diff options
| -rw-r--r-- | EqualProps.thy | 192 | ||||
| -rw-r--r-- | HoTT_Methods.thy | 67 | ||||
| -rw-r--r-- | Nat.thy | 2 | ||||
| -rw-r--r-- | Proj.thy | 32 | ||||
| -rw-r--r-- | ex/Methods.thy | 43 |
5 files changed, 215 insertions, 121 deletions
diff --git a/EqualProps.thy b/EqualProps.thy index d645fb6..3d99456 100644 --- a/EqualProps.thy +++ b/EqualProps.thy @@ -1,6 +1,6 @@ (* Title: HoTT/EqualProps.thy Author: Josh Chen - Date: Jun 2018 + Date: Aug 2018 Properties of equality. *) @@ -18,22 +18,28 @@ section \<open>Symmetry / Path inverse\<close> axiomatization inv :: "Term \<Rightarrow> Term" ("_\<inverse>" [1000] 1000) where inv_def: "inv \<equiv> \<lambda>p. ind\<^sub>= (\<lambda>x. refl(x)) p" +text " + In the proof below we begin by using path induction on \<open>p\<close> with the application of \<open>rule Equal_elim\<close>, telling Isabelle the specific substitutions to use. + The proof is finished with a standard application of the relevant type rules. +" lemma inv_type: assumes "A : U(i)" and "x : A" and "y : A" and "p: x =\<^sub>A y" shows "p\<inverse>: y =\<^sub>A x" unfolding inv_def -proof (rule Equal_elim[where ?x=x and ?y=y]) \<comment> \<open>Path induction\<close> - show "\<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> y =\<^sub>A x: U(i)" using assms(1) .. - show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" .. -qed (fact assms)+ +by (rule Equal_elim[where ?x=x and ?y=y]) (simple lem: assms) + \<comment> \<open>The type doesn't depend on \<open>p\<close> so we don't need to specify \<open>?p\<close> in the \<open>where\<close> clause above.\<close> +text " + The next proof requires explicitly telling Isabelle what to substitute on the RHS of the typing judgment after the initial application of the type rules. + (If viewing this inside Isabelle, place the cursor after the \<open>proof\<close> statement and observe the second subgoal.) +" -lemma inv_comp: assumes "A : U(i)" and "a : A" shows "(refl a)\<inverse> \<equiv> refl(a)" +lemma inv_comp: + assumes "A : U(i)" and "a : A" shows "(refl a)\<inverse> \<equiv> refl(a)" unfolding inv_def proof show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" .. - show "\<And>x. x: A \<Longrightarrow> x =\<^sub>A x: U(i)" using assms(1) .. -qed (fact assms) +qed (simple lem: assms) section \<open>Transitivity / Path composition\<close> @@ -45,67 +51,173 @@ text " 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" +text " + More complicated proofs---the nested path inductions require more explicit step-by-step rule applications: +" lemma rcompose_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 proof - show "\<And>x. x: A \<Longrightarrow> - \<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z p. ind\<^sub>= refl p) p: \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" + 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)" proof - show "\<And>x y. \<lbrakk>x: A ; y: A\<rbrakk> \<Longrightarrow> - \<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z p. ind\<^sub>= refl p) p: x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" + fix y assume 2: "y: A" + show "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" proof - { fix x y p assume asm: "x: A" "y: A" "p: x =\<^sub>A y" - show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z p. ind\<^sub>= refl p) p: \<Prod>z:A. y =[A] z \<rightarrow> x =[A] z" + fix p assume 3: "p: x =\<^sub>A y" + show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z" proof (rule Equal_elim[where ?x=x and ?y=y]) - show "\<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z: U(i)" + show "\<And>u. u: A \<Longrightarrow> \<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A z \<rightarrow> u =\<^sub>A z" proof - show "\<And>x y z. \<lbrakk>x: A; y: A; z: A\<rbrakk> \<Longrightarrow> y =\<^sub>A z \<rightarrow> x =\<^sub>A z: U(i)" - by (rule Prod_form Equal_form assms | assumption)+ - qed (rule assms) - - show "\<And>x. x: A \<Longrightarrow> \<^bold>\<lambda>z p. ind\<^sub>= refl p: \<Prod>z:A. x =\<^sub>A z \<rightarrow> x =\<^sub>A z" - proof - show "\<And>x z. \<lbrakk>x: A; z: A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>p. ind\<^sub>= refl p: x =\<^sub>A z \<rightarrow> x =\<^sub>A z" + show "\<And>u z. \<lbrakk>u: A; z: A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z" proof - { fix x z p assume asm: "x: A" "z: A" "p: x =\<^sub>A z" - show "ind\<^sub>= refl p: x =[A] z" - proof (rule Equal_elim[where ?x=x and ?y=z]) - show "\<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> x =\<^sub>A y: U(i)" by standard (rule assms) - show "\<And>x. x: A \<Longrightarrow> refl x: x =\<^sub>A x" .. - qed (fact asm)+ } - show "\<And>x z. \<lbrakk>x: A; z: A\<rbrakk> \<Longrightarrow> x =\<^sub>A z: U(i)" by standard (rule assms) - qed + fix u z q assume asm: "u: A" "z: A" "q: u =\<^sub>A z" + show "ind\<^sub>= refl q: u =\<^sub>A z" + by (rule Equal_elim[where ?x=u and ?y=z]) (simple lem: assms asm) + qed (simple lem: assms) qed (rule assms) - qed (rule asm)+ } - show "\<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> x =\<^sub>A y: U(i)" by standard (rule assms) - qed + qed (simple lem: assms 1 2 3) + qed (simple lem: assms 1 2) qed (rule assms) -qed (fact assms) +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 standard+ (rule rcompose_type assms)+ + by (simple lem: assms rcompose_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: + assumes "A: U(i)" and "a: A" + shows "rcompose`a`a`refl(a)`a`refl(a) \<equiv> refl(a)" +unfolding rcompose_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)" + proof + fix y assume 2: "y: A" + show "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" + proof + fix p assume 3: "p: x =\<^sub>A y" + show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z" + proof (rule Equal_elim[where ?x=x and ?y=y]) + show "\<And>u. u: A \<Longrightarrow> \<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A z \<rightarrow> u =\<^sub>A z" + proof + show "\<And>u z. \<lbrakk>u: A; z: A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z" + proof + fix u z assume asm: "u: A" "z: A" + show "\<And>q. q: u =\<^sub>A z \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z" + by (rule Equal_elim[where ?x=u and ?y=z]) (simple lem: assms asm) + qed (simple lem: assms) + qed (rule assms) + qed (simple lem: assms 1 2 3) + qed (simple lem: assms 1 2) + qed (rule assms) } + + show "(\<^bold>\<lambda>y p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p)`a`refl(a)`a`refl(a) \<equiv> refl(a)" + proof (subst comp) + { fix y assume 1: "y: A" + show "\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: a =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> a =\<^sub>A z)" + proof + fix p assume 2: "p: a =\<^sub>A y" + show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q) p: \<Prod>z:A. y =\<^sub>A z \<rightarrow> a =\<^sub>A z" + proof (rule Equal_elim[where ?x=a and ?y=y]) + fix u assume 3: "u: A" + show "\<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =[A] z \<rightarrow> u =[A] z" + proof + fix z assume 4: "z: A" + show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A z \<rightarrow> u =\<^sub>A z" + proof + show "\<And>q. q: u =\<^sub>A z \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z" + by (rule Equal_elim[where ?x=u and ?y=z]) (simple lem: assms 3 4) + qed (simple lem: assms 3 4) + qed fact + qed (simple lem: assms 1 2) + qed (simple lem: assms 1) } + + show "(\<^bold>\<lambda>p. ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p)`refl(a)`a`refl(a) \<equiv> refl(a)" + proof (subst comp) + { fix p assume 1: "p: a =\<^sub>A a" + show "ind\<^sub>= (\<lambda>_. \<^bold>\<lambda>z. \<^bold>\<lambda>q. ind\<^sub>= refl q) p: \<Prod>z:A. a =\<^sub>A a \<rightarrow> a =\<^sub>A z" + proof (rule Equal_elim[where ?x=a and ?y=a]) + fix u assume 2: "u: A" + show "\<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A u\<rightarrow> u =\<^sub>A z" + proof + fix z assume 3: "z: A" + show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A u \<rightarrow> u =\<^sub>A z" + proof + show "\<And>q. q: u =\<^sub>A u \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z" + by (rule Equal_elim[where ?x=u and ?y=z]) (simple lem: assms 2 3) + qed (simple lem: assms 2) + qed fact + qed (simple lem: assms 1) } + + show "(ind\<^sub>=(\<lambda>_. \<^bold>\<lambda>z q. ind\<^sub>= refl q)(refl(a)))`a`refl(a) \<equiv> refl(a)" + proof (subst comp) + { fix u assume 1: "u: A" + show "\<^bold>\<lambda>z q. ind\<^sub>= refl q: \<Prod>z:A. u =\<^sub>A u\<rightarrow> u =\<^sub>A z" + proof + fix z assume 2: "z: A" + show "\<^bold>\<lambda>q. ind\<^sub>= refl q: u =\<^sub>A u \<rightarrow> u =\<^sub>A z" + proof + show "\<And>q. q: u =\<^sub>A u \<Longrightarrow> ind\<^sub>= refl q: u =\<^sub>A z" + by (rule Equal_elim[where ?x=u and ?y=z]) (simple lem: assms 1 2) + qed (simple lem: assms 1) + qed fact } + + show "(\<^bold>\<lambda>z q. ind\<^sub>= refl q)`a`refl(a) \<equiv> refl(a)" + proof (subst comp) + { fix a assume 1: "a: A" + show "\<^bold>\<lambda>q. ind\<^sub>= refl q: a =\<^sub>A a \<rightarrow> a =\<^sub>A a" + proof + show "\<And>q. q: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl q: a =\<^sub>A a" + by (rule Equal_elim[where ?x=a and ?y=a]) (simple lem: assms 1) + qed (simple lem: assms 1) } + + show "(\<^bold>\<lambda>q. ind\<^sub>= refl q)`refl(a) \<equiv> refl(a)" + proof (subst comp) + show "\<And>p. p: a =\<^sub>A a \<Longrightarrow> ind\<^sub>= refl p: a =\<^sub>A a" + by (rule Equal_elim[where ?x=a and ?y=a]) (simple lem: assms) + show "ind\<^sub>= refl (refl(a)) \<equiv> refl(a)" + proof + show "\<And>x. x: A \<Longrightarrow> refl(x): x =\<^sub>A x" .. + qed (simple lem: assms) + qed (simple lem: assms) + qed fact + qed (simple lem: assms) + qed (simple lem: assms) + qed fact +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_comp: "\<lbrakk> + compose_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" +lemma compose_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) + 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) + lemma compose_comp: - assumes "A : U(i)" and "a : A" shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" - unfolding rcompose_def - by (simplify lems: assms) + 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) -lemmas EqualProps_rules [intro] = inv_type inv_comp compose_type compose_comp +lemmas EqualProps_rules [intro] = inv_type compose_type lemmas EqualProps_comps [comp] = inv_comp compose_comp diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy index f80b99b..e2eb4f1 100644 --- a/HoTT_Methods.thy +++ b/HoTT_Methods.thy @@ -13,69 +13,52 @@ theory HoTT_Methods begin -section \<open>Setup\<close> - -text "Import the \<open>subst\<close> method, used by \<open>simplify\<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 "~~/src/Tools/eqsubst.ML" - - section \<open>Method definitions\<close> subsection \<open>Simple type rule proof search\<close> -text "Prove type judgments \<open>A : U\<close> and inhabitation judgments \<open>a : A\<close> using rules declared [intro] and [elim], as well as additional provided lemmas. - -Can also perform typechecking, and search for elements of a type." +text " + Prove type judgments \<open>A : U\<close> and inhabitation judgments \<open>a : A\<close> using the type rules declared [intro] and [elim] (in the respective theory files), as well as additional provided lemmas. + + Can also perform typechecking, and search for elements of a type. +" -method simple uses lems = (assumption | rule lems | standard)+ +method simple uses lem = (assumption | rule lem | standard)+ subsection \<open>Wellformedness checker\<close> -text "Find a proof of any valid typing judgment derivable from a given wellformed judgment. -The named theorem \<open>wellform\<close> is declared in HoTT_Base.thy." +text " + \<open>wellformed\<close> finds a proof of any valid typing judgment derivable from the judgments passed as \<open>lem\<close>. + The named theorem \<open>wellform\<close> is declared in HoTT_Base.thy. +" -method wellformed uses jdgmt declares wellform = +method wellformed' uses jmt declares wellform = match wellform in rl: "PROP ?P" \<Rightarrow> \<open>( - catch \<open>rule rl[OF jdgmt]\<close> \<open>fail\<close> | - catch \<open>wellformed jdgmt: rl[OF jdgmt]\<close> \<open>fail\<close> + catch \<open>rule rl[OF jmt]\<close> \<open>fail\<close> | + catch \<open>wellformed' jmt: rl[OF jmt]\<close> \<open>fail\<close> )\<close> +method wellformed uses lem = + match lem in lem: "?X : ?Y" \<Rightarrow> \<open>wellformed' jmt: lem\<close> -subsection \<open>Derivation search\<close> - -text "Combine \<open>simple\<close>, \<open>wellformed\<close> and the universe hierarchy rules to search for derivations of judgments. -\<open>wellformed\<close> uses the facts passed as \<open>lems\<close> to derive any required typing judgments. -Definitions passed as \<open>unfolds\<close> are unfolded throughout." - -method derive uses lems unfolds = ( - unfold unfolds | - simple lems: lems | - match lems in lem: "?X : ?Y" \<Rightarrow> \<open>wellformed jdgmt: lem\<close> | - rule U_hierarchy (*| - (rule U_cumulative, simple lems: lems) | - (rule U_cumulative, match lems in lem: "?X : ?Y" \<Rightarrow> \<open>wellformed jdgmt: lem\<close>)*) - )+ +subsection \<open>Derivation search\<close> -subsection \<open>Simplification\<close> +text " Combine \<open>simple\<close> and \<open>wellformed\<close> to search for derivations of judgments." -text "The methods \<open>rsimplify\<close> and \<open>simplify\<close> search for lambda applications to simplify and are suitable for solving definitional equalities, as well as harder proofs where \<open>derive\<close> fails. +method derive uses lem = (simple lem: lem | wellformed lem: lem)+ -It is however not true that these methods are strictly stronger; if they fail to find a suitable substitution they will fail where \<open>derive\<close> may succeed. -\<open>simplify\<close> is allowed to use the Pure Simplifier and is hence unsuitable for simplifying lambda expressions using only the type rules. -In this case use \<open>rsimplify\<close> instead." +subsection \<open>Substitution and simplification\<close> -method rsimplify uses lems = (subst (0) comp, derive lems: lems)+ - \<comment> \<open>\<open>subst\<close> performs an equational rewrite according to some computation rule declared as [comp], after which \<open>derive\<close> attempts to solve any new assumptions that arise.\<close> +text "Import the \<open>subst\<close> method, used for substituting definitional equalities." -method simplify uses lems = (simp add: lems | rsimplify lems: lems)+ +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 "~~/src/Tools/eqsubst.ML" end
\ No newline at end of file @@ -42,8 +42,8 @@ and n: \<nat> \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(succ n) \<equiv> f(n)(ind\<^sub>\<nat> f a n)" -lemmas Nat_rules [intro] = Nat_form Nat_intro1 Nat_intro2 Nat_elim Nat_comp1 Nat_comp2 lemmas Nat_intro = Nat_intro1 Nat_intro2 +lemmas Nat_rules [intro] = Nat_form Nat_intro Nat_elim Nat_comp1 Nat_comp2 lemmas Nat_comps [comp] = Nat_comp1 Nat_comp2 @@ -21,17 +21,14 @@ text "Typing judgments and computation rules for the dependent and non-dependent lemma fst_type: assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "fst(p): A" -unfolding fst_def -proof - show "A: U(i)" using assms(1) by (rule Sum_wellform) -qed (fact assms | assumption)+ +unfolding fst_def by (derive lem: assms) lemma fst_comp: assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "a: A" and "b: B(a)" shows "fst(<a,b>) \<equiv> a" unfolding fst_def proof - show "\<And>x. x: A \<Longrightarrow> x: A" . + show "a: A" and "b: B(a)" by fact+ qed (rule assms)+ @@ -39,20 +36,16 @@ lemma snd_type: assumes "\<Sum>x:A. B(x): U(i)" and "p: \<Sum>x:A. B(x)" shows "snd(p): B(fst p)" unfolding snd_def proof - show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" - proof - - have "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> fst(p): A" using assms(1) by (rule fst_type) - with assms(1) show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (rule Sum_wellform) - qed - + show "\<And>p. p: \<Sum>x:A. B(x) \<Longrightarrow> B(fst p): U(i)" by (derive lem: assms fst_type) + fix x y assume asm: "x: A" "y: B(x)" show "y: B(fst <x,y>)" proof (subst fst_comp) - show "A: U(i)" using assms(1) by (rule Sum_wellform) - show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" using assms(1) by (rule Sum_wellform) - qed (rule asm)+ -qed (fact assms) + show "A: U(i)" by (wellformed lem: assms(1)) + show "\<And>x. x: A \<Longrightarrow> B(x): U(i)" by (wellformed lem: assms(1)) + qed fact+ +qed fact lemma snd_comp: @@ -60,13 +53,12 @@ lemma snd_comp: unfolding snd_def proof show "\<And>x y. y: B(x) \<Longrightarrow> y: B(x)" . - show "a: A" by (fact assms) - show "b: B(a)" by (fact assms) - show *: "B(a): U(i)" using assms(3) by (rule assms(2)) - show "B(a): U(i)" by (fact *) -qed + show "a: A" by fact + show "b: B(a)" by fact +qed (simple lem: assms) +lemmas Proj_types [intro] = fst_type snd_type lemmas Proj_comps [intro] = fst_comp snd_comp diff --git a/ex/Methods.thy b/ex/Methods.thy index d174dde..b0c5f92 100644 --- a/ex/Methods.thy +++ b/ex/Methods.thy @@ -1,6 +1,6 @@ (* Title: HoTT/ex/Methods.thy Author: Josh Chen - Date: Jul 2018 + Date: Aug 2018 HoTT method usage examples *) @@ -9,33 +9,40 @@ theory Methods imports "../HoTT" begin + lemma - assumes "A : U" "B: A \<rightarrow> U" "\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U" - shows "\<Sum>x:A. \<Prod>y:B x. \<Sum>z:C x y. \<Prod>w:A. x =\<^sub>A w : U" -by (simple lems: assms) + assumes "A : U(i)" "B: A \<longrightarrow> U(i)" "\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)" + shows "\<Sum>x:A. \<Prod>y:B x. \<Sum>z:C x y. \<Prod>w:A. x =\<^sub>A w : U(i)" +by (simple lem: assms) lemma - assumes "f : \<Sum>x:A. \<Prod>y: B x. \<Sum>z: C x y. D x y z" + assumes "\<Sum>x:A. \<Prod>y: B x. \<Sum>z: C x y. D x y z: U(i)" shows - "A : U" and - "B: A \<rightarrow> U" and - "\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U" and - "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<rightarrow> U" + "A : U(i)" and + "B: A \<longrightarrow> U(i)" and + "\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)" and + "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<longrightarrow> U(i)" proof - - show "A : U" by (wellformed jdgmt: assms) - show "B: A \<rightarrow> U" by (wellformed jdgmt: assms) - show "\<And>x. x : A \<Longrightarrow> C x: B x \<rightarrow> U" by (wellformed jdgmt: assms) - show "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<rightarrow> U" by (wellformed jdgmt: assms) + show + "A : U(i)" and + "B: A \<longrightarrow> U(i)" and + "\<And>x. x : A \<Longrightarrow> C x: B x \<longrightarrow> U(i)" and + "\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> D x y: C x y \<longrightarrow> U(i)" + by (derive lem: assms) qed -text "Typechecking:" +text "Typechecking and constructing inhabitants:" \<comment> \<open>Correctly determines the type of the pair\<close> -schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> (a, b) : ?A" by simple - -\<comment> \<open>Finds element\<close> -schematic_goal "\<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> ?x : A \<times> B" by simple +schematic_goal "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> <a, b> : ?A" +by simple + +\<comment> \<open>Finds pair (too easy).\<close> +schematic_goal "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> ?x : A \<times> B" +apply (rule Sum_intro) +apply assumption+ +done end
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