diff options
author | Josh Chen | 2020-04-02 17:57:48 +0200 |
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committer | Josh Chen | 2020-04-02 17:57:48 +0200 |
commit | c2dfffffb7586662c67e44a2d255a1a97ab0398b (patch) | |
tree | ed949f5ab7dc64541c838694b502555a275b0995 /spartan/theories | |
parent | b01b8ee0f3472cb728f09463d0620ac8b8066bcb (diff) |
Brand-spanking new version using Spartan infrastructure
Diffstat (limited to 'spartan/theories')
-rw-r--r-- | spartan/theories/Equivalence.thy | 431 | ||||
-rw-r--r-- | spartan/theories/Identity.thy | 433 | ||||
-rw-r--r-- | spartan/theories/Spartan.thy | 463 |
3 files changed, 1327 insertions, 0 deletions
diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy new file mode 100644 index 0000000..44b77dd --- /dev/null +++ b/spartan/theories/Equivalence.thy @@ -0,0 +1,431 @@ +theory Equivalence +imports Identity + +begin + +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" + +translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g" + +Lemma homotopy_type [typechk]: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x" + shows "f ~ g: U i" + unfolding homotopy_def by typechk + +Lemma (derive) homotopy_refl: + assumes + "A: U i" + "f: \<Prod>x: A. B x" + shows "f ~ f" + unfolding homotopy_def by intros + +Lemma (derive) hsym: + assumes + "f: \<Prod>x: A. B x" + "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 intros + apply (rule pathinv) + \<guillemotright> by (elim H) + \<guillemotright> by typechk + done + +lemmas homotopy_symmetric = hsym[rotated 4] + +text \<open>\<open>hsym\<close> attribute for homotopies:\<close> + +ML \<open> +structure HSym_Attr = Sym_Attr ( + struct + val name = "hsym" + val symmetry_rule = @{thm homotopy_symmetric} + end +) +\<close> + +setup \<open>HSym_Attr.setup\<close> + +Lemma (derive) htrans: + assumes + "f: \<Prod>x: A. B x" + "g: \<Prod>x: A. B x" + "h: \<Prod>x: A. B x" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "\<lbrakk>H1: f ~ g; H2: g ~ h\<rbrakk> \<Longrightarrow> f ~ h" + unfolding homotopy_def + apply intro + \<guillemotright> vars x + apply (rule pathcomp[where ?y="g x"]) + \<^item> by (elim H1) + \<^item> by (elim H2) + done + \<guillemotright> by typechk + done + +lemmas homotopy_transitive = htrans[rotated 5] + +Lemma (derive) commute_homotopy: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + "f: A \<rightarrow> B" "g: A \<rightarrow> B" + "H: homotopy A (\<lambda>_. B) 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] + apply (equality \<open>p:_\<close>) + focus vars x + apply reduce + \<comment> \<open>Here it would really be nice to have automation for transport and + propositional equality.\<close> + apply (rule use_transport[where ?y="H x \<bullet> refl (g x)"]) + \<guillemotright> by (rule pathcomp_right_refl) + \<guillemotright> by (rule pathinv) (rule pathcomp_left_refl) + \<guillemotright> by typechk + done + done + +Corollary (derive) commute_homotopy': + assumes + "A: U i" + "x: A" + "f: A \<rightarrow> A" + "H: homotopy A (\<lambda>_. A) f (id A)" + shows "H (f x) = f [H x]" +oops + +Lemma homotopy_id_left [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "homotopy_refl A f: (id B) \<circ> f ~ f" + unfolding homotopy_refl_def homotopy_def by reduce + +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 + +Lemma homotopy_funcomp_left: + assumes + "H: homotopy B C g g'" + "f: A \<rightarrow> B" + "g: \<Prod>x: B. C x" + "g': \<Prod>x: B. C x" + "A: U i" "B: U i" + "\<And>x. x: B \<Longrightarrow> C x: U i" + shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g' \<circ>\<^bsub>A\<^esub> f)" + unfolding homotopy_def + apply (intro; reduce) + apply (insert \<open>H: _\<close>[unfolded homotopy_def]) + apply (elim H) + done + +Lemma homotopy_funcomp_right: + assumes + "H: homotopy A (\<lambda>_. B) f f'" + "f: A \<rightarrow> B" + "f': A \<rightarrow> B" + "g: B \<rightarrow> C" + "A: U i" "B: U i" "C: U i" + shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g \<circ>\<^bsub>A\<^esub> f')" + unfolding homotopy_def + apply (intro; reduce) + apply (insert \<open>H: _\<close>[unfolded homotopy_def]) + apply (dest PiE, assumption) + apply (rule ap, assumption) + done + + +section \<open>Quasi-inverse and bi-invertibility\<close> + +subsection \<open>Quasi-inverses\<close> + +definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A. + homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)" + +lemma qinv_type [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "qinv A B f: U i" + unfolding qinv_def by typechk + +definition qinv_i ("qinv") + where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f" + +translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f" + +Lemma (derive) id_qinv: + assumes "A: U i" + shows "qinv (id A)" + unfolding qinv_def + apply intro defer + apply intro defer + apply (rule homotopy_id_right) + apply (rule homotopy_id_left) + done + +Lemma (derive) quasiinv_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "prf: qinv f \<Longrightarrow> qinv (fst prf)" + unfolding qinv_def + apply intro + \<guillemotright> by (rule \<open>f:_\<close>) + \<guillemotright> apply (elim "prf") + focus vars g HH + apply intro + \<^item> by reduce (snd HH) + \<^item> by reduce (fst HH) + done + done + done + +Lemma (derive) funcomp_qinv: + assumes + "A: U i" "B: U i" "C: U i" + "f: A \<rightarrow> B" "g: B \<rightarrow> C" + shows "qinv f \<rightarrow> qinv g \<rightarrow> qinv (g \<circ> f)" + apply (intros, unfold qinv_def, elims) + focus + premises hyps + vars _ _ finv _ ginv _ HfA HfB HgB HgC + + apply intro + apply (rule funcompI[where ?f=ginv and ?g=finv]) + + text \<open>Now a whole bunch of rewrites and we're done.\<close> +oops + +subsection \<open>Bi-invertible maps\<close> + +definition "biinv A B f \<equiv> + (\<Sum>g: B \<rightarrow> A. homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A)) + \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))" + +lemma biinv_type [typechk]: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "biinv A B f: U i" + unfolding biinv_def by typechk + +definition biinv_i ("biinv") + where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f" + +translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f" + +Lemma (derive) qinv_imp_biinv: + assumes + "A: U i" "B: U i" + "f: A \<rightarrow> B" + shows "?prf: qinv f \<rightarrow> biinv f" + apply intros + unfolding qinv_def biinv_def + by (rule Sig_dist_exp) + +text \<open> + Show that bi-invertible maps are quasi-inverses, as a demonstration of how to + work in this system. +\<close> + +Lemma (derive) biinv_imp_qinv: + assumes "A: U i" "B: U i" "f: A \<rightarrow> B" + shows "biinv f \<rightarrow> qinv f" + + text \<open>Split the hypothesis \<^term>\<open>biinv f\<close> into its components:\<close> + apply intro + unfolding biinv_def + apply elims + + text \<open>Name the components:\<close> + focus premises vars _ _ _ g H1 h H2 + thm \<open>g:_\<close> \<open>h:_\<close> \<open>H1:_\<close> \<open>H2:_\<close> + + text \<open> + \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>, so the proof will be a triple + \<^term>\<open><g, <?H1, ?H2>>\<close>. + \<close> + unfolding qinv_def + apply intro + \<guillemotright> by (rule \<open>g: _\<close>) + \<guillemotright> apply intro + text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close> + apply (rule \<open>H1: _\<close>) + + text \<open> + It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>, + then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof + block is used to calculate "forward". + \<close> + proof - + have "g \<circ> (id B) ~ g \<circ> f \<circ> h" + by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[hsym]) + + moreover have "g \<circ> f \<circ> h ~ (id A) \<circ> h" + by (subst funcomp_assoc[symmetric]) + (rule homotopy_funcomp_left, rule \<open>H1:_\<close>) + + ultimately have "g ~ h" + apply (rewrite to "g \<circ> (id B)" id_right[symmetric]) + apply (rewrite to "(id A) \<circ> h" id_left[symmetric]) + by (rule homotopy_transitive) (assumption, typechk) + + then have "f \<circ> g ~ f \<circ> h" + by (rule homotopy_funcomp_right) + + with \<open>H2:_\<close> + show "f \<circ> g ~ id B" + by (rule homotopy_transitive) (assumption+, typechk) + qed + done + done + +Lemma (derive) id_biinv: + "A: U i \<Longrightarrow> biinv (id A)" + by (rule qinv_imp_biinv) (rule id_qinv) + +Lemma (derive) funcomp_biinv: + assumes + "A: U i" "B: U i" "C: U i" + "f: A \<rightarrow> B" "g: B \<rightarrow> C" + shows "biinv f \<rightarrow> biinv g \<rightarrow> biinv (g \<circ> f)" + apply intros + focus vars biinv\<^sub>f biinv\<^sub>g + + text \<open>Follows from \<open>funcomp_qinv\<close>.\<close> +oops + + +section \<open>Equivalence\<close> + +text \<open> + Following the HoTT book, we first define equivalence in terms of + bi-invertibility. +\<close> + +definition equivalence (infix "\<simeq>" 110) + where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f" + +lemma equivalence_type [typechk]: + assumes "A: U i" "B: U i" + shows "A \<simeq> B: U i" + unfolding equivalence_def by typechk + +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 + +text \<open> + The following could perhaps be easier with transport (but then I think we need + univalence?)... +\<close> + +Lemma (derive) equivalence_symmetric: + assumes "A: U i" "B: U i" + shows "A \<simeq> B \<rightarrow> B \<simeq> A" + apply intros + unfolding equivalence_def + apply elim + \<guillemotright> vars _ f "prf" + apply (dest (4) biinv_imp_qinv) + apply intro + \<^item> unfolding qinv_def by (rule fst[of "(biinv_imp_qinv A B f) prf"]) + \<^item> by (rule qinv_imp_biinv) (rule quasiinv_qinv) + done + done + +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" + apply intros + unfolding equivalence_def + + text \<open>Use \<open>funcomp_biinv\<close>.\<close> +oops + +text \<open> + Equal types are equivalent. We give two proofs: the first by induction, and + the second by following the HoTT book and showing that transport is an + equivalence. +\<close> + +Lemma + assumes + "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B" + shows "A \<simeq> B" + by (equality \<open>p:_\<close>) (rule equivalence_refl) + +text \<open> + The following proof is wordy because (1) the typechecker doesn't rewrite, and + (2) we don't yet have universe automation. +\<close> + +Lemma (derive) id_imp_equiv: + assumes + "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B" + 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 (equality \<open>p:_\<close>) + \<^item> premises vars A B + apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric]) + 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) + done + \<^item> premises vars A + apply (subst transport_comp) + \<^enum> by (rule U_in_U) + \<^enum> by reduce (rule lift_universe) + \<^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]) + 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) + done + done + +(*Uncomment this to see all implicits from here on. +no_translations + "f x" \<leftharpoondown> "f `x" + "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y" + "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f" + "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p" + "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q" + "fst" \<leftharpoondown> "CONST Spartan.fst A B" + "snd" \<leftharpoondown> "CONST Spartan.snd A B" + "f[p]" \<leftharpoondown> "CONST ap A B x y f p" + "trans P p" \<leftharpoondown> "CONST transport A P x y p" + "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p" + "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u" + "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p" + "f ~ g" \<leftharpoondown> "CONST homotopy A B f g" + "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f" + "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f" +*) + + +end diff --git a/spartan/theories/Identity.thy b/spartan/theories/Identity.thy new file mode 100644 index 0000000..0edf20e --- /dev/null +++ b/spartan/theories/Identity.thy @@ -0,0 +1,433 @@ +chapter \<open>The identity type\<close> + +text \<open> + The identity type, the higher groupoid structure of types, + and type families as fibrations. +\<close> + +theory Identity +imports Spartan + +begin + +axiomatization + Id :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and + refl :: \<open>o \<Rightarrow> o\<close> and + IdInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> + +syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ =\<^bsub>_\<^esub>/ _)" [111, 0, 111] 110) + +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 + + IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and + + IdE: "\<lbrakk> + p: a =\<^bsub>A\<^esub> b; + a: A; + b: A; + \<And>x y p. \<lbrakk>p: x =\<^bsub>A\<^esub> y; x: A; y: A\<rbrakk> \<Longrightarrow> C x y p: U i; + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) + \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a b p: C a b p" and + + Id_comp: "\<lbrakk> + a: A; + \<And>x y p. \<lbrakk>x: A; y: A; p: x =\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> C x y p: U i; + \<And>x. x: A \<Longrightarrow> f x: C x x (refl x) + \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a a (refl a) \<equiv> f a" + +lemmas + [intros] = IdF IdI and + [elims] = IdE and + [comps] = Id_comp + + +section \<open>Induction\<close> + +ML_file \<open>../lib/equality.ML\<close> + +method_setup equality = \<open>Scan.lift Parse.thm >> (fn (fact, _) => fn ctxt => + CONTEXT_METHOD (K (Equality.equality_context_tac fact ctxt)))\<close> + + +section \<open>Symmetry and transitivity\<close> + +Lemma (derive) pathinv: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "y =\<^bsub>A\<^esub> x" + by (equality \<open>p:_\<close>) intro + +lemma pathinv_comp [comps]: + assumes "x: A" "A: U i" + shows "pathinv A x x (refl x) \<equiv> refl x" + unfolding pathinv_def by reduce + +Lemma (derive) pathcomp: + assumes + "A: U i" "x: A" "y: A" "z: A" + "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" + shows + "x =\<^bsub>A\<^esub> z" + apply (equality \<open>p:_\<close>) + focus premises vars x p + apply (equality \<open>p:_\<close>) + apply intro + done + done + +lemma pathcomp_comp [comps]: + assumes "a: A" "A: U i" + shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a" + unfolding pathcomp_def by reduce + +text \<open>Set up \<open>sym\<close> attribute for propositional equalities:\<close> + +ML \<open> +structure Id_Sym_Attr = Sym_Attr ( + struct + val name = "sym" + val symmetry_rule = @{thm pathinv[rotated 3]} + end +) +\<close> + +setup \<open>Id_Sym_Attr.setup\<close> + + +section \<open>Notation\<close> + +definition Id_i (infix "=" 110) + where [implicit]: "Id_i x y \<equiv> x =\<^bsub>?\<^esub> y" + +definition pathinv_i ("_\<inverse>" [1000]) + where [implicit]: "pathinv_i p \<equiv> pathinv ? ? ? p" + +definition pathcomp_i (infixl "\<bullet>" 121) + where [implicit]: "pathcomp_i p q \<equiv> pathcomp ? ? ? ? p q" + +translations + "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y" + "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p" + "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q" + + +section \<open>Basic propositional equalities\<close> + +(*TODO: Better integration of equality type directly into reasoning...*) + +Lemma (derive) pathcomp_left_refl: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "(refl x) \<bullet> p = p" + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + +Lemma (derive) pathcomp_right_refl: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "p \<bullet> (refl y) = p" + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + +Lemma (derive) pathcomp_left_inv: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "p\<inverse> \<bullet> p = refl y" + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + +Lemma (derive) pathcomp_right_inv: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "p \<bullet> p\<inverse> = refl x" + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + +Lemma (derive) pathinv_pathinv: + assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y" + shows "p\<inverse>\<inverse> = p" + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + +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" + shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r" + apply (equality \<open>p:_\<close>) + focus premises vars x p + apply (equality \<open>p:_\<close>) + focus premises vars x p + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + done + done + + +section \<open>Functoriality of functions\<close> + +Lemma (derive) ap: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "f: A \<rightarrow> B" + "p: x =\<^bsub>A\<^esub> y" + shows "f x = f y" + apply (equality \<open>p:_\<close>) + apply intro + done + +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]: + 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 + +Lemma (derive) ap_pathcomp: + assumes + "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" + shows + "f[p \<bullet> q] = f[p] \<bullet> f[q]" + apply (equality \<open>p:_\<close>) + focus premises vars x p + apply (equality \<open>p:_\<close>) + apply (reduce; intro) + done + done + +Lemma (derive) ap_pathinv: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "f: A \<rightarrow> B" + "p: x =\<^bsub>A\<^esub> y" + shows "f[p\<inverse>] = f[p]\<inverse>" + by (equality \<open>p:_\<close>) (reduce; intro) + +text \<open>The next two proofs currently use some low-level rewriting.\<close> + +Lemma (derive) ap_funcomp: + assumes + "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" + shows "(g \<circ> f)[p] = g[f[p]]" + apply (equality \<open>p:_\<close>) + apply (simp only: funcomp_apply_comp[symmetric]) + apply (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 (equality \<open>p:_\<close>) + apply (rewrite at "\<hole> = _" id_comp[symmetric]) + apply (rewrite at "_ = \<hole>" id_comp[symmetric]) + apply (reduce; intros) + done + + +section \<open>Transport\<close> + +Lemma (derive) transport: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + shows "P x \<rightarrow> P y" + by (equality \<open>p:_\<close>) intro + +definition transport_i ("trans") + where [implicit]: "trans P p \<equiv> transport ? P ? ? p" + +translations "trans P p" \<leftharpoondown> "CONST transport A P x y p" + +Lemma transport_comp [comps]: + assumes + "a: A" + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + shows "trans P (refl a) \<equiv> id (P a)" + unfolding transport_def id_def by reduce + +\<comment> \<open>TODO: Build transport automation!\<close> + +Lemma use_transport: + assumes + "p: y =\<^bsub>A\<^esub> x" + "u: P x" + "x: A" "y: A" + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + shows "trans P p\<inverse> u: P y" + by typechk + +Lemma (derive) transport_left_inv: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)" + by (equality \<open>p:_\<close>) (reduce; intro) + +Lemma (derive) transport_right_inv: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)" + by (equality \<open>p:_\<close>) (reduce; intros) + +Lemma (derive) transport_pathcomp: + assumes + "A: U i" + "\<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" + shows "trans P q (trans P p u) = trans P (p \<bullet> q) u" + apply (equality \<open>p:_\<close>) + focus premises vars x p + apply (equality \<open>p:_\<close>) + apply (reduce; intros) + done + done + +Lemma (derive) transport_compose_typefam: + assumes + "A: U i" "B: U i" + "\<And>x. x: B \<Longrightarrow> P x: U i" + "f: A \<rightarrow> B" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + "u: P (f x)" + shows "trans (\<lambda>x. P (f x)) p u = trans P f[p] u" + by (equality \<open>p:_\<close>) (reduce; intros) + +Lemma (derive) transport_function_family: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "\<And>x. x: A \<Longrightarrow> Q x: U i" + "f: \<Prod>x: A. P x \<rightarrow> Q x" + "x: A" "y: A" + "u: P x" + "p: x =\<^bsub>A\<^esub> y" + shows "trans Q p ((f x) u) = (f y) (trans P p u)" + by (equality \<open>p:_\<close>) (reduce; intros) + +Lemma (derive) transport_const: + assumes + "A: U i" "B: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + shows "\<Prod>b: B. trans (\<lambda>_. B) p b = b" + by (intro, equality \<open>p:_\<close>) (reduce; intro) + +definition transport_const_i ("trans'_const") + where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p" + +translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p" + +Lemma transport_const_comp [comps]: + 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 + +Lemma (derive) pathlift: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + "u: P x" + shows "<x, u> = <y, trans P p u>" + by (equality \<open>p:_\<close>) (reduce; intros) + +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]: + 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 + +Lemma (derive) pathlift_fst: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> P x: U i" + "x: A" "y: A" + "u: P x" + "p: x =\<^bsub>A\<^esub> y" + shows "fst[lift P p u] = p" + apply (equality \<open>p:_\<close>) + 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 intro + done + + +section \<open>Dependent paths\<close> + +Lemma (derive) apd: + assumes + "A: U i" + "\<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" + shows "trans P p (f x) = f y" + by (equality \<open>p:_\<close>) (reduce; intros; typechk) + +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]: + 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 + +Lemma (derive) apd_ap: + assumes + "A: U i" "B: U i" + "f: A \<rightarrow> B" + "x: A" "y: A" + "p: x =\<^bsub>A\<^esub> y" + shows "apd f p = trans_const B p (f x) \<bullet> f[p]" + by (equality \<open>p:_\<close>) (reduce; intro) + +end diff --git a/spartan/theories/Spartan.thy b/spartan/theories/Spartan.thy new file mode 100644 index 0000000..fb901d5 --- /dev/null +++ b/spartan/theories/Spartan.thy @@ -0,0 +1,463 @@ +text \<open>Spartan type theory\<close> + +theory Spartan +imports + Pure + "HOL-Eisbach.Eisbach" + "HOL-Eisbach.Eisbach_Tools" +keywords + "Theorem" "Lemma" "Corollary" "Proposition" :: thy_goal_stmt and + "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and + "derive" "vars":: quasi_command and + "print_coercions" :: thy_decl + +begin + + +section \<open>Preamble\<close> + +declare [[eta_contract=false]] + + +section \<open>Metatype setup\<close> + +typedecl o + + +section \<open>Judgments\<close> + +judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999) + + +section \<open>Universes\<close> + +typedecl lvl \<comment> \<open>Universe levels\<close> + +axiomatization + O :: \<open>lvl\<close> and + S :: \<open>lvl \<Rightarrow> lvl\<close> and + lt :: \<open>lvl \<Rightarrow> lvl \<Rightarrow> prop\<close> (infix "<" 900) + where + O_min: "O < S i" and + lt_S: "i < S i" and + lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k" + +axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where + U_hierarchy: "i < j \<Longrightarrow> U i: U j" and + U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j" + +lemma U_in_U: + "U i: U (S i)" + by (rule U_hierarchy, rule lt_S) + +lemma lift_universe: + "A: U i \<Longrightarrow> A: U (S i)" + by (erule U_cumulative, rule lt_S) + + +section \<open>\<Prod>-type\<close> + +axiomatization + Pi :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + lam :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + app :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(1_ `_)" [120, 121] 120) + +syntax + "_Pi" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Prod>_: _./ _)" 30) + "_lam" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30) +translations + "\<Prod>x: A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)" + "\<lambda>x: A. b" \<rightleftharpoons> "CONST lam A (\<lambda>x. b)" + +abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B" + +axiomatization where + PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and + + PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and + + PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and + + beta: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> (\<lambda>x: A. b x) `a \<equiv> b a" and + + eta: "f: \<Prod>x: A. B x \<Longrightarrow> \<lambda>x: A. f `x \<equiv> f" and + + Pi_cong: "\<lbrakk> + A: U i; + \<And>x. x: A \<Longrightarrow> B x: U i; + \<And>x. x: A \<Longrightarrow> B' x: U i; + \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x + \<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and + + lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x" + + +section \<open>\<Sum>-type\<close> + +axiomatization + Sig :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and + pair :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(2<_,/ _>)") and + SigInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close> + +syntax "_Sum" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Sum>_: _./ _)" 20) + +translations "\<Sum>x: A. B" \<rightleftharpoons> "CONST Sig A (\<lambda>x. B)" + +abbreviation Prod (infixl "\<times>" 50) + where "A \<times> B \<equiv> \<Sum>_: A. B" + +axiomatization where + SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and + + SigI: "\<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 + + SigE: "\<lbrakk> + p: \<Sum>x: A. B x; + A: U i; + \<And>x. x : A \<Longrightarrow> B x: U i; + \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i; + \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y> + \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f p: C p" and + + Sig_comp: "\<lbrakk> + a: A; + b: B a; + \<And>x. x: A \<Longrightarrow> B x: U i; + \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i; + \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y> + \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f <a, b> \<equiv> f a b" and + + Sig_cong: "\<lbrakk> + \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x; + A: U i; + \<And>x. x : A \<Longrightarrow> B x: U i; + \<And>x. x : A \<Longrightarrow> B' x: U i + \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x" + + + + + +section \<open>Proof commands\<close> + +named_theorems typechk + +ML_file \<open>../lib/lib.ML\<close> +ML_file \<open>../lib/goals.ML\<close> +ML_file \<open>../lib/focus.ML\<close> + + +section \<open>Congruence automation\<close> + +ML_file \<open>../lib/congruence.ML\<close> + + +section \<open>Methods\<close> + +ML_file \<open>../lib/elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close> + +named_theorems intros and comps +lemmas + [intros] = PiF PiI SigF SigI and + [elims] = PiE SigE and + [comps] = beta Sig_comp and + [cong] = Pi_cong lam_cong Sig_cong + +ML_file \<open>../lib/tactics.ML\<close> + +method_setup assumptions = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (TRYALL (assumptions_tac ctxt))))\<close> + +method_setup known = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (TRYALL (known_tac ctxt))))\<close> + +method_setup intro = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close> + +method_setup intros = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close> + +method_setup elim = + \<open>Scan.option Args.term >> (fn tm => fn ctxt => + SIMPLE_METHOD' (SIDE_CONDS (elims_tac tm ctxt) ctxt))\<close> + +method elims = elim+ + +method_setup typechk = + \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD ( + CHANGED (ALLGOALS (TRY o typechk_tac ctxt))))\<close> + +method_setup rule = + \<open>Attrib.thms >> (fn ths => fn ctxt => + SIMPLE_METHOD (HEADGOAL (rule_tac ths ctxt)))\<close> + +method_setup dest = + \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms + >> (fn (opt_n, ths) => fn ctxt => + SIMPLE_METHOD (HEADGOAL (dest_tac opt_n ths ctxt)))\<close> + +subsection \<open>Rewriting\<close> + +\<comment> \<open>\<open>subst\<close> method\<close> +ML_file "~~/src/Tools/misc_legacy.ML" +ML_file "~~/src/Tools/IsaPlanner/isand.ML" +ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML" +ML_file "~~/src/Tools/IsaPlanner/zipper.ML" +ML_file "../lib/eqsubst.ML" + +\<comment> \<open>\<open>rewrite\<close> method\<close> +consts rewrite_HOLE :: "'a::{}" ("\<hole>") + +lemma eta_expand: + fixes f :: "'a::{} \<Rightarrow> 'b::{}" + shows "f \<equiv> \<lambda>x. f x" . + +lemma rewr_imp: + assumes "PROP A \<equiv> PROP B" + shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)" + apply (rule Pure.equal_intr_rule) + apply (drule equal_elim_rule2[OF assms]; assumption) + apply (drule equal_elim_rule1[OF assms]; assumption) + done + +lemma imp_cong_eq: + "(PROP A \<Longrightarrow> (PROP B \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP C')) \<equiv> + ((PROP B \<Longrightarrow> PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP A \<Longrightarrow> PROP C'))" + apply (Pure.intro Pure.equal_intr_rule) + apply (drule (1) cut_rl; drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; assumption)+ + apply (drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; assumption)+ + done + +ML_file \<open>~~/src/HOL/Library/cconv.ML\<close> +ML_file \<open>../lib/rewrite.ML\<close> + +\<comment> \<open>\<open>reduce\<close> method computes terms via judgmental equalities\<close> +setup \<open> + map_theory_simpset (fn ctxt => + ctxt addSolver (mk_solver "" typechk_tac)) +\<close> + +method reduce uses add = (simp add: comps add | subst comps, reduce add: add)+ + + +section \<open>Implicit notations\<close> + +text \<open> + \<open>?\<close> is used to mark implicit arguments in definitions, while \<open>{}\<close> is expanded + immediately for elaboration in statements. +\<close> + +consts + iarg :: \<open>'a\<close> ("?") + hole :: \<open>'b\<close> ("{}") + +ML_file \<open>../lib/implicits.ML\<close> + +attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close> + +ML \<open> +val _ = Context.>> + (Syntax_Phases.term_check 1 "" (fn ctxt => map (Implicits.make_holes ctxt))) +\<close> + +text \<open>Automatically insert inhabitation judgments where needed:\<close> + +consts inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)") +translations "CONST inhabited A" \<rightharpoonup> "CONST has_type {} A" + + +section \<open>Lambda coercion\<close> + +\<comment> \<open>Coerce object lambdas to meta-lambdas\<close> +abbreviation (input) lambda :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> + where "lambda f \<equiv> \<lambda>x. f `x" + +ML_file \<open>~~/src/Tools/subtyping.ML\<close> +declare [[coercion_enabled, coercion lambda]] + +translations "f x" \<leftharpoondown> "f `x" + + +section \<open>Functions\<close> + +lemma eta_exp: + assumes "f: \<Prod>x: A. B x" + shows "f \<equiv> \<lambda>x: A. f x" + by (rule eta[symmetric]) + +lemma lift_universe_codomain: + assumes "A: U i" "f: A \<rightarrow> U j" + shows "f: A \<rightarrow> U (S j)" + apply (sub eta_exp) + apply known + apply (Pure.rule intros; rule lift_universe) + done + +subsection \<open>Function composition\<close> + +definition "funcomp A g f \<equiv> \<lambda>x: A. g `(f `x)" + +syntax + "_funcomp" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ \<circ>\<^bsub>_\<^esub>/ _)" [111, 0, 110] 110) +translations + "g \<circ>\<^bsub>A\<^esub> f" \<rightleftharpoons> "CONST funcomp A g f" + +lemma funcompI [typechk]: + assumes + "A: U i" + "B: U i" + "\<And>x. x: B \<Longrightarrow> C x: U i" + "f: A \<rightarrow> B" + "g: \<Prod>x: B. C x" + shows + "g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)" + unfolding funcomp_def by typechk + +lemma funcomp_assoc [comps]: + assumes + "f: A \<rightarrow> B" + "g: B \<rightarrow> C" + "h: \<Prod>x: C. D x" + "A: U i" + shows + "(h \<circ>\<^bsub>B\<^esub> g) \<circ>\<^bsub>A\<^esub> f \<equiv> h \<circ>\<^bsub>A\<^esub> g \<circ>\<^bsub>A\<^esub> f" + unfolding funcomp_def by reduce + +lemma funcomp_lambda_comp [comps]: + assumes + "A: U i" + "\<And>x. x: A \<Longrightarrow> b x: B" + "\<And>x. x: B \<Longrightarrow> c x: C x" + shows + "(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)" + unfolding funcomp_def by reduce + +lemma funcomp_apply_comp [comps]: + assumes + "f: A \<rightarrow> B" "g: \<Prod>x: B. C x" + "x: A" + "A: U i" "B: U i" + "\<And>x y. x: B \<Longrightarrow> C x: U i" + shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)" + unfolding funcomp_def by reduce + +text \<open>Notation:\<close> + +definition funcomp_i (infixr "\<circ>" 120) + where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>?\<^esub> f" + +translations "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f" + +subsection \<open>Identity function\<close> + +definition id where "id A \<equiv> \<lambda>x: A. x" + +lemma + idI [typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and + id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" + unfolding id_def by reduce + +lemma id_left [comps]: + assumes "f: A \<rightarrow> B" "A: U i" "B: U i" + shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f" + unfolding id_def + by (subst eta_exp[of f]) (reduce, rule eta) + +lemma id_right [comps]: + assumes "f: A \<rightarrow> B" "A: U i" "B: U i" + shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f" + unfolding id_def + by (subst eta_exp[of f]) (reduce, rule eta) + +lemma id_U [typechk]: + "id (U i): U i \<rightarrow> U i" + by typechk (fact U_in_U) + + +section \<open>Pairs\<close> + +definition "fst A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>_. A) (\<lambda>x y. x) p" +definition "snd A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>p. B (fst A B p)) (\<lambda>x y. y) p" + +lemma fst_type [typechk]: + assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A" + unfolding fst_def by typechk + +lemma fst_comp [comps]: + assumes + "a: A" + "b: B a" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst A B <a, b> \<equiv> a" + unfolding fst_def by reduce + +lemma snd_type [typechk]: + assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)" + unfolding snd_def by typechk reduce + +lemma snd_comp [comps]: + assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd A B <a, b> \<equiv> b" + unfolding snd_def by reduce + +subsection \<open>Notation\<close> + +definition fst_i ("fst") + where [implicit]: "fst \<equiv> Spartan.fst ? ?" + +definition snd_i ("snd") + where [implicit]: "snd \<equiv> Spartan.snd ? ?" + +translations + "fst" \<leftharpoondown> "CONST Spartan.fst A B" + "snd" \<leftharpoondown> "CONST Spartan.snd A B" + +subsection \<open>Projections\<close> + +Lemma fst [typechk]: + assumes + "p: \<Sum>x: A. B x" + "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "fst p: A" + by typechk + +Lemma snd [typechk]: + assumes + "p: \<Sum>x: A. B x" + "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i" + shows "snd p: B (fst p)" + by typechk + +method fst for p::o = rule fst[of p] +method snd for p::o = rule snd[of p] + +subsection \<open>Properties of \<Sigma>\<close> + +Lemma (derive) Sig_dist_exp: + assumes + "p: \<Sum>x: A. B x \<times> C x" + "A: U i" + "\<And>x. x: A \<Longrightarrow> B x: U i" + "\<And>x. x: A \<Longrightarrow> C x: U i" + shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)" + apply (elim p) + focus vars x y + apply intro + \<guillemotright> apply intro + apply assumption + apply (fst y) + done + \<guillemotright> apply intro + apply assumption + apply (snd y) + done + done + done + + +end |