--- a/Tutorial/Tutorial1.thy Wed Jan 19 23:58:12 2011 +0100
+++ b/Tutorial/Tutorial1.thy Thu Jan 20 23:19:30 2011 +0100
@@ -1,21 +1,18 @@
header {*
- Nominal Isabelle Tutorial
- =========================
-
- There will be hands-on exercises throughout the tutorial. Therefore
- it would be good if you have your own laptop.
+ Nominal Isabelle Tutorial at POPL'11
+ ====================================
Nominal Isabelle is a definitional extension of Isabelle/HOL, which
means it does not add any new axioms to higher-order logic. It merely
adds new definitions and an infrastructure for 'nominal resoning'.
- The Interface
- -------------
+ The jEdit Interface
+ -------------------
- The Isabelle theorem prover comes with an interface for jEdit interface.
+ The Isabelle theorem prover comes with an interface for the jEdit.
Unlike many other theorem prover interfaces (e.g. ProofGeneral) where you
try to advance a 'checked' region in a proof script, this interface immediately
checks the whole buffer. The text you type is also immediately checked
@@ -64,6 +61,7 @@
\<bullet> bullet (permutations)
*}
+
theory Tutorial1
imports Lambda
begin
@@ -72,12 +70,12 @@
text {*
All formal developments in Isabelle are part of a theory. A theory
- needs to have a name and must import some pre-existing theory (as indicated
- above). The imported theory will normally be the theory Nominal2 (which
- contains many useful concepts like set-theory, lists, nominal theory etc).
-
+ needs to have a name and must import some pre-existing theory. The
+ imported theory will normally be the theory Nominal2 (which contains
+ many useful concepts like set-theory, lists, nominal theory etc).
For the purpose of the tutorial we import the theory Lambda.thy which
- contains already some useful definitions for (alpha-equated) lambda terms.
+ contains already some useful definitions for (alpha-equated) lambda
+ terms.
*}
@@ -85,11 +83,11 @@
section {* Types *}
text {*
- Isabelle is based on simple types including some polymorphism. It also includes
- some overloading, which means that sometimes explicit type annotations need to
- be given.
+ Isabelle is based on simple types including some polymorphism. It also
+ includes overloading, which means that sometimes explicit type annotations
+ need to be given.
- - Base types include: nat, bool, string
+ - Base types include: nat, bool, string, lam (defined in the Lambda theory)
- Type formers include: 'a list, ('a \<times> 'b), 'c set
@@ -101,15 +99,15 @@
typ nat
typ bool
typ string
-typ lam -- {* alpha-equated lambda terms defined in Lambda.thy *}
-typ name -- {* type of (object) variables defined in Lambda.thy *}
-typ "('a \<times> 'b)" -- {* pair type *}
-typ "'c set" -- {* set type *}
-typ "'a list" -- {* list type *}
-typ "nat \<Rightarrow> bool" -- {* type of functions from natural numbers to booleans *}
+typ lam -- {* alpha-equated lambda terms defined in Lambda.thy *}
+typ name -- {* type of (object) variables defined in Lambda.thy *}
+typ "('a \<times> 'b)" -- {* pair type *}
+typ "'c set" -- {* set type *}
+typ "'a list" -- {* list type *}
+typ "lam \<Rightarrow> nat" -- {* type of functions from lambda terms to natural numbers *}
-text {* Some malformed types: *}
+text {* Some malformed types - note the "stop" signal on the left margin *}
(*
typ boolean -- {* undeclared type *}
@@ -139,10 +137,11 @@
text {*
Lam [x].Var is the syntax we made up for lambda abstractions. You can have
- your own syntax. We also came up with "name". If you prefer, you can call
- it identifiers or have more than one type for (object languag) variables.
+ your own syntax, if you prefer (but \<lambda> is already taken up for Isabelle's
+ functions). We also came up with "name". If you prefer, you can call
+ it "ident" or have more than one type for (object language) variables.
- Isabelle provides some useful colour feedback about constants (black),
+ Isabelle provides some useful colour feedback about its constants (black),
free variables (blue) and bound variables (green).
*}
@@ -160,7 +159,7 @@
term "{1,2,3} = {3,2,1}"
term "\<forall>x. P x"
term "A \<longrightarrow> B"
-term "atom a \<sharp> t"
+term "atom a \<sharp> t" -- {* freshness in Nominal *}
text {*
When working with Isabelle, one deals with an object logic (that is HOL) and
@@ -179,10 +178,12 @@
section {* Inductive Definitions: Transitive Closures of Beta-Reduction *}
text {*
- The theory Lmabda alraedy contains a definition for beta-reduction, written
+ The theory Lmabda already contains a definition for beta-reduction, written
+*}
- t \<longrightarrow>b t'
+term "t \<longrightarrow>b t'"
+text {*
In this section we want to define inductively the transitive closure of
beta-reduction.
@@ -1195,358 +1196,6 @@
by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
(auto simp add: fresh_fact forget)
-text {******************************************************************
-
- The CBV Reduction Relation (Small-Step Semantics)
- -------------------------------------------------
-
- In order to establish the property that the CK Machine
- calculates a nomrmalform which corresponds to the
- evaluation relation, we introduce the call-by-value
- small-step semantics.
-
-*}
-
-inductive
- cbv :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<longrightarrow>cbv _" [60, 60] 60)
-where
- cbv1: "\<lbrakk>val v; atom x \<sharp> v\<rbrakk> \<Longrightarrow> App (Lam [x].t) v \<longrightarrow>cbv t[x::=v]"
-| cbv2[intro]: "t \<longrightarrow>cbv t' \<Longrightarrow> App t t2 \<longrightarrow>cbv App t' t2"
-| cbv3[intro]: "t \<longrightarrow>cbv t' \<Longrightarrow> App t2 t \<longrightarrow>cbv App t2 t'"
-
-equivariance val
-equivariance cbv
-nominal_inductive cbv
- avoids cbv1: "x"
- unfolding fresh_star_def
- by (simp_all add: lam.fresh Abs_fresh_iff fresh_Pair fresh_fact)
-
-text {*
- In order to satisfy the vc-condition we have to formulate
- this relation with the additional freshness constraint
- x\<sharp>v. Though this makes the definition vc-ompatible, it
- makes the definition less useful. We can with some pain
- show that the more restricted rule is equivalent to the
- usual rule. *}
-
-lemma subst_rename:
- assumes a: "atom y \<sharp> t"
- shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet>t)[y ::= s]"
-using a
-apply (nominal_induct t avoiding: x y s rule: lam.strong_induct)
-apply (auto simp add: lam.fresh fresh_at_base)
-done
-
-thm subst_rename
-
-lemma better_cbv1[intro]:
- assumes a: "val v"
- shows "App (Lam [x].t) v \<longrightarrow>cbv t[x::=v]"
-proof -
- obtain y::"name" where fs: "atom y \<sharp> (x, t, v)" by (rule obtain_fresh)
- have "App (Lam [x].t) v = App (Lam [y].((y \<leftrightarrow> x) \<bullet> t)) v" using fs
- by (auto simp add: lam.eq_iff Abs1_eq_iff' flip_def fresh_Pair fresh_at_base)
- also have "\<dots> \<longrightarrow>cbv ((y \<leftrightarrow> x) \<bullet> t)[y ::= v]" using fs a by (auto intro: cbv1)
- also have "\<dots> = t[x ::= v]" using fs by (simp add: subst_rename[symmetric])
- finally show "App (Lam [x].t) v \<longrightarrow>cbv t[x ::= v]" by simp
-qed
-
-text {*
- The transitive closure of the cbv-reduction relation: *}
-
-inductive
- "cbvs" :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>cbv* _" [60, 60] 60)
-where
- cbvs1[intro]: "e \<longrightarrow>cbv* e"
-| cbvs2[intro]: "\<lbrakk>e1\<longrightarrow>cbv e2; e2 \<longrightarrow>cbv* e3\<rbrakk> \<Longrightarrow> e1 \<longrightarrow>cbv* e3"
-
-lemma cbvs3[intro]:
- assumes a: "e1 \<longrightarrow>cbv* e2" "e2 \<longrightarrow>cbv* e3"
- shows "e1 \<longrightarrow>cbv* e3"
-using a by (induct) (auto)
-
-text {******************************************************************
-
- 8.) Exercise
- ------------
-
- If more simple exercises are needed, then complete the following proof.
-
-*}
-
-lemma cbv_in_ctx:
- assumes a: "t \<longrightarrow>cbv t'"
- shows "E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>"
-using a
-proof (induct E)
- case Hole
- have "t \<longrightarrow>cbv t'" by fact
- then show "\<box>\<lbrakk>t\<rbrakk> \<longrightarrow>cbv \<box>\<lbrakk>t'\<rbrakk>" sorry
-next
- case (CAppL E s)
- have ih: "t \<longrightarrow>cbv t' \<Longrightarrow> E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by fact
- have a: "t \<longrightarrow>cbv t'" by fact
- show "(CAppL E s)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppL E s)\<lbrakk>t'\<rbrakk>" sorry
-next
- case (CAppR s E)
- have ih: "t \<longrightarrow>cbv t' \<Longrightarrow> E\<lbrakk>t\<rbrakk> \<longrightarrow>cbv E\<lbrakk>t'\<rbrakk>" by fact
- have a: "t \<longrightarrow>cbv t'" by fact
- show "(CAppR s E)\<lbrakk>t\<rbrakk> \<longrightarrow>cbv (CAppR s E)\<lbrakk>t'\<rbrakk>" sorry
-qed
-
-
-text {******************************************************************
-
- 9.) Exercise
- ------------
-
- The point of the cbv-reduction was that we can easily relatively
- establish the follwoing property:
-
-*}
-
-lemma machine_implies_cbvs_ctx:
- assumes a: "<e, Es> \<mapsto> <e', Es'>"
- shows "(Es\<down>)\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* (Es'\<down>)\<lbrakk>e'\<rbrakk>"
-using a
-proof (induct)
- case (m1 t1 t2 Es)
-
- show "Es\<down>\<lbrakk>App t1 t2\<rbrakk> \<longrightarrow>cbv* ((CAppL \<box> t2) # Es)\<down>\<lbrakk>t1\<rbrakk>" sorry
-next
- case (m2 v t2 Es)
- have "val v" by fact
-
- show "((CAppL \<box> t2) # Es)\<down>\<lbrakk>v\<rbrakk> \<longrightarrow>cbv* (CAppR v \<box> # Es)\<down>\<lbrakk>t2\<rbrakk>" sorry
-next
- case (m3 v x t Es)
- have "val v" by fact
-
- show "(((CAppR (Lam [x].t) \<box>) # Es)\<down>)\<lbrakk>v\<rbrakk> \<longrightarrow>cbv* (Es\<down>)\<lbrakk>(t[x ::= v])\<rbrakk>" sorry
-qed
-
-text {*
- It is not difficult to extend the lemma above to
- arbitrary reductions sequences of the CK machine. *}
-
-lemma machines_implies_cbvs_ctx:
- assumes a: "<e, Es> \<mapsto>* <e', Es'>"
- shows "(Es\<down>)\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* (Es'\<down>)\<lbrakk>e'\<rbrakk>"
-using a
-by (induct) (auto dest: machine_implies_cbvs_ctx)
-
-text {*
- So whenever we let the CL machine start in an initial
- state and it arrives at a final state, then there exists
- a corresponding cbv-reduction sequence. *}
-
-corollary machines_implies_cbvs:
- assumes a: "<e, []> \<mapsto>* <e', []>"
- shows "e \<longrightarrow>cbv* e'"
-using a by (auto dest: machines_implies_cbvs_ctx)
-
-text {*
- We now want to relate the cbv-reduction to the evaluation
- relation. For this we need two auxiliary lemmas. *}
-
-lemma eval_val:
- assumes a: "val t"
- shows "t \<Down> t"
-using a by (induct) (auto)
-
-lemma e_App_elim:
- assumes a: "App t1 t2 \<Down> v"
- shows "\<exists>x t v'. t1 \<Down> Lam [x].t \<and> t2 \<Down> v' \<and> t[x::=v'] \<Down> v"
-using a by (cases) (auto simp add: lam.eq_iff lam.distinct)
-
-text {******************************************************************
-
- 10.) Exercise
- -------------
-
- Complete the first case in the proof below.
-
-*}
-
-lemma cbv_eval:
- assumes a: "t1 \<longrightarrow>cbv t2" "t2 \<Down> t3"
- shows "t1 \<Down> t3"
-using a
-proof(induct arbitrary: t3)
- case (cbv1 v x t t3)
- have a1: "val v" by fact
- have a2: "t[x ::= v] \<Down> t3" by fact
-
- show "App (Lam [x].t) v \<Down> t3" sorry
-next
- case (cbv2 t t' t2 t3)
- have ih: "\<And>t3. t' \<Down> t3 \<Longrightarrow> t \<Down> t3" by fact
- have "App t' t2 \<Down> t3" by fact
- then obtain x t'' v'
- where a1: "t' \<Down> Lam [x].t''"
- and a2: "t2 \<Down> v'"
- and a3: "t''[x ::= v'] \<Down> t3" using e_App_elim by blast
- have "t \<Down> Lam [x].t''" using ih a1 by auto
- then show "App t t2 \<Down> t3" using a2 a3 by auto
-qed (auto dest!: e_App_elim)
-
-
-text {*
- Next we extend the lemma above to arbitray initial
- sequences of cbv-reductions. *}
-
-lemma cbvs_eval:
- assumes a: "t1 \<longrightarrow>cbv* t2" "t2 \<Down> t3"
- shows "t1 \<Down> t3"
-using a by (induct) (auto intro: cbv_eval)
-
-text {*
- Finally, we can show that if from a term t we reach a value
- by a cbv-reduction sequence, then t evaluates to this value. *}
-
-lemma cbvs_implies_eval:
- assumes a: "t \<longrightarrow>cbv* v" "val v"
- shows "t \<Down> v"
-using a
-by (induct) (auto intro: eval_val cbvs_eval)
-
-text {*
- All facts tied together give us the desired property about
- K machines. *}
-
-theorem machines_implies_eval:
- assumes a: "<t1, []> \<mapsto>* <t2, []>"
- and b: "val t2"
- shows "t1 \<Down> t2"
-proof -
- have "t1 \<longrightarrow>cbv* t2" using a by (simp add: machines_implies_cbvs)
- then show "t1 \<Down> t2" using b by (simp add: cbvs_implies_eval)
-qed
-
-lemma valid_elim:
- assumes a: "valid ((x, T) # \<Gamma>)"
- shows "atom x \<sharp> \<Gamma> \<and> valid \<Gamma>"
-using a by (cases) (auto)
-
-lemma valid_insert:
- assumes a: "valid (\<Delta> @ [(x, T)] @ \<Gamma>)"
- shows "valid (\<Delta> @ \<Gamma>)"
-using a
-by (induct \<Delta>)
- (auto simp add: fresh_append fresh_Cons dest!: valid_elim)
-
-lemma fresh_list:
- shows "atom y \<sharp> xs = (\<forall>x \<in> set xs. atom y \<sharp> x)"
-by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
-
-lemma context_unique:
- assumes a1: "valid \<Gamma>"
- and a2: "(x, T) \<in> set \<Gamma>"
- and a3: "(x, U) \<in> set \<Gamma>"
- shows "T = U"
-using a1 a2 a3
-by (induct) (auto simp add: fresh_list fresh_Pair fresh_at_base)
-
-lemma type_substitution_aux:
- assumes a: "(\<Delta> @ [(x, T')] @ \<Gamma>) \<turnstile> e : T"
- and b: "\<Gamma> \<turnstile> e' : T'"
- shows "(\<Delta> @ \<Gamma>) \<turnstile> e[x ::= e'] : T"
-using a b
-proof (nominal_induct \<Gamma>'\<equiv>"\<Delta> @ [(x, T')] @ \<Gamma>" e T avoiding: x e' \<Delta> rule: typing.strong_induct)
- case (t_Var y T x e' \<Delta>)
- have a1: "valid (\<Delta> @ [(x, T')] @ \<Gamma>)" by fact
- have a2: "(y,T) \<in> set (\<Delta> @ [(x, T')] @ \<Gamma>)" by fact
- have a3: "\<Gamma> \<turnstile> e' : T'" by fact
- from a1 have a4: "valid (\<Delta> @ \<Gamma>)" by (rule valid_insert)
- { assume eq: "x = y"
- from a1 a2 have "T = T'" using eq by (auto intro: context_unique)
- with a3 have "\<Delta> @ \<Gamma> \<turnstile> Var y[x::=e'] : T" using eq a4 by (auto intro: weakening)
- }
- moreover
- { assume ineq: "x \<noteq> y"
- from a2 have "(y, T) \<in> set (\<Delta> @ \<Gamma>)" using ineq by simp
- then have "\<Delta> @ \<Gamma> \<turnstile> Var y[x::=e'] : T" using ineq a4 by auto
- }
- ultimately show "\<Delta> @ \<Gamma> \<turnstile> Var y[x::=e'] : T" by blast
-qed (force simp add: fresh_append fresh_Cons)+
-
-corollary type_substitution:
- assumes a: "(x,T') # \<Gamma> \<turnstile> e : T"
- and b: "\<Gamma> \<turnstile> e' : T'"
- shows "\<Gamma> \<turnstile> e[x::=e'] : T"
-using a b type_substitution_aux[where \<Delta>="[]"]
-by (auto)
-
-lemma t_App_elim:
- assumes a: "\<Gamma> \<turnstile> App t1 t2 : T"
- shows "\<exists>T'. \<Gamma> \<turnstile> t1 : T' \<rightarrow> T \<and> \<Gamma> \<turnstile> t2 : T'"
-using a
-by (cases) (auto simp add: lam.eq_iff lam.distinct)
-
-lemma t_Lam_elim:
- assumes ty: "\<Gamma> \<turnstile> Lam [x].t : T"
- and fc: "atom x \<sharp> \<Gamma>"
- shows "\<exists>T1 T2. T = T1 \<rightarrow> T2 \<and> (x, T1) # \<Gamma> \<turnstile> t : T2"
-using ty fc
-apply(cases)
-apply(auto simp add: lam.eq_iff lam.distinct ty.eq_iff)
-apply(auto simp add: Abs1_eq_iff)
-apply(rule_tac p="(x \<leftrightarrow> xa)" in permute_boolE)
-apply(perm_simp)
-apply(simp add: flip_def swap_fresh_fresh ty_fresh)
-done
-
-theorem cbv_type_preservation:
- assumes a: "t \<longrightarrow>cbv t'"
- and b: "\<Gamma> \<turnstile> t : T"
- shows "\<Gamma> \<turnstile> t' : T"
-using a b
-by (nominal_induct avoiding: \<Gamma> T rule: cbv.strong_induct)
- (auto dest!: t_Lam_elim t_App_elim simp add: type_substitution ty.eq_iff)
-
-corollary cbvs_type_preservation:
- assumes a: "t \<longrightarrow>cbv* t'"
- and b: "\<Gamma> \<turnstile> t : T"
- shows "\<Gamma> \<turnstile> t' : T"
-using a b
-by (induct) (auto intro: cbv_type_preservation)
-
-text {*
- The Type-Preservation Property for the Machine and Evaluation Relation. *}
-
-theorem machine_type_preservation:
- assumes a: "<t, []> \<mapsto>* <t', []>"
- and b: "\<Gamma> \<turnstile> t : T"
- shows "\<Gamma> \<turnstile> t' : T"
-proof -
- from a have "t \<longrightarrow>cbv* t'" by (simp add: machines_implies_cbvs)
- then show "\<Gamma> \<turnstile> t' : T" using b by (simp add: cbvs_type_preservation)
-qed
-
-theorem eval_type_preservation:
- assumes a: "t \<Down> t'"
- and b: "\<Gamma> \<turnstile> t : T"
- shows "\<Gamma> \<turnstile> t' : T"
-proof -
- from a have "<t, []> \<mapsto>* <t', []>" by (simp add: eval_implies_machines)
- then show "\<Gamma> \<turnstile> t' : T" using b by (simp add: machine_type_preservation)
-qed
-
-text {* The Progress Property *}
-
-lemma canonical_tArr:
- assumes a: "[] \<turnstile> t : T1 \<rightarrow> T2"
- and b: "val t"
- shows "\<exists>x t'. t = Lam [x].t'"
-using b a by (induct) (auto)
-
-theorem progress:
- assumes a: "[] \<turnstile> t : T"
- shows "(\<exists>t'. t \<longrightarrow>cbv t') \<or> (val t)"
-using a
-by (induct \<Gamma>\<equiv>"[]::ty_ctx" t T)
- (auto intro: cbv.intros dest!: canonical_tArr)
-
end