diff options
| -rw-r--r-- | Coprod.thy | 1 | ||||
| -rw-r--r-- | Empty.thy | 1 | ||||
| -rw-r--r-- | Equal.thy | 1 | ||||
| -rw-r--r-- | HoTT.thy | 9 | ||||
| -rw-r--r-- | HoTT_Base.thy | 29 | ||||
| -rw-r--r-- | HoTT_Methods.thy | 25 | ||||
| -rw-r--r-- | Nat.thy | 1 | ||||
| -rw-r--r-- | Prod.thy | 1 | ||||
| -rw-r--r-- | Sum.thy | 1 | ||||
| -rw-r--r-- | Unit.thy | 1 | ||||
| -rw-r--r-- | ex/HoTT book/Ch1.thy | 47 | ||||
| -rw-r--r-- | ex/Methods.thy | 73 | ||||
| -rw-r--r-- | ex/Synthesis.thy | 94 | ||||
| -rw-r--r-- | tests/Subgoal.thy | 63 | ||||
| -rw-r--r-- | tests/Test.thy | 103 |
15 files changed, 177 insertions, 273 deletions
@@ -42,6 +42,7 @@ where \<And>x. x: A \<Longrightarrow> c x: C (inl x); \<And>y. y: B \<Longrightarrow> d y: C (inr y) \<rbrakk> \<Longrightarrow> ind\<^sub>+ (\<lambda>x. c x) (\<lambda>y. d y) (inr b) \<equiv> d b" +lemmas Coprod_form [form] lemmas Coprod_routine [intro] = Coprod_form Coprod_intro_inl Coprod_intro_inr Coprod_elim lemmas Coprod_comp [comp] = Coprod_comp_inl Coprod_comp_inr @@ -20,6 +20,7 @@ where Empty_elim: "\<lbrakk>a: \<zero>; C: \<zero> \<longrightarrow> U i\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero> a: C a" +lemmas Empty_form [form] lemmas Empty_routine [intro] = Empty_form Empty_elim @@ -44,6 +44,7 @@ axiomatization where \<And>x. x: A \<Longrightarrow> f x: C x x (refl x); \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i \<rbrakk> \<Longrightarrow> ind\<^sub>= (\<lambda>x. f x) (refl a) \<equiv> f a" +lemmas Equal_form [form] lemmas Equal_routine [intro] = Equal_form Equal_intro Equal_elim lemmas Equal_comp [comp] @@ -1,5 +1,7 @@ -(* Title: HoTT/HoTT.thy - Author: Josh Chen +(* +Title: HoTT.thy +Author: Joshua Chen +Date: 2018 Homotopy type theory *) @@ -26,8 +28,7 @@ Proj begin -lemmas forms = - Nat_form Prod_form Sum_form Coprod_form Equal_form Unit_form Empty_form +text \<open>Rule bundles:\<close> lemmas intros = Nat_intro_0 Nat_intro_succ Prod_intro Sum_intro Equal_intro Coprod_intro_inl Coprod_intro_inr Unit_intro diff --git a/HoTT_Base.thy b/HoTT_Base.thy index 7453883..2ad0ac5 100644 --- a/HoTT_Base.thy +++ b/HoTT_Base.thy @@ -7,7 +7,7 @@ Basic setup of a homotopy type theory object logic with a cumulative Russell-sty *) theory HoTT_Base -imports Pure "HOL-Eisbach.Eisbach" +imports Pure begin @@ -18,17 +18,14 @@ typedecl t \<comment> \<open>Type of object types and terms\<close> typedecl ord \<comment> \<open>Type of meta-level numerals\<close> axiomatization - O :: ord and + O :: ord and Suc :: "ord \<Rightarrow> ord" and - lt :: "[ord, ord] \<Rightarrow> prop" (infix "<" 999) + lt :: "[ord, ord] \<Rightarrow> prop" (infix "<" 999) and + leq :: "[ord, ord] \<Rightarrow> prop" (infix "\<le>" 999) where lt_Suc [intro]: "n < (Suc n)" and lt_trans [intro]: "\<lbrakk>m1 < m2; m2 < m3\<rbrakk> \<Longrightarrow> m1 < m3" and - Suc_monotone [simp]: "m < n \<Longrightarrow> (Suc m) < (Suc n)" - -method proveSuc = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+) - -text \<open>Method @{method proveSuc} proves statements of the form \<open>n < (Suc (... (Suc n) ...))\<close>.\<close> + leq_min [intro]: "O \<le> n" section \<open>Judgment\<close> @@ -42,15 +39,15 @@ axiomatization U :: "ord \<Rightarrow> t" where U_hierarchy: "i < j \<Longrightarrow> U i: U j" and - U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j" + U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j" and + U_cumulative': "\<lbrakk>A: U i; i \<le> j\<rbrakk> \<Longrightarrow> A: U j" text \<open> Using method @{method rule} with @{thm U_cumulative} is unsafe, if applied blindly it will typically lead to non-termination. One should instead use @{method elim}, or instantiate @{thm U_cumulative} suitably. -\<close> -method cumulativity = (elim U_cumulative, proveSuc) \<comment> \<open>Proves \<open>A: U i \<Longrightarrow> A: U (Suc (... (Suc i) ...))\<close>\<close> -method hierarchy = (rule U_hierarchy, proveSuc) \<comment> \<open>Proves \<open>U i: U (Suc (... (Suc i) ...))\<close>\<close> +@{thm U_cumulative'} is an alternative rule used by the method \<open>lift\<close> in @{file HoTT_Methods.thy}. +\<close> section \<open>Type families\<close> @@ -68,11 +65,15 @@ type_synonym tf = "t \<Rightarrow> t" \<comment> \<open>Type of type families\< section \<open>Named theorems\<close> named_theorems comp +named_theorems form text \<open> Declare named theorems to be used by proof methods defined in @{file HoTT_Methods.thy}. -@{attribute comp} declares computation rules. -These are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting. + +@{attribute comp} declares computation rules, which are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting. + +@{attribute form} declares type formation rules. +These are mainly used by the \<open>cumulativity\<close> method, which lifts types into higher universes. \<close> (* Todo: Set up the Simplifier! *) diff --git a/HoTT_Methods.thy b/HoTT_Methods.thy index 8929f69..f0cee6c 100644 --- a/HoTT_Methods.thy +++ b/HoTT_Methods.thy @@ -12,6 +12,26 @@ imports HoTT_Base "HOL-Eisbach.Eisbach" "HOL-Eisbach.Eisbach_Tools" begin +section \<open>Handling universes\<close> + +method provelt = (rule lt_Suc | (rule lt_trans, (rule lt_Suc)+)+) + +method hierarchy = (rule U_hierarchy, provelt) + +method cumulativity declares form = ( + ((elim U_cumulative' | (rule U_cumulative', rule form)), rule leq_min) | + ((elim U_cumulative | (rule U_cumulative, rule form)), provelt) +) + +text \<open> +Methods @{method provelt}, @{method hierarchy}, and @{method cumulativity} prove statements of the form +\<^item> \<open>n < (Suc (... (Suc n) ...))\<close>, +\<^item> \<open>U i: U (Suc (... (Suc i) ...))\<close>, and +\<^item> @{prop "A: U i \<Longrightarrow> A: U j"}, where @{prop "i \<le> j"} +respectively. +\<close> + + section \<open>Deriving typing judgments\<close> method routine uses add = (assumption | rule add | rule)+ @@ -38,14 +58,15 @@ Method @{method compute} performs single-step simplifications, using any rules d Premises of the rule(s) applied are added as new subgoals. \<close> + section \<open>Derivation search\<close> text \<open> Combine @{method routine} and @{method compute} to search for derivations of judgments. -Also handle universes using methods @{method hierarchy} and @{method cumulativity} defined in @{file HoTT_Base.thy}. +Also handle universes using @{method hierarchy} and @{method cumulativity}. \<close> -method derive uses lems = (routine add: lems | compute comp: lems | cumulativity | hierarchy)+ +method derive uses lems = (routine add: lems | compute comp: lems | cumulativity form: lems | hierarchy)+ section \<open>Induction\<close> @@ -41,6 +41,7 @@ where C: \<nat> \<longrightarrow> U i; \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> (\<lambda>n c. f n c) a (succ n) \<equiv> f n (ind\<^sub>\<nat> f a n)" +lemmas Nat_form [form] lemmas Nat_routine [intro] = Nat_form Nat_intro_0 Nat_intro_succ Nat_elim lemmas Nat_comps [comp] = Nat_comp_0 Nat_comp_succ @@ -61,6 +61,7 @@ Note that this is a separate rule from function extensionality. Note that the bold lambda symbol \<open>\<^bold>\<lambda>\<close> used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation). \<close> +lemmas Prod_form [form] lemmas Prod_routine [intro] = Prod_form Prod_intro Prod_elim lemmas Prod_comps [comp] = Prod_comp Prod_uniq @@ -51,6 +51,7 @@ axiomatization where Sum_form_eq: "\<lbrakk>A: U i; B: A \<longrightarrow> U i; C: A \<longrightarrow> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x \<equiv> \<Sum>x:A. C x" +lemmas Sum_form [form] lemmas Sum_routine [intro] = Sum_form Sum_intro Sum_elim lemmas Sum_comp [comp] @@ -25,6 +25,7 @@ where Unit_comp: "\<lbrakk>c: C \<star>; C: \<one> \<longrightarrow> U i\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> c \<star> \<equiv> c" +lemmas Unit_form [form] lemmas Unit_routine [intro] = Unit_form Unit_intro Unit_elim lemmas Unit_comp [comp] 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 diff --git a/tests/Subgoal.thy b/tests/Subgoal.thy deleted file mode 100644 index 82d7e5d..0000000 --- a/tests/Subgoal.thy +++ /dev/null @@ -1,63 +0,0 @@ -theory Subgoal - imports "../HoTT" -begin - - -text " - Proof of \<open>rpathcomp_type\<close> (see EqualProps.thy) in apply-style. - Subgoaling can be used to fix variables and apply the elimination rules. -" - -lemma rpathcomp_type: - assumes "A: U(i)" - shows "rpathcomp: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" -unfolding rpathcomp_def -apply standard - subgoal premises 1 for x \<comment> \<open>\<open>subgoal\<close> is the proof script version of \<open>fix-assume-show\<close>.\<close> - apply standard - subgoal premises 2 for y - apply standard - subgoal premises 3 for p - apply (rule Equal_elim[where ?x=x and ?y=y and ?A=A]) - \<comment> \<open>specifying \<open>?A=A\<close> is crucial here to prevent the next \<open>subgoal\<close> from binding a schematic ?A which should be instantiated to \<open>A\<close>.\<close> - prefer 4 - apply standard - apply (rule Prod_intro) - subgoal premises 4 for u z q - apply (rule Equal_elim[where ?x=u and ?y=z]) - apply (routine lems: assms 4) - done - apply (routine lems: assms 1 2 3) - done - apply (routine lems: assms 1 2) - done - apply fact - done -apply fact -done - - -text " - \<open>subgoal\<close> converts schematic variables to fixed free variables, making it unsuitable for use in \<open>schematic_goal\<close> proofs. - This is the same thing as being unable to start a ``sub schematic-goal'' inside an ongoing proof. - - This is a problem for syntheses which need to use induction (elimination rules), as these often have to be applied to fixed variables, while keeping any schematic variables intact. -" - -schematic_goal rpathcomp_synthesis: - assumes "A: U(i)" - shows "?a: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)" - -text " - Try (and fail) to synthesize the constant for path composition, following the proof of \<open>rpathcomp_type\<close> below. -" - -apply (rule intros) - apply (rule intros) - apply (rule intros) - subgoal 123 for x y p - apply (rule Equal_elim[where ?x=x and ?y=y and ?A=A]) - oops - - -end diff --git a/tests/Test.thy b/tests/Test.thy index de65dbd..6f9f996 100644 --- a/tests/Test.thy +++ b/tests/Test.thy @@ -1,121 +1,110 @@ -(* Title: HoTT/tests/Test.thy - Author: Josh Chen - Date: Aug 2018 +(* +Title: tests/Test.thy +Author: Joshua Chen +Date: 2018 -This is an old "test suite" from early implementations of the theory. -It is not always guaranteed to be up to date, or reflect most recent versions of the theory. +This is an old test suite from early implementations. +It is not always guaranteed to be up to date or to reflect most recent versions of the theory. *) theory Test - imports "../HoTT" +imports "../HoTT" + begin -text " - A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified. - - Things that *should* be automated: - - Checking that \<open>A\<close> is a well-formed type, when writing things like \<open>x : A\<close> and \<open>A : U\<close>. - - Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair? -" +text \<open> +A bunch of theorems and other statements for sanity-checking, as well as things that should be automatically simplified. + +Things that *should* be automated: +\<^item> Checking that @{term A} is a well-formed type, when writing things like @{prop "x: A"} and @{prop "A: U i"}. +\<^item> Checking that the argument to a (dependent/non-dependent) function matches the type? Also the arguments to a pair? +\<close> declare[[unify_trace_simp, unify_trace_types, simp_trace, simp_trace_depth_limit=5]] - \<comment> \<open>Turn on trace for unification and the simplifier, for debugging.\<close> +\<comment> \<open>Turn on trace for unification and the Simplifier, for debugging.\<close> section \<open>\<Prod>-type\<close> subsection \<open>Typing functions\<close> -text " - Declaring \<open>Prod_intro\<close> with the \<open>intro\<close> attribute (in HoTT.thy) enables \<open>standard\<close> to prove the following. -" +text \<open>Declaring @{thm Prod_intro} with the @{attribute intro} attribute enables @{method rule} to prove the following.\<close> -proposition assumes "A : U(i)" shows "\<^bold>\<lambda>x. x: A \<rightarrow> A" by (routine lems: assms) +proposition assumes "A : U(i)" shows "\<^bold>\<lambda>x. x: A \<rightarrow> A" +by (routine add: assms) proposition assumes "A : U(i)" and "A \<equiv> B" shows "\<^bold>\<lambda>x. x : B \<rightarrow> A" proof - have "A \<rightarrow> A \<equiv> B \<rightarrow> A" using assms by simp - moreover have "\<^bold>\<lambda>x. x : A \<rightarrow> A" by (routine lems: assms) + moreover have "\<^bold>\<lambda>x. x : A \<rightarrow> A" by (routine add: assms) ultimately show "\<^bold>\<lambda>x. x : B \<rightarrow> A" by simp qed proposition assumes "A : U(i)" and "B : U(i)" shows "\<^bold>\<lambda>x y. x : A \<rightarrow> B \<rightarrow> A" -by (routine lems: assms) - +by (routine add: assms) subsection \<open>Function application\<close> -proposition assumes "a : A" shows "(\<^bold>\<lambda>x. x)`a \<equiv> a" by (derive lems: assms) - -text "Currying:" +proposition assumes "a : A" shows "(\<^bold>\<lambda>x. x)`a \<equiv> a" +by (derive lems: assms) lemma assumes "a : A" and "\<And>x. x: A \<Longrightarrow> B(x): U(i)" shows "(\<^bold>\<lambda>x y. y)`a \<equiv> \<^bold>\<lambda>y. y" -proof compute - show "\<And>x. x : A \<Longrightarrow> \<^bold>\<lambda>y. y : B(x) \<rightarrow> B(x)" by (routine lems: assms) -qed fact +by (derive lems: assms) -lemma "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. y)`a`b \<equiv> b" by derive +lemma "\<lbrakk>A: U(i); B: U(i); a : A; b : B\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. y)`a`b \<equiv> b" +by derive -lemma "\<lbrakk>A: U(i); a : A \<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. f x y)`a \<equiv> \<^bold>\<lambda>y. f a y" -proof compute - show "\<And>x. \<lbrakk>A: U(i); x: A\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>y. f x y: \<Prod>y:B(x). C x y" - proof - oops +lemma "\<lbrakk>A: U(i); a : A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x y. f x y)`a \<equiv> \<^bold>\<lambda>y. f a y" +proof derive +oops \<comment> \<open>Missing some premises.\<close> lemma "\<lbrakk>a : A; b : B(a); c : C(a)(b)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. \<^bold>\<lambda>y. \<^bold>\<lambda>z. f x y z)`a`b`c \<equiv> f a b c" - oops +proof derive +oops subsection \<open>Currying functions\<close> proposition curried_function_formation: - fixes A B C - assumes - "A : U(i)" and - "B: A \<longrightarrow> U(i)" and - "\<And>x. C(x): B(x) \<longrightarrow> U(i)" + assumes "A : U(i)" and "B: A \<longrightarrow> U(i)" and "\<And>x. C(x): B(x) \<longrightarrow> U(i)" shows "\<Prod>x:A. \<Prod>y:B(x). C x y : U(i)" - by (routine lems: assms) - +by (routine add: assms) proposition higher_order_currying_formation: assumes - "A: U(i)" and - "B: A \<longrightarrow> U(i)" and + "A: U(i)" and "B: A \<longrightarrow> U(i)" and "\<And>x y. y: B(x) \<Longrightarrow> C(x)(y): U(i)" and "\<And>x y z. z : C(x)(y) \<Longrightarrow> D(x)(y)(z): U(i)" shows "\<Prod>x:A. \<Prod>y:B(x). \<Prod>z:C(x)(y). D(x)(y)(z): U(i)" - by (routine lems: assms) - +by (routine add: assms) lemma curried_type_judgment: - assumes "A: U(i)" "B: A \<longrightarrow> U(i)" "\<And>x y. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y" + assumes "A: U(i)" and "B: A \<longrightarrow> U(i)" and "\<And>x y. \<lbrakk>x : A; y : B(x)\<rbrakk> \<Longrightarrow> f x y : C x y" shows "\<^bold>\<lambda>x y. f x y : \<Prod>x:A. \<Prod>y:B(x). C x y" - by (routine lems: assms) +by (routine add: assms) -text " - Polymorphic identity function is now trivial due to lambda expression polymorphism. - (Was more involved in previous monomorphic incarnations.) -" +text \<open> +Polymorphic identity function is now trivial due to lambda expression polymorphism. +It was more involved in previous monomorphic incarnations. +\<close> -definition Id :: "Term" where "Id \<equiv> \<^bold>\<lambda>x. x" - -lemma "\<lbrakk>x: A\<rbrakk> \<Longrightarrow> Id`x \<equiv> x" -unfolding Id_def by (compute, routine) +lemma "x: A \<Longrightarrow> id`x \<equiv> x" +by derive section \<open>Natural numbers\<close> -text "Automatic proof methods recognize natural numbers." +text \<open>Automatic proof methods recognize natural numbers.\<close> + +proposition "succ(succ(succ 0)): \<nat>" by routine -proposition "succ(succ(succ 0)): Nat" by routine end |