diff options
Diffstat (limited to '')
| -rw-r--r-- | ex/HoTT book/Ch1.thy | 47 | ||||
| -rw-r--r-- | ex/Methods.thy | 73 | ||||
| -rw-r--r-- | ex/Synthesis.thy | 94 |
3 files changed, 81 insertions, 133 deletions
diff --git a/ex/HoTT book/Ch1.thy b/ex/HoTT book/Ch1.thy index a577fca..263f43d 100644 --- a/ex/HoTT book/Ch1.thy +++ b/ex/HoTT book/Ch1.thy @@ -1,55 +1,50 @@ -(* Title: HoTT/ex/HoTT book/Ch1.thy - Author: Josh Chen +(* +Title: ex/HoTT book/Ch1.thy +Author: Josh Chen +Date: 2018 A formalization of some content of Chapter 1 of the Homotopy Type Theory book. *) theory Ch1 - imports "../../HoTT" +imports "../../HoTT" + begin chapter \<open>HoTT Book, Chapter 1\<close> -section \<open>1.6 Dependent pair types (\<Sigma>-types)\<close> +section \<open>1.6 Dependent pair types (\<Sum>-types)\<close> -text "Propositional uniqueness principle:" +paragraph \<open>Propositional uniqueness principle.\<close> schematic_goal - assumes "(\<Sum>x:A. B(x)): U(i)" and "p: \<Sum>x:A. B(x)" - shows "?a: p =[\<Sum>x:A. B(x)] <fst p, snd p>" + assumes "A: U i" and "B: A \<longrightarrow> U i" and "p: \<Sum>x:A. B x" + shows "?a: p =[\<Sum>x:A. B x] <fst p, snd p>" -text "Proof by induction on \<open>p: \<Sum>x:A. B(x)\<close>:" +text \<open>Proof by induction on @{term "p: \<Sum>x:A. B x"}:\<close> proof (rule Sum_elim[where ?p=p]) - text "We just need to prove the base case; the rest will be taken care of automatically." - - fix x y assume asm: "x: A" "y: B(x)" show - "refl(<x,y>): <x,y> =[\<Sum>x:A. B(x)] <fst <x,y>, snd <x,y>>" - proof (subst (0 1) comp) - text " - The computation rules for \<open>fst\<close> and \<open>snd\<close> require that \<open>x\<close> and \<open>y\<close> have appropriate types. - The automatic proof methods have trouble picking the appropriate types, so we state them explicitly, - " - show "x: A" and "y: B(x)" by (fact asm)+ - - text "...twice, once each for the substitutions of \<open>fst\<close> and \<open>snd\<close>." - show "x: A" and "y: B(x)" by (fact asm)+ + text \<open>We prove the base case.\<close> + fix x y assume asm: "x: A" "y: B x" show "refl <x,y>: <x,y> =[\<Sum>x:A. B x] <fst <x,y>, snd <x,y>>" + proof compute + show "x: A" and "y: B x" by (fact asm)+ \<comment> \<open>Hint the correct types.\<close> + text \<open>And now @{method derive} takes care of the rest. +\<close> qed (derive lems: assms asm) - qed (derive lems: assms) section \<open>Exercises\<close> -text "Exercise 1.13" +paragraph \<open>Exercise 1.13\<close> -abbreviation "not" ("\<not>'(_')") where "\<not>(A) \<equiv> A \<rightarrow> \<zero>" +abbreviation "not" ("\<not>_") where "\<not>A \<equiv> A \<rightarrow> \<zero>" text "This proof requires the use of universe cumulativity." -proposition assumes "A: U(i)" shows "\<^bold>\<lambda>f. f`(inr(\<^bold>\<lambda>a. f`inl(a))): \<not>(\<not>(A + \<not>(A)))" -by (derive lems: assms U_cumulative[where ?A=\<zero> and ?i=O and ?j=i]) +proposition assumes "A: U i" shows "\<^bold>\<lambda>f. f`(inr(\<^bold>\<lambda>a. f`(inl a))): \<not>(\<not>(A + \<not>A))" +by (derive lems: assms) end diff --git a/ex/Methods.thy b/ex/Methods.thy index c78af14..09975b0 100644 --- a/ex/Methods.thy +++ b/ex/Methods.thy @@ -1,76 +1,49 @@ -(* Title: HoTT/ex/Methods.thy - Author: Josh Chen +(* +Title: ex/Methods.thy +Author: Joshua Chen +Date: 2018 -HoTT method usage examples +Basic HoTT method usage examples. *) theory Methods - imports "../HoTT" -begin +imports "../HoTT" +begin -text "Wellformedness results, metatheorems written into the object theory using the wellformedness rules." lemma 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 (routine lems: assms) - - -lemma - assumes "\<Sum>x:A. \<Prod>y: B x. \<Sum>z: C x y. D x y z: U(i)" - shows - "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(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 lems: assms) -qed - - -text "Typechecking and constructing inhabitants:" + shows "\<Sum>x:A. \<Prod>y:B x. \<Sum>z:C x y. \<Prod>w:A. x =\<^sub>A w: U(i)" +by (routine add: assms) -\<comment> \<open>Correctly determines the type of the pair\<close> +\<comment> \<open>Correctly determines the type of the pair.\<close> schematic_goal "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> <a, b> : ?A" by routine \<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 (rule intros) apply assumption+ done - -text " - Function application. - The proof methods are not yet automated as well as I would like; we still often have to explicitly specify types. -" - -lemma - assumes "A: U(i)" "a: A" - shows "(\<^bold>\<lambda>x. <x,0>)`a \<equiv> <a,0>" +\<comment> \<open>Function application. We still often have to explicitly specify types.\<close> +lemma "\<lbrakk>A: U i; a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. <x,0>)`a \<equiv> <a,0>" proof compute show "\<And>x. x: A \<Longrightarrow> <x,0>: A \<times> \<nat>" by routine -qed (routine lems: assms) - +qed -lemma - assumes "A: U(i)" "B: A \<longrightarrow> U(i)" "a: A" "b: B(a)" - shows "(\<^bold>\<lambda>x y. <x,y>)`a`b \<equiv> <a,b>" -proof compute - show "\<And>x. x: A \<Longrightarrow> \<^bold>\<lambda>y. <x,y>: \<Prod>y:B(x). \<Sum>x:A. B(x)" by (routine lems: assms) +text \<open> +The proof below takes a little more work than one might expect; it would be nice to have a one-line method or proof. +\<close> - show "(\<^bold>\<lambda>b. <a,b>)`b \<equiv> <a, b>" +lemma "\<lbrakk>A: U i; B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. <x,y>)`a`b \<equiv> <a,b>" +proof (compute, routine) + show "\<lbrakk>A: U i; B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>y. <a,y>)`b \<equiv> <a,b>" proof compute - show "\<And>b. b: B(a) \<Longrightarrow> <a, b>: \<Sum>x:A. B(x)" by (routine lems: assms) - qed fact -qed fact + show "\<And>b. \<lbrakk>A: U i; B: A \<longrightarrow> U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B x" by routine + qed +qed end diff --git a/ex/Synthesis.thy b/ex/Synthesis.thy index a5e77ec..3ee973c 100644 --- a/ex/Synthesis.thy +++ b/ex/Synthesis.thy @@ -1,78 +1,58 @@ -(* Title: HoTT/ex/Synthesis.thy - Author: Josh Chen +(* +Title: ex/Synthesis.thy +Author: Joshua Chen +Date: 2018 -Examples of synthesis. +Examples of synthesis *) theory Synthesis - imports "../HoTT" +imports "../HoTT" + begin section \<open>Synthesis of the predecessor function\<close> -text " - In this example we construct, with the help of Isabelle, a predecessor function for the natural numbers. - - This is also done in \<open>CTT.thy\<close>; there the work is easier as the equality type is extensional, and also the methods are set up a little more nicely. -" +text \<open> +In this example we construct a predecessor function for the natural numbers. +This is also done in @{file "~~/src/CTT/ex/Synthesis.thy"}, there the work is much easier as the equality type is extensional. +\<close> -text "First we show that the property we want is well-defined." +text \<open>First we show that the property we want is well-defined.\<close> -lemma pred_welltyped: "\<Sum>pred:\<nat>\<rightarrow>\<nat> . ((pred`0) =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. (pred`(succ n)) =\<^sub>\<nat> n): U(O)" +lemma pred_welltyped: "\<Sum>pred: \<nat>\<rightarrow>\<nat>. (pred`0 =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. pred`(succ n) =\<^sub>\<nat> n): U O" by routine -text " - Now we look for an inhabitant of this type. - Observe that we're looking for a lambda term \<open>pred\<close> satisfying \<open>(pred`0) =\<^sub>\<nat> 0\<close> and \<open>\<Prod>n:\<nat>. (pred`(succ n)) =\<^sub>\<nat> n\<close>. - What if we require **definitional** equality instead of just propositional equality? -" +text \<open> +Now we look for an inhabitant of this type. +Observe that we're looking for a lambda term @{term pred} satisfying @{term "pred`0 =\<^sub>\<nat> 0"} and @{term "\<Prod>n:\<nat>. pred`(succ n) =\<^sub>\<nat> n"}. +What if we require *definitional* instead of just propositional equality? +\<close> schematic_goal "?p`0 \<equiv> 0" and "\<And>n. n: \<nat> \<Longrightarrow> (?p`(succ n)) \<equiv> n" apply compute prefer 4 apply compute -prefer 3 apply compute -apply (rule Nat_routine Nat_elim | compute | assumption)+ -done - -text " - The above proof finds a candidate, namely \<open>\<^bold>\<lambda>n. ind\<^sub>\<nat> (\<lambda>a b. a) 0 n\<close>. - We prove this has the required type and properties. -" - -definition pred :: Term where "pred \<equiv> \<^bold>\<lambda>n. ind\<^sub>\<nat> (\<lambda>a b. a) 0 n" - -lemma pred_type: "pred: \<nat> \<rightarrow> \<nat>" unfolding pred_def by routine - -lemma pred_props: "<refl(0), \<^bold>\<lambda>n. refl(n)>: ((pred`0) =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. (pred`(succ n)) =\<^sub>\<nat> n)" -proof (routine lems: pred_type) - have *: "pred`0 \<equiv> 0" unfolding pred_def - proof compute - show "\<And>n. n: \<nat> \<Longrightarrow> ind\<^sub>\<nat> (\<lambda>a b. a) 0 n: \<nat>" by routine - show "ind\<^sub>\<nat> (\<lambda>a b. a) 0 0 \<equiv> 0" - proof compute - show "\<nat>: U(O)" .. - qed routine - qed rule - then show "refl(0): (pred`0) =\<^sub>\<nat> 0" by (subst *) routine - - show "\<^bold>\<lambda>n. refl(n): \<Prod>n:\<nat>. (pred`(succ(n))) =\<^sub>\<nat> n" - unfolding pred_def proof - show "\<And>n. n: \<nat> \<Longrightarrow> refl(n): ((\<^bold>\<lambda>n. ind\<^sub>\<nat> (\<lambda>a b. a) 0 n)`succ(n)) =\<^sub>\<nat> n" - proof compute - show "\<And>n. n: \<nat> \<Longrightarrow> ind\<^sub>\<nat> (\<lambda>a b. a) 0 n: \<nat>" by routine - show "\<And>n. n: \<nat> \<Longrightarrow> refl(n): ind\<^sub>\<nat> (\<lambda>a b. a) 0 (succ n) =\<^sub>\<nat> n" - proof compute - show "\<nat>: U(O)" .. - qed routine - qed rule - qed rule -qed - -theorem - "<pred, <refl(0), \<^bold>\<lambda>n. refl(n)>>: \<Sum>pred:\<nat>\<rightarrow>\<nat> . ((pred`0) =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. (pred`(succ n)) =\<^sub>\<nat> n)" -by (routine lems: pred_welltyped pred_type pred_props) +apply (rule Nat_routine | compute)+ +oops +\<comment> \<open>Something in the original proof broke when I revamped the theory. The completion of this derivation is left as an exercise to the reader.\<close> + +text \<open> +The above proof finds a candidate, namely @{term "\<lambda>n. ind\<^sub>\<nat> (\<lambda>a b. a) 0 n"}. +We prove this has the required type and properties. +\<close> + +definition pred :: t where "pred \<equiv> \<^bold>\<lambda>n. ind\<^sub>\<nat> (\<lambda>a b. a) 0 n" + +lemma pred_type: "pred: \<nat> \<rightarrow> \<nat>" +unfolding pred_def by routine + +lemma pred_props: "<refl 0, \<^bold>\<lambda>n. refl n>: (pred`0 =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. pred`(succ n) =\<^sub>\<nat> n)" +unfolding pred_def by derive + +theorem "<pred, <refl(0), \<^bold>\<lambda>n. refl(n)>>: \<Sum>pred:\<nat>\<rightarrow>\<nat> . ((pred`0) =\<^sub>\<nat> 0) \<times> (\<Prod>n:\<nat>. (pred`(succ n)) =\<^sub>\<nat> n)" +by (derive lems: pred_type pred_props) end |