_whisker: have "s \<bullet> refl x = s" by (rule pathcomp_refl) also have ".. = t" by fact also have ".. = t \<bullet> refl x" by (rule pathcomp_refl[symmetric]) - finally show "{}" by this + finally show "?" by this qed done @@ -502,15 +502,15 @@ Lemma (def) left_whisker: have "refl x \<bullet> s = s" by (rule refl_pathcomp) also have ".. = t" by fact also have ".. = refl x \<bullet> t" by (rule refl_pathcomp[symmetric]) - finally show "{}" by this + finally show "?" by this qed done definition right_whisker_i (infix "\<bullet>\<^sub>r" 121) - where [implicit]: "\<alpha> \<bullet>\<^sub>r r \<equiv> right_whisker ? ? ? ? ? ? r \<alpha>" + where [implicit]: "\<alpha> \<bullet>\<^sub>r r \<equiv> right_whisker {} {} {} {} {} {} r \<alpha>" definition left_whisker_i (infix "\<bullet>\<^sub>l" 121) - where [implicit]: "r \<bullet>\<^sub>l \<alpha> \<equiv> left_whisker ? ? ? ? ? ? r \<alpha>" + where [implicit]: "r \<bullet>\<^sub>l \<alpha> \<equiv> left_whisker {} {} {} {} {} {} r \<alpha>" translations "\<alpha> \<bullet>\<^sub>r r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>" @@ -532,8 +532,8 @@ method right_whisker = (rule right_whisker) section \<open>Horizontal path-composition\<close> -text \<open>Conditions under which horizontal path-composition is defined.\<close> locale horiz_pathcomposable = +\<comment> \<open>Conditions under which horizontal path-composition is defined.\<close> fixes i A a b c p q r s assumes [type]: @@ -615,17 +615,17 @@ Lemma (def) pathcomp_eq_horiz_pathcomp: have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl) also have ".. = \<alpha>" by (rule refl_pathcomp) - finally have eq\<alpha>: "{} = \<alpha>" by this + finally have eq\<alpha>: "? = \<alpha>" by this have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl) also have ".. = \<beta>" by (rule refl_pathcomp) - finally have eq\<beta>: "{} = \<beta>" by this + finally have eq\<beta>: "? = \<beta>" by this have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a)) - = \<alpha> \<bullet> {}" by right_whisker (fact eq\<alpha>) + = \<alpha> \<bullet> ?" by right_whisker (fact eq\<alpha>) also have ".. = \<alpha> \<bullet> \<beta>" by left_whisker (fact eq\<beta>) - finally show "{} = \<alpha> \<bullet> \<beta>" by this + finally show "? = \<alpha> \<bullet> \<beta>" by this qed Lemma (def) pathcomp_eq_horiz_pathcomp': @@ -637,17 +637,17 @@ Lemma (def) pathcomp_eq_horiz_pathcomp': have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<beta>" by (rule pathcomp_refl) also have ".. = \<beta>" by (rule refl_pathcomp) - finally have eq\<beta>: "{} = \<beta>" by this + finally have eq\<beta>: "? = \<beta>" by this have "refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a) = refl (refl a) \<bullet> \<alpha>" by (rule pathcomp_refl) also have ".. = \<alpha>" by (rule refl_pathcomp) - finally have eq\<alpha>: "{} = \<alpha>" by this + finally have eq\<alpha>: "? = \<alpha>" by this have "refl (refl a) \<bullet> \<beta> \<bullet> refl (refl a) \<bullet> (refl (refl a) \<bullet> \<alpha> \<bullet> refl (refl a)) - = \<beta> \<bullet> {}" by right_whisker (fact eq\<beta>) + = \<beta> \<bullet> ?" by right_whisker (fact eq\<beta>) also have ".. = \<beta> \<bullet> \<alpha>" by left_whisker (fact eq\<alpha>) - finally show "{} = \<beta> \<bullet> \<alpha>" by this + finally show "? = \<beta> \<bullet> \<alpha>" by this qed Lemma (def) eckmann_hilton: diff --git a/hott/List+.thy b/hott/List_HoTT.thy index 869d667..9bd1517 100644 --- a/hott/List+.thy +++ b/hott/List_HoTT.thy @@ -1,4 +1,4 @@ -theory "List+" +theory List_HoTT imports Spartan.List Nat @@ -7,7 +7,7 @@ begin section \<open>Length\<close> -definition [implicit]: "len \<equiv> ListRec ? Nat 0 (fn _ _ rec. suc rec)" +definition [implicit]: "len \<equiv> ListRec {} Nat 0 (fn _ _ rec. suc rec)" experiment begin Lemma "len [] \<equiv> ?n" by (subst comp; typechk?) diff --git a/hott/Nat.thy b/hott/Nat.thy index f45387c..1aa7932 100644 --- a/hott/Nat.thy +++ b/hott/Nat.thy @@ -115,7 +115,7 @@ Lemma (def) add_comm: proof reduce have "suc (m + n) = suc (n + m)" by (eq ih) refl also have ".. = suc n + m" by (transport eq: suc_add) refl - finally show "{}" by this + finally show "?" by this qed done @@ -152,7 +152,7 @@ Lemma (def) zero_mul: proof reduce have "0 + 0 * n = 0 + 0 " by (eq ih) refl also have ".. = 0" by reduce refl - finally show "{}" by this + finally show "?" by this qed done @@ -163,9 +163,9 @@ Lemma (def) suc_mul: \<^item> by reduce refl \<^item> vars n ih proof (reduce, transport eq: \<open>ih:_\<close>) - have "suc m + (m * n + n) = suc (m + {})" by (rule suc_add) + have "suc m + (m * n + n) = suc (m + ?)" by (rule suc_add) also have ".. = suc (m + m * n + n)" by (transport eq: add_assoc) refl - finally show "{}" by this + finally show "?" by this qed done @@ -180,7 +180,7 @@ Lemma (def) mul_dist_add: also have ".. = l + l * m + l * n" by (rule add_assoc) also have ".. = l * m + l + l * n" by (transport eq: add_comm) refl also have ".. = l * m + (l + l * n)" by (transport eq: add_assoc) refl - finally show "{}" by this + finally show "?" by this qed done @@ -193,7 +193,7 @@ Lemma (def) mul_assoc: proof reduce have "l * (m + m * n) = l * m + l * (m * n)" by (rule mul_dist_add) also have ".. = l * m + l * m * n" by (transport eq: \<open>ih:_\<close>) refl - finally show "{}" by this + finally show "?" by this qed done @@ -207,7 +207,7 @@ Lemma (def) mul_comm: have "suc n * m = n * m + m" by (rule suc_mul) also have ".. = m + m * n" by (transport eq: \<open>ih:_\<close>, transport eq: add_comm) refl - finally show "{}" by this + finally show "?" by this qed done |