diff options
Diffstat (limited to 'hott')
| -rw-r--r-- | hott/Equivalence.thy | 52 | ||||
| -rw-r--r-- | hott/Identity.thy | 147 | ||||
| -rw-r--r-- | hott/More_List.thy | 4 | ||||
| -rw-r--r-- | hott/More_Nat.thy | 6 | ||||
| -rw-r--r-- | hott/Nat.thy | 17 |
5 files changed, 109 insertions, 117 deletions
diff --git a/hott/Equivalence.thy b/hott/Equivalence.thy index 66248a9..88adc8b 100644 --- a/hott/Equivalence.thy +++ b/hott/Equivalence.thy @@ -35,7 +35,7 @@ Lemma (derive) hsym: "\<And>x. x: A \<Longrightarrow> B x: U i" shows "H: f ~ g \<Longrightarrow> g ~ f" unfolding homotopy_def - apply intros + apply intro apply (rule pathinv) \<guillemotright> by (elim H) \<guillemotright> by typechk @@ -106,7 +106,8 @@ Lemma homotopy_id_left [typechk]: Lemma homotopy_id_right [typechk]: assumes "A: U i" "B: U i" "f: A \<rightarrow> B" shows "homotopy_refl A f: f \<circ> (id A) ~ f" - unfolding homotopy_refl_def homotopy_def by reduce + unfolding homotopy_refl_def homotopy_def + by reduce Lemma homotopy_funcomp_left: assumes @@ -165,11 +166,9 @@ Lemma (derive) id_qinv: assumes "A: U i" shows "qinv (id A)" unfolding qinv_def - apply intro defer - apply intro defer - apply htpy_id - apply id_htpy - done + proof intro + show "id A: A \<rightarrow> A" by typechk + qed (reduce, intro; refl) Lemma (derive) quasiinv_qinv: assumes "A: U i" "B: U i" "f: A \<rightarrow> B" @@ -272,7 +271,7 @@ Lemma (derive) biinv_imp_qinv: also have ".. ~ (id A) \<circ> h" by (subst funcomp_assoc[symmetric]) (htpy_o, fact) also have ".. ~ h" by reduce refl finally have "g ~ h" by this - then have "f \<circ> g ~ f \<circ> h" by o_htpy + then have "f \<circ> g ~ f \<circ> h" by (o_htpy, this) also note \<open>H2:_\<close> finally show "f \<circ> g ~ id B" by this qed @@ -313,10 +312,9 @@ Lemma (derive) equivalence_refl: assumes "A: U i" shows "A \<simeq> A" unfolding equivalence_def - apply intro defer - apply (rule qinv_imp_biinv) defer - apply (rule id_qinv) - done + proof intro + show "biinv (id A)" by (rule qinv_imp_biinv) (rule id_qinv) + qed typechk text \<open> The following could perhaps be easier with transport (but then I think we need @@ -340,11 +338,14 @@ Lemma (derive) equivalence_symmetric: Lemma (derive) equivalence_transitive: assumes "A: U i" "B: U i" "C: U i" shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C" + (* proof intros + fix AB BC assume "AB: A \<simeq> B" "BC: B \<simeq> C" + let "?f: {}" = "(fst AB) :: o" *) apply intros unfolding equivalence_def focus vars p q apply (elim p, elim q) focus vars f biinv\<^sub>f g biinv\<^sub>g apply intro - \<guillemotright> apply (rule funcompI) defer by assumption known + \<guillemotright> apply (rule funcompI) defer by assumption+ known \<guillemotright> by (rule funcomp_biinv) done done @@ -372,32 +373,31 @@ Lemma (derive) id_imp_equiv: shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B" unfolding equivalence_def apply intros defer - - \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close> - supply [[auto_typechk=false]] - - \<guillemotright> apply (eq p, typechk) + \<guillemotright> apply (eq p) \<^item> prems vars A B apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + using [[solve_side_conds=1]] apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) - apply (rule transport, rule U_in_U) - apply (rule lift_universe_codomain, rule U_in_U) - apply (typechk, rule U_in_U) + apply (rule transport, rule Ui_in_USi) + apply (rule lift_universe_codomain, rule Ui_in_USi) + apply (typechk, rule Ui_in_USi) done \<^item> prems vars A + using [[solve_side_conds=1]] apply (subst transport_comp) - \<^enum> by (rule U_in_U) - \<^enum> by reduce (rule lift_U) + \<^enum> by (rule Ui_in_USi) + \<^enum> by reduce (rule in_USi_if_in_Ui) \<^enum> by reduce (rule id_biinv) done done \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close> apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + using [[solve_side_conds=1]] apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric]) - apply (rule transport, rule U_in_U) - apply (rule lift_universe_codomain, rule U_in_U) - apply (typechk, rule U_in_U) + apply (rule transport, rule Ui_in_USi) + apply (rule lift_universe_codomain, rule Ui_in_USi) + apply (typechk, rule Ui_in_USi) done done diff --git a/hott/Identity.thy b/hott/Identity.thy index dd2d6a0..1cb3946 100644 --- a/hott/Identity.thy +++ b/hott/Identity.thy @@ -20,7 +20,8 @@ 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 - IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and + \<comment> \<open>Here `A: U i` 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 @@ -39,35 +40,36 @@ axiomatization where \<rbrakk> \<Longrightarrow> IdInd A (fn x y p. C x y p) (fn x. f x) a a (refl a) \<equiv> f a" lemmas - [intros] = IdF IdI and - [elims "?p" "?a" "?b"] = IdE and - [comps] = Id_comp and + [form] = IdF and + [intro, intros] = IdI and + [elim "?p" "?a" "?b"] = IdE and + [comp] = Id_comp and [refl] = IdI section \<open>Path induction\<close> -method_setup eq = \<open> -Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K ( - CONTEXT_SUBGOAL (fn (goal, i) => - let - val facts = Proof_Context.facts_of ctxt - val prems = Logic.strip_assums_hyp goal - val template = \<^const>\<open>has_type\<close> - $ tm - $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>) - $ Var (("*?x", 0), \<^typ>\<open>o\<close>) - $ Var (("*?y", 0), \<^typ>\<open>o\<close>)) - val types = - map (Thm.prop_of o #1) (Facts.could_unify facts template) - @ filter (fn prem => Term.could_unify (template, prem)) prems - |> map Lib.type_of_typing - in case types of - (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ => - elim_context_tac [tm, x, y] ctxt i - | _ => Context_Tactic.CONTEXT_TACTIC no_tac - end) 1))) -\<close> +method_setup eq = + \<open>Args.term >> (fn tm => K (CONTEXT_METHOD ( + CHEADGOAL o SIDE_CONDS ( + CONTEXT_SUBGOAL (fn (goal, i) => fn cst as (ctxt, st) => + let + val facts = Proof_Context.facts_of ctxt + val prems = Logic.strip_assums_hyp goal + val template = \<^const>\<open>has_type\<close> + $ tm + $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>) + $ Var (("*?x", 0), \<^typ>\<open>o\<close>) + $ Var (("*?y", 0), \<^typ>\<open>o\<close>)) + val types = + map (Thm.prop_of o #1) (Facts.could_unify facts template) + @ filter (fn prem => Term.could_unify (template, prem)) prems + |> map Lib.type_of_typing + in case types of + (Const (\<^const_name>\<open>Id\<close>, _) $ _ $ x $ y) :: _ => + elim_ctac [tm, x, y] i cst + | _ => no_ctac cst + end)))))\<close> section \<open>Symmetry and transitivity\<close> @@ -77,7 +79,7 @@ Lemma (derive) pathinv: shows "y =\<^bsub>A\<^esub> x" by (eq p) intro -lemma pathinv_comp [comps]: +lemma pathinv_comp [comp]: assumes "x: A" "A: U i" shows "pathinv A x x (refl x) \<equiv> refl x" unfolding pathinv_def by reduce @@ -95,7 +97,7 @@ Lemma (derive) pathcomp: done done -lemma pathcomp_comp [comps]: +lemma pathcomp_comp [comp]: assumes "a: A" "A: U i" shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a" unfolding pathcomp_def by reduce @@ -133,14 +135,14 @@ Lemma (derive) refl_pathcomp: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "(refl x) \<bullet> p = p" apply (eq p) - apply (reduce; intros) + apply (reduce; intro) done Lemma (derive) pathcomp_refl: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "p \<bullet> (refl y) = p" apply (eq p) - apply (reduce; intros) + apply (reduce; intro) done definition [implicit]: "ru p \<equiv> pathcomp_refl ? ? ? p" @@ -150,36 +152,30 @@ translations "CONST ru p" \<leftharpoondown> "CONST pathcomp_refl A x y p" "CONST lu p" \<leftharpoondown> "CONST refl_pathcomp A x y p" -Lemma lu_refl [comps]: +Lemma lu_refl [comp]: assumes "A: U i" "x: A" shows "lu (refl x) \<equiv> refl (refl x)" - unfolding refl_pathcomp_def by reduce + unfolding refl_pathcomp_def by reduce+ -Lemma ru_refl [comps]: +Lemma ru_refl [comp]: assumes "A: U i" "x: A" shows "ru (refl x) \<equiv> refl (refl x)" - unfolding pathcomp_refl_def by reduce + unfolding pathcomp_refl_def by reduce+ Lemma (derive) inv_pathcomp: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "p\<inverse> \<bullet> p = refl y" - apply (eq p) - apply (reduce; intros) - done + by (eq p) (reduce; intro) Lemma (derive) pathcomp_inv: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "p \<bullet> p\<inverse> = refl x" - apply (eq p) - apply (reduce; intros) - done + by (eq p) (reduce; intro) Lemma (derive) pathinv_pathinv: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "p\<inverse>\<inverse> = p" - apply (eq p) - apply (reduce; intros) - done + by (eq p) (reduce; intro) Lemma (derive) pathcomp_assoc: assumes @@ -191,7 +187,7 @@ Lemma (derive) pathcomp_assoc: apply (eq p) focus prems vars x p apply (eq p) - apply (reduce; intros) + apply (reduce; intro) done done done @@ -206,16 +202,14 @@ Lemma (derive) ap: "f: A \<rightarrow> B" "p: x =\<^bsub>A\<^esub> y" shows "f x = f y" - apply (eq p) - apply intro - done + by (eq p) intro definition ap_i ("_[_]" [1000, 0]) where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p" translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p" -Lemma ap_refl [comps]: +Lemma ap_refl [comp]: assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i" shows "f[refl x] \<equiv> refl (f x)" unfolding ap_def by reduce @@ -254,17 +248,16 @@ Lemma (derive) ap_funcomp: "p: x =\<^bsub>A\<^esub> y" shows "(g \<circ> f)[p] = g[f[p]]" apply (eq p) - apply (simp only: funcomp_apply_comp[symmetric]) - apply (reduce; intro) + \<guillemotright> by reduce + \<guillemotright> by reduce intro done Lemma (derive) ap_id: assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "(id A)[p] = p" apply (eq p) - apply (rewrite at "\<hole> = _" id_comp[symmetric]) - apply (rewrite at "_ = \<hole>" id_comp[symmetric]) - apply (reduce; intros) + \<guillemotright> by reduce + \<guillemotright> by reduce intro done @@ -284,7 +277,7 @@ definition transport_i ("trans") translations "trans P p" \<leftharpoondown> "CONST transport A P x y p" -Lemma transport_comp [comps]: +Lemma transport_comp [comp]: assumes "a: A" "A: U i" @@ -302,6 +295,7 @@ Lemma use_transport: "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" shows "trans P p\<inverse> u: P y" + unfolding transport_def pathinv_def by typechk method transport uses eq = (rule use_transport[OF eq]) @@ -322,7 +316,7 @@ Lemma (derive) transport_right_inv: "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)" - by (eq p) (reduce; intros) + by (eq p) (reduce; intro) Lemma (derive) transport_pathcomp: assumes @@ -335,7 +329,7 @@ Lemma (derive) transport_pathcomp: apply (eq p) focus prems vars x p apply (eq p) - apply (reduce; intros) + apply (reduce; intro) done done @@ -348,7 +342,7 @@ Lemma (derive) transport_compose_typefam: "p: x =\<^bsub>A\<^esub> y" "u: P (f x)" shows "trans (fn x. P (f x)) p u = trans P f[p] u" - by (eq p) (reduce; intros) + by (eq p) (reduce; intro) Lemma (derive) transport_function_family: assumes @@ -360,7 +354,7 @@ Lemma (derive) transport_function_family: "u: P x" "p: x =\<^bsub>A\<^esub> y" shows "trans Q p ((f x) u) = (f y) (trans P p u)" - by (eq p) (reduce; intros) + by (eq p) (reduce; intro) Lemma (derive) transport_const: assumes @@ -375,12 +369,12 @@ definition transport_const_i ("trans'_const") translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p" -Lemma transport_const_comp [comps]: +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" - unfolding transport_const_def by reduce + unfolding transport_const_def by reduce+ Lemma (derive) pathlift: assumes @@ -390,21 +384,21 @@ Lemma (derive) pathlift: "p: x =\<^bsub>A\<^esub> y" "u: P x" shows "<x, u> = <y, trans P p u>" - by (eq p) (reduce; intros) + by (eq p) (reduce; intro) definition pathlift_i ("lift") 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" -Lemma pathlift_comp [comps]: +Lemma pathlift_comp [comp]: assumes "u: P x" "x: A" "\<And>x. x: A \<Longrightarrow> P x: U i" "A: U i" shows "lift P (refl x) u \<equiv> refl <x, u>" - unfolding pathlift_def by reduce + unfolding pathlift_def by reduce+ Lemma (derive) pathlift_fst: assumes @@ -415,13 +409,7 @@ Lemma (derive) pathlift_fst: "p: x =\<^bsub>A\<^esub> y" shows "fst[lift P p u] = p" apply (eq p) - text \<open>Some rewriting needed here:\<close> - \<guillemotright> vars x y - (*Here an automatic reordering tactic would be helpful*) - apply (rewrite at x in "x = y" fst_comp[symmetric]) - prefer 4 - apply (rewrite at y in "_ = y" fst_comp[symmetric]) - done + \<guillemotright> by reduce \<guillemotright> by reduce intro done @@ -436,21 +424,21 @@ Lemma (derive) apd: "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" shows "trans P p (f x) = f y" - by (eq p) (reduce; intros; typechk) + by (eq p) (reduce; intro) definition apd_i ("apd") where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p" translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p" -Lemma dependent_map_comp [comps]: +Lemma dependent_map_comp [comp]: assumes "f: \<Prod>x: A. P x" "x: A" "A: U i" "\<And>x. x: A \<Longrightarrow> P x: U i" shows "apd f (refl x) \<equiv> refl (f x)" - unfolding apd_def by reduce + unfolding apd_def by reduce+ Lemma (derive) apd_ap: assumes @@ -467,6 +455,9 @@ 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> apply (eq r) focus prems vars x s t proof - @@ -500,17 +491,17 @@ translations "\<alpha> \<bullet>\<^sub>r\<^bsub>a\<^esub> r" \<leftharpoondown> "CONST right_whisker A a b c p q r \<alpha>" "r \<bullet>\<^sub>l\<^bsub>c\<^esub> \<alpha>" \<leftharpoondown> "CONST left_whisker A a b c p q r \<alpha>" -Lemma whisker_refl [comps]: +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>" - unfolding right_whisker_def by reduce + unfolding right_whisker_def by reduce+ -Lemma refl_whisker [comps]: +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>" - unfolding left_whisker_def by reduce + unfolding left_whisker_def by reduce+ method left_whisker = (rule left_whisker) method right_whisker = (rule right_whisker) @@ -632,12 +623,12 @@ Lemma (derive) eckmann_hilton: proof - have "\<alpha> \<bullet> \<beta> = \<alpha> \<star> \<beta>" by (rule pathcomp_eq_horiz_pathcomp) - also have [simplified comps]: ".. = \<alpha> \<star>' \<beta>" + also have [simplified comp]: ".. = \<alpha> \<star>' \<beta>" by (transport eq: \<Omega>2.horiz_pathcomp_eq_horiz_pathcomp') refl also have ".. = \<beta> \<bullet> \<alpha>" by (rule pathcomp_eq_horiz_pathcomp') finally show "\<alpha> \<bullet> \<beta> = \<beta> \<bullet> \<alpha>" - by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def) + by this (reduce add: \<Omega>2.horiz_pathcomp_def \<Omega>2.horiz_pathcomp'_def)+ qed end diff --git a/hott/More_List.thy b/hott/More_List.thy index 460dc7b..1b8034f 100644 --- a/hott/More_List.thy +++ b/hott/More_List.thy @@ -10,8 +10,8 @@ section \<open>Length\<close> definition [implicit]: "len \<equiv> ListRec ? Nat 0 (fn _ _ rec. suc rec)" experiment begin - Lemma "len [] \<equiv> ?n" by (subst comps)+ - Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comps)+ + Lemma "len [] \<equiv> ?n" by (subst comp)+ + Lemma "len [0, suc 0, suc (suc 0)] \<equiv> ?n" by (subst comp)+ end diff --git a/hott/More_Nat.thy b/hott/More_Nat.thy index 8177170..59c8d54 100644 --- a/hott/More_Nat.thy +++ b/hott/More_Nat.thy @@ -15,15 +15,15 @@ Lemma (derive) eq: shows "Nat \<rightarrow> Nat \<rightarrow> U O" apply intro focus vars n apply elim \<guillemotright> by (rule TopF) \<comment> \<open>n \<equiv> 0\<close> \<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> suc _\<close> - \<guillemotright> by (rule U_in_U) + \<guillemotright> by (rule Ui_in_USi) done \<comment> \<open>m \<equiv> suc k\<close> apply intro focus vars k k_eq n apply (elim n) \<guillemotright> by (rule BotF) \<comment> \<open>n \<equiv> 0\<close> \<guillemotright> prems vars l proof - show "k_eq l: U O" by typechk qed - \<guillemotright> by (rule U_in_U) + \<guillemotright> by (rule Ui_in_USi) done - by (rule U_in_U) + by (rule Ui_in_USi) done text \<open> diff --git a/hott/Nat.thy b/hott/Nat.thy index 4abd315..177ec47 100644 --- a/hott/Nat.thy +++ b/hott/Nat.thy @@ -38,9 +38,10 @@ where f n (NatInd (fn n. C n) c\<^sub>0 (fn k rec. f k rec) n)" lemmas - [intros] = NatF Nat_zero Nat_suc and - [elims "?n"] = NatE and - [comps] = Nat_comp_zero Nat_comp_suc + [form] = NatF and + [intro, intros] = Nat_zero Nat_suc and + [elim "?n"] = NatE and + [comp] = Nat_comp_zero Nat_comp_suc abbreviation "NatRec C \<equiv> NatInd (fn _. C)" @@ -66,12 +67,12 @@ lemma add_type [typechk]: shows "m + n: Nat" unfolding add_def by typechk -lemma add_zero [comps]: +lemma add_zero [comp]: assumes "m: Nat" shows "m + 0 \<equiv> m" unfolding add_def by reduce -lemma add_suc [comps]: +lemma add_suc [comp]: assumes "m: Nat" "n: Nat" shows "m + suc n \<equiv> suc (m + n)" unfolding add_def by reduce @@ -127,17 +128,17 @@ lemma mul_type [typechk]: shows "m * n: Nat" unfolding mul_def by typechk -lemma mul_zero [comps]: +lemma mul_zero [comp]: assumes "n: Nat" shows "n * 0 \<equiv> 0" unfolding mul_def by reduce -lemma mul_one [comps]: +lemma mul_one [comp]: assumes "n: Nat" shows "n * 1 \<equiv> n" unfolding mul_def by reduce -lemma mul_suc [comps]: +lemma mul_suc [comp]: assumes "m: Nat" "n: Nat" shows "m * suc n \<equiv> m + m * n" unfolding mul_def by reduce |