--- a/Tutorial/Tutorial1.thy Sat Jan 22 12:46:01 2011 -0600
+++ b/Tutorial/Tutorial1.thy Sat Jan 22 15:07:36 2011 -0600
@@ -60,6 +60,7 @@
\<bullet> bullet (permutation application)
*}
+
theory Tutorial1
imports Lambda
begin
@@ -460,13 +461,15 @@
proof(induct)
case (ms1 e1 Es1)
have c: "<e1, Es1> \<mapsto>* <e3, Es3>" by fact
- then show "<e1, Es1> \<mapsto>* <e3, Es3>" by simp
+
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" sorry
next
case (ms2 e1 Es1 e2 Es2 e2' Es2')
have ih: "<e2', Es2'> \<mapsto>* <e3, Es3> \<Longrightarrow> <e2, Es2> \<mapsto>* <e3, Es3>" by fact
have d1: "<e2', Es2'> \<mapsto>* <e3, Es3>" by fact
have d2: "<e1, Es1> \<mapsto> <e2, Es2>" by fact
- show "<e1, Es1> \<mapsto>* <e3, Es3>" using d1 d2 ih by blast
+
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" sorry
qed
text {*
@@ -584,41 +587,6 @@
qed
-theorem
- assumes a: "t \<Down> t'"
- shows "<t, Es> \<mapsto>* <t', Es>"
-using a
-proof (induct arbitrary: Es)
- case (e_Lam x t Es)
- -- {* no assumptions *}
- show "<Lam [x].t, Es> \<mapsto>* <Lam [x].t, Es>" by auto
-next
- case (e_App t1 x t t2 v' v Es)
- -- {* all assumptions in this case *}
- have a1: "t1 \<Down> Lam [x].t" by fact
- have ih1: "\<And>Es. <t1, Es> \<mapsto>* <Lam [x].t, Es>" by fact
- have a2: "t2 \<Down> v'" by fact
- have ih2: "\<And>Es. <t2, Es> \<mapsto>* <v', Es>" by fact
- have a3: "t[x::=v'] \<Down> v" by fact
- have ih3: "\<And>Es. <t[x::=v'], Es> \<mapsto>* <v, Es>" by fact
- -- {* your reasoning *}
- have "<App t1 t2, Es> \<mapsto>* <t1, CAppL \<box> t2 # Es>" by auto
- moreover
- have "<t1, CAppL \<box> t2 # Es> \<mapsto>* <Lam [x].t, CAppL \<box> t2 # Es>" using ih1 by auto
- moreover
- have "<Lam [x].t, CAppL \<box> t2 # Es> \<mapsto>* <t2, CAppR (Lam [x].t) \<box> # Es>" by auto
- moreover
- have "<t2, CAppR (Lam [x].t) \<box> # Es> \<mapsto>* <v', CAppR (Lam [x].t) \<box> # Es>"
- using ih2 by auto
- moreover
- have "val v'" using a2 eval_to_val by auto
- then have "<v', CAppR (Lam [x].t) \<box> # Es> \<mapsto>* <t[x::=v'], Es>" by auto
- moreover
- have "<t[x::=v'], Es> \<mapsto>* <v, Es>" using ih3 by auto
- ultimately show "<App t1 t2, Es> \<mapsto>* <v, Es>" by blast
-qed
-
-
text {*
Again the automatic tools in Isabelle can discharge automatically
of the routine work in these proofs. We can write:
@@ -637,7 +605,6 @@
using a eval_implies_machines_ctx by simp
-
section {* Function Definitions: Filling a Lambda-Term into a Context *}
text {*
@@ -666,7 +633,7 @@
by simp
fun
- ctx_compose :: "ctx \<Rightarrow> ctx \<Rightarrow> ctx" ("_ \<odot> _" [99,98] 99)
+ ctx_compose :: "ctx \<Rightarrow> ctx \<Rightarrow> ctx" (infixr "\<odot>" 99)
where
"\<box> \<odot> E' = E'"
| "(CAppL E t') \<odot> E' = CAppL (E \<odot> E') t'"
@@ -715,11 +682,19 @@
show "((CAppR t' E1) \<odot> E2)\<lbrakk>t\<rbrakk> = (CAppR t' E1)\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" sorry
qed
+
+subsection {* EXERCISE 6 *}
+
+text {*
+ Remove the sorries in the ctx_append proof below. You can make
+ use of the following two properties.
+*}
+
lemma neut_hole:
shows "E \<odot> \<box> = E"
by (induct E) (simp_all)
-lemma odot_assoc: (* fixme call compose *)
+lemma compose_assoc:
shows "(E1 \<odot> E2) \<odot> E3 = E1 \<odot> (E2 \<odot> E3)"
by (induct E1) (simp_all)
@@ -735,7 +710,6 @@
show "((E # Es1) @ Es2)\<down> = Es2\<down> \<odot> (E # Es1)\<down>" sorry
qed
-
text {*
The last proof involves several steps of equational reasoning.
Isar provides some convenient means to express such equational
@@ -743,7 +717,6 @@
construction.
*}
-
lemma
shows "(Es1 @ Es2)\<down> = (Es2\<down>) \<odot> (Es1\<down>)"
proof (induct Es1)
@@ -752,14 +725,14 @@
next
case (Cons E Es1)
have ih: "(Es1 @ Es2)\<down> = Es2\<down> \<odot> Es1\<down>" by fact
- have "((E # Es1) @ Es2)\<down> = (Es1 @ Es2)\<down> \<odot> E" by simp
+ have "((E # Es1) @ Es2)\<down> = (E # (Es1 @ Es2))\<down>" by simp
+ also have "\<dots> = (Es1 @ Es2)\<down> \<odot> E" by simp
also have "\<dots> = (Es2\<down> \<odot> Es1\<down>) \<odot> E" using ih by simp
- also have "\<dots> = Es2\<down> \<odot> (Es1\<down> \<odot> E)" using odot_assoc by simp
+ also have "\<dots> = Es2\<down> \<odot> (Es1\<down> \<odot> E)" using compose_assoc by simp
also have "\<dots> = Es2\<down> \<odot> (E # Es1)\<down>" by simp
finally show "((E # Es1) @ Es2)\<down> = Es2\<down> \<odot> (E # Es1)\<down>" by simp
qed
-
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Tutorial/Tutorial1s.thy Sat Jan 22 15:07:36 2011 -0600
@@ -0,0 +1,752 @@
+
+header {*
+
+ 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 just
+ adds new definitions and an infrastructure for 'nominal resoning'.
+
+
+ The jEdit Interface
+ -------------------
+
+ The Isabelle theorem prover comes with an interface for jEdit.
+ Unlike many other theorem prover interfaces (e.g. ProofGeneral) where you
+ advance a 'checked' region in a proof script, this interface immediately
+ checks the whole buffer. The text you type is also immediately checked.
+ Malformed text or unfinished proofs are highlighted in red with a little
+ red 'stop' signal on the left-hand side. If you drag the 'red-box' cursor
+ over a line, the Output window gives further feedback.
+
+ Note: If a section is not parsed correctly, the work-around is to cut it
+ out and paste it back into the text (cut-out: Ctrl + X; paste in: Ctrl + V;
+ Cmd is Ctrl on the Mac)
+
+ Nominal Isabelle and jEdit can be started by typing on the command line
+
+ isabelle jedit -l HOL-Nominal2
+ isabelle jedit -l HOL-Nominal2 A.thy B.thy ...
+
+
+ Symbols
+ -------
+
+ Symbols can considerably improve the readability of your statements and proofs.
+ They can be input by just typing 'name-of-symbol' where 'name-of-symbol' is the
+ usual latex name of that symbol. A little window will then appear in which
+ you can select the symbol. With `Escape' you can ignore any suggestion.
+
+ There are some handy short-cuts for frequently used symbols.
+ For example
+
+ short-cut symbol
+
+ => \<Rightarrow>
+ ==> \<Longrightarrow>
+ --> \<longrightarrow>
+ ! \<forall>
+ ? \<exists>
+ /\ \<and>
+ \/ \<or>
+ ~ \<not>
+ ~= \<noteq>
+ : \<in>
+ ~: \<notin>
+
+ For nominal the following two symbols have a special meaning
+
+ \<sharp> sharp (freshness)
+ \<bullet> bullet (permutation application)
+*}
+
+theory Tutorial1s
+imports Lambda
+begin
+
+section {* Theories *}
+
+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. The
+ imported theory will normally be Nominal2 (which provides many useful
+ concepts like set-theory, lists, nominal things etc). For the purpose
+ of this tutorial we import the theory Lambda.thy, which contains
+ already some useful definitions for (alpha-equated) lambda terms.
+*}
+
+
+
+section {* Types *}
+
+text {*
+ 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, lam (defined in the Lambda theory)
+
+ - Type formers include: 'a list, ('a \<times> 'b), 'c set
+
+ - Type variables are written like in ML with an apostrophe: 'a, 'b, \<dots>
+
+ Types known to Isabelle can be queried using the command "typ".
+*}
+
+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 "lam \<Rightarrow> nat" -- {* type of functions from lambda terms to natural numbers *}
+
+
+text {* Some malformed types - note the "stop" signal on the left margin *}
+
+(*
+typ boolean -- {* undeclared type *}
+typ set -- {* type argument missing *}
+*)
+
+
+section {* Terms *}
+
+text {*
+ Every term in Isabelle needs to be well-typed. However they can have
+ polymorphic type. Whether a term is accepted can be queried using
+ the command "term".
+*}
+
+term c -- {* a variable of polymorphic type *}
+term "1::nat" -- {* the constant 1 in natural numbers *}
+term 1 -- {* the constant 1 with polymorphic type *}
+term "{1, 2, 3::nat}" -- {* the set containing natural numbers 1, 2 and 3 *}
+term "[1, 2, 3]" -- {* the list containing the polymorphic numbers 1, 2 and 3 *}
+term "(True, ''c'')" -- {* a pair consisting of the boolean true and the string "c" *}
+term "Suc 0" -- {* successor of 0, in other words 1::nat *}
+term "Lam [x].Var x" -- {* a lambda-term *}
+term "App t1 t2" -- {* another lambda-term *}
+term "x::name" -- {* an (object) variable of type name *}
+term "atom (x::name)" -- {* atom is an overloded function *}
+
+text {*
+ Lam [x].Var is the syntax we made up for lambda abstractions. If you
+ prefer, you can have your own syntax (but \<lambda> is already taken up for
+ Isabelle's functions). We also came up with the type "name" for variables.
+ You can introduce your own types of object variables using the
+ command atom_decl:
+*}
+
+atom_decl ident
+atom_decl ty_var
+
+text {*
+ Isabelle provides some useful colour feedback about its constants (black),
+ free variables (blue) and bound variables (green).
+*}
+
+term "True" -- {* a constant defined somewhere...written in black *}
+term "true" -- {* not recognised as a constant, therefore it is interpreted
+ as a free variable, written in blue *}
+term "\<forall>x. P x" -- {* x is bound (green), P is free (blue) *}
+
+
+text {* Formulae in Isabelle/HOL are terms of type bool *}
+
+term "True"
+term "True \<and> False"
+term "True \<or> B"
+term "{1,2,3} = {3,2,1}"
+term "\<forall>x. P x"
+term "A \<longrightarrow> B"
+term "atom a \<sharp> t" -- {* freshness in Nominal *}
+
+text {*
+ When working with Isabelle, one deals with an object logic (that is HOL) and
+ Isabelle's rule framework (called Pure). Occasionally one has to pay attention
+ to this fact. But for the moment we ignore this completely.
+*}
+
+term "A \<longrightarrow> B" -- {* versus *}
+term "A \<Longrightarrow> B"
+
+term "\<forall>x. P x" -- {* versus *}
+term "\<And>x. P x"
+
+
+section {* Inductive Definitions: Evaluation Relation *}
+
+text {*
+ In this section we want to define inductively the evaluation
+ relation and for cbv-reduction relation.
+
+ Inductive definitions in Isabelle start with the keyword "inductive"
+ and a predicate name. One can optionally provide a type for the predicate.
+ Clauses of the inductive predicate are introduced by "where" and more than
+ two clauses need to be separated by "|". One can also give a name to each
+ clause and indicate that it should be added to the hints database ("[intro]").
+ A typical clause has some premises and a conclusion. This is written in
+ Isabelle as:
+
+ "premise \<Longrightarrow> conclusion"
+ "premise1 \<Longrightarrow> premise2 \<Longrightarrow> \<dots> premisen \<Longrightarrow> conclusion"
+
+ There is an alternative way of writing the latter clause as
+
+ "\<lbrakk>premise1; premise2; \<dots> premisen\<rbrakk> \<Longrightarrow> conclusion"
+
+ If no premise is present, then one just writes
+
+ "conclusion"
+
+ Below we give two definitions for the transitive closure
+*}
+
+inductive
+ eval :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<Down> _" [60, 60] 60)
+where
+ e_Lam[intro]: "Lam [x].t \<Down> Lam [x].t"
+| e_App[intro]: "\<lbrakk>t1 \<Down> Lam [x].t; t2 \<Down> v'; t[x::=v'] \<Down> v\<rbrakk> \<Longrightarrow> App t1 t2 \<Down> v"
+
+text {*
+ Values and cbv are also defined using inductive.
+*}
+
+inductive
+ val :: "lam \<Rightarrow> bool"
+where
+ v_Lam[intro]: "val (Lam [x].t)"
+
+section {* Theorems *}
+
+text {*
+ A central concept in Isabelle is that of theorems. Isabelle's theorem
+ database can be queried using
+*}
+
+thm e_App
+thm e_Lam
+thm conjI
+thm conjunct1
+
+text {*
+ Notice that theorems usually contain schematic variables (e.g. ?P, ?Q, \<dots>).
+ These schematic variables can be substituted with any term (of the right type
+ of course).
+
+ When defining the predicates beta_star, Isabelle provides us automatically
+ with the following theorems that state how they can be introduced and what
+ constitutes an induction over them.
+*}
+
+thm eval.intros
+thm eval.induct
+
+
+section {* Lemmas / Theorems / Corollaries *}
+
+text {*
+ Whether to use lemma, theorem or corollary makes no semantic difference
+ in Isabelle.
+
+ A lemma starts with "lemma" and consists of a statement ("shows \<dots>") and
+ optionally a lemma name, some type-information for variables ("fixes \<dots>")
+ and some assumptions ("assumes \<dots>").
+
+ Lemmas also need to have a proof, but ignore this 'detail' for the moment.
+
+ Examples are
+*}
+
+lemma alpha_equ:
+ shows "Lam [x].Var x = Lam [y].Var y"
+ by (simp add: lam.eq_iff Abs1_eq_iff lam.fresh fresh_at_base)
+
+lemma Lam_freshness:
+ assumes a: "atom y \<sharp> Lam [x].t"
+ shows "(y = x) \<or> (y \<noteq> x \<and> atom y \<sharp> t)"
+ using a by (auto simp add: lam.fresh Abs_fresh_iff)
+
+lemma neutral_element:
+ fixes x::"nat"
+ shows "x + 0 = x"
+ by simp
+
+text {*
+ Note that in the last statement, the explicit type annotation is important
+ in order for Isabelle to be able to figure out what 0 stands for (e.g. a
+ natural number, a vector, etc) and which lemmas to apply.
+*}
+
+
+section {* Isar Proofs *}
+
+text {*
+ Isar is a language for writing formal proofs that can be understood by humans
+ and by Isabelle. An Isar proof can be thought of as a sequence of 'stepping stones'
+ that start with some assumptions and lead to the goal to be established. Every such
+ stepping stone is introduced by "have" followed by the statement of the stepping
+ stone. An exception is the goal to be proved, which need to be introduced with "show".
+
+ have "statement" \<dots>
+ show "goal_to_be_proved" \<dots>
+
+ Since proofs usually do not proceed in a linear fashion, labels can be given
+ to every stepping stone and they can be used later to explicitly refer to this
+ corresponding stepping stone ("using").
+
+ have label: "statement1" \<dots>
+ \<dots>
+ have "later_statement" using label \<dots>
+
+ Each stepping stone (or have-statement) needs to have a justification. The
+ simplest justification is "sorry" which admits any stepping stone, even false
+ ones (this is good during the development of proofs).
+
+ have "outrageous_false_statement" sorry
+
+ Assumptions can be 'justified' using "by fact".
+
+ have "assumption" by fact
+
+ Derived facts can be justified using
+
+ have "statement" by simp -- simplification
+ have "statement" by auto -- proof search and simplification
+ have "statement" by blast -- only proof search
+
+ If facts or lemmas are needed in order to justify a have-statement, then
+ one can feed these facts into the proof by using "using label \<dots>" or
+ "using theorem-name \<dots>". More than one label at a time is allowed. For
+ example
+
+ have "statement" using label1 label2 theorem_name by auto
+
+ Induction proofs in Isar are set up by indicating over which predicate(s)
+ the induction proceeds ("using a b") followed by the command "proof (induct)".
+ In this way, Isabelle uses defaults for which induction should be performed.
+ These defaults can be overridden by giving more information, like the variable
+ over which a structural induction should proceed, or a specific induction principle,
+ such as well-founded inductions.
+
+ After the induction is set up, the proof proceeds by cases. In Isar these
+ cases can be given in any order. Most commonly they are started with "case" and the
+ name of the case, and optionally some legible names for the variables used inside
+ the case.
+
+ In each "case", we need to establish a statement introduced by "show". Once
+ this has been done, the next case can be started using "next". When all cases
+ are completed, the proof can be finished using "qed".
+
+ This means a typical induction proof has the following pattern
+
+ proof (induct)
+ case \<dots>
+ \<dots>
+ show \<dots>
+ next
+ case \<dots>
+ \<dots>
+ show \<dots>
+ \<dots>
+ qed
+
+ Two very simple example proofs are as follows.
+*}
+
+subsection {* EXERCISE 1 *}
+
+lemma eval_val:
+ assumes a: "val t"
+ shows "t \<Down> t"
+using a
+proof (induct)
+ case (v_Lam x t)
+ show "Lam [x]. t \<Down> Lam [x]. t" by auto
+qed
+
+subsection {* EXERCISE 2 *}
+
+text {* Fill the gaps in the proof below. *}
+
+lemma eval_to_val:
+ assumes a: "t \<Down> t'"
+ shows "val t'"
+using a
+proof (induct)
+ case (e_Lam x t)
+ show "val (Lam [x].t)" by auto
+next
+ case (e_App t1 x t t2 v v')
+ -- {* all assumptions in this case *}
+ have "t1 \<Down> Lam [x].t" by fact
+ have ih1: "val (Lam [x]. t)" by fact
+ have "t2 \<Down> v" by fact
+ have ih2: "val v" by fact
+ have "t [x ::= v] \<Down> v'" by fact
+ have ih3: "val v'" by fact
+
+ show "val v'" using ih3 by auto
+qed
+
+
+section {* Datatypes: Evaluation Contexts*}
+
+text {*
+ Datatypes can be defined in Isabelle as follows: we have to provide the name
+ of the datatype and a list its type-constructors. Each type-constructor needs
+ to have the information about the types of its arguments, and optionally
+ can also contain some information about pretty syntax. For example, we like
+ to write "\<box>" for holes.
+*}
+
+datatype ctx =
+ Hole ("\<box>")
+ | CAppL "ctx" "lam"
+ | CAppR "lam" "ctx"
+
+text {* Now Isabelle knows about: *}
+
+typ ctx
+term "\<box>"
+term "CAppL"
+term "CAppL \<box> (Var x)"
+
+subsection {* MINI EXERCISE *}
+
+text {*
+ Try and see what happens if you apply a Hole to a Hole?
+*}
+
+
+section {* Machines for Evaluation *}
+
+type_synonym ctxs = "ctx list"
+
+inductive
+ machine :: "lam \<Rightarrow> ctxs \<Rightarrow>lam \<Rightarrow> ctxs \<Rightarrow> bool" ("<_,_> \<mapsto> <_,_>" [60, 60, 60, 60] 60)
+where
+ m1: "<App t1 t2, Es> \<mapsto> <t1, (CAppL \<box> t2) # Es>"
+| m2: "val v \<Longrightarrow> <v, (CAppL \<box> t2) # Es> \<mapsto> <t2, (CAppR v \<box>) # Es>"
+| m3: "val v \<Longrightarrow> <v, (CAppR (Lam [x].t) \<box>) # Es> \<mapsto> <t[x ::= v], Es>"
+
+text {*
+ Since the machine defined above only performs a single reduction,
+ we need to define the transitive closure of this machine. *}
+
+inductive
+ machines :: "lam \<Rightarrow> ctxs \<Rightarrow> lam \<Rightarrow> ctxs \<Rightarrow> bool" ("<_,_> \<mapsto>* <_,_>" [60, 60, 60, 60] 60)
+where
+ ms1: "<t,Es> \<mapsto>* <t,Es>"
+| ms2: "\<lbrakk><t1,Es1> \<mapsto> <t2,Es2>; <t2,Es2> \<mapsto>* <t3,Es3>\<rbrakk> \<Longrightarrow> <t1,Es1> \<mapsto>* <t3,Es3>"
+
+declare machine.intros[intro] machines.intros[intro]
+
+section {* EXERCISE 3 *}
+
+text {*
+ We need to show that the machines-relation is transitive.
+ Fill in the gaps below.
+*}
+
+lemma ms3[intro]:
+ assumes a: "<e1, Es1> \<mapsto>* <e2, Es2>"
+ and b: "<e2, Es2> \<mapsto>* <e3, Es3>"
+ shows "<e1, Es1> \<mapsto>* <e3, Es3>"
+using a b
+proof(induct)
+ case (ms1 e1 Es1)
+ have c: "<e1, Es1> \<mapsto>* <e3, Es3>" by fact
+ then show "<e1, Es1> \<mapsto>* <e3, Es3>" by simp
+next
+ case (ms2 e1 Es1 e2 Es2 e2' Es2')
+ have ih: "<e2', Es2'> \<mapsto>* <e3, Es3> \<Longrightarrow> <e2, Es2> \<mapsto>* <e3, Es3>" by fact
+ have d1: "<e2', Es2'> \<mapsto>* <e3, Es3>" by fact
+ have d2: "<e1, Es1> \<mapsto> <e2, Es2>" by fact
+ have d3: "<e2, Es2> \<mapsto>* <e3, Es3>" using ih d1 by auto
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" using d2 d3 by auto
+qed
+
+text {*
+ Just like gotos in the Basic programming language, labels often reduce
+ the readability of proofs. Therefore one can use in Isar the notation
+ "then have" in order to feed a have-statement to the proof of
+ the next have-statement. This is used in teh second case below.
+*}
+
+lemma
+ assumes a: "<e1, Es1> \<mapsto>* <e2, Es2>"
+ and b: "<e2, Es2> \<mapsto>* <e3, Es3>"
+ shows "<e1, Es1> \<mapsto>* <e3, Es3>"
+using a b
+proof(induct)
+ case (ms1 e1 Es1)
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" by fact
+next
+ case (ms2 e1 Es1 e2 Es2 e2' Es2')
+ have ih: "<e2', Es2'> \<mapsto>* <e3, Es3> \<Longrightarrow> <e2, Es2> \<mapsto>* <e3, Es3>" by fact
+ have "<e2', Es2'> \<mapsto>* <e3, Es3>" by fact
+ then have d3: "<e2, Es2> \<mapsto>* <e3, Es3>" using ih by simp
+ have d2: "<e1, Es1> \<mapsto> <e2, Es2>" by fact
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" using d2 d3 by auto
+qed
+
+text {*
+ The label ih2 cannot be got rid of in this way, because it is used
+ two lines below and we cannot rearange them. We can still avoid the
+ label by feeding a sequence of facts into a proof using the
+ "moreover"-chaining mechanism:
+
+ have "statement_1" \<dots>
+ moreover
+ have "statement_2" \<dots>
+ \<dots>
+ moreover
+ have "statement_n" \<dots>
+ ultimately have "statement" \<dots>
+
+ In this chain, all "statement_i" can be used in the proof of the final
+ "statement". With this we can simplify our proof further to:
+*}
+
+lemma
+ assumes a: "<e1, Es1> \<mapsto>* <e2, Es2>"
+ and b: "<e2, Es2> \<mapsto>* <e3, Es3>"
+ shows "<e1, Es1> \<mapsto>* <e3, Es3>"
+using a b
+proof(induct)
+ case (ms1 e1 Es1)
+ show "<e1, Es1> \<mapsto>* <e3, Es3>" by fact
+next
+ case (ms2 e1 Es1 e2 Es2 e2' Es2')
+ have ih: "<e2', Es2'> \<mapsto>* <e3, Es3> \<Longrightarrow> <e2, Es2> \<mapsto>* <e3, Es3>" by fact
+ have "<e2', Es2'> \<mapsto>* <e3, Es3>" by fact
+ then have "<e2, Es2> \<mapsto>* <e3, Es3>" using ih by simp
+ moreover
+ have "<e1, Es1> \<mapsto> <e2, Es2>" by fact
+ ultimately show "<e1, Es1> \<mapsto>* <e3, Es3>" by auto
+qed
+
+
+text {*
+ While automatic proof procedures in Isabelle are not able to prove statements
+ like "P = NP" assuming usual definitions for P and NP, they can automatically
+ discharge the lemmas we just proved. For this we only have to set up the induction
+ and auto will take care of the rest. This means we can write:
+*}
+
+lemma
+ assumes a: "val t"
+ shows "t \<Down> t"
+using a by (induct) (auto)
+
+lemma
+ assumes a: "t \<Down> t'"
+ shows "val t'"
+using a by (induct) (auto)
+
+lemma
+ assumes a: "<e1, Es1> \<mapsto>* <e2, Es2>"
+ and b: "<e2, Es2> \<mapsto>* <e3, Es3>"
+ shows "<e1, Es1> \<mapsto>* <e3, Es3>"
+using a b by (induct) (auto)
+
+
+section {* EXERCISE 4 *}
+
+text {*
+ The point of the machine is that it simulates the evaluation
+ relation. Therefore we like to prove the following:
+*}
+
+theorem
+ assumes a: "t \<Down> t'"
+ shows "<t, Es> \<mapsto>* <t', Es>"
+using a
+proof (induct arbitrary: Es)
+ case (e_Lam x t Es)
+ -- {* no assumptions *}
+ show "<Lam [x].t, Es> \<mapsto>* <Lam [x].t, Es>" by auto
+next
+ case (e_App t1 x t t2 v' v Es)
+ -- {* all assumptions in this case *}
+ have a1: "t1 \<Down> Lam [x].t" by fact
+ have ih1: "\<And>Es. <t1, Es> \<mapsto>* <Lam [x].t, Es>" by fact
+ have a2: "t2 \<Down> v'" by fact
+ have ih2: "\<And>Es. <t2, Es> \<mapsto>* <v', Es>" by fact
+ have a3: "t[x::=v'] \<Down> v" by fact
+ have ih3: "\<And>Es. <t[x::=v'], Es> \<mapsto>* <v, Es>" by fact
+ -- {* your reasoning *}
+ have "<App t1 t2, Es> \<mapsto>* <t1, CAppL \<box> t2 # Es>" by auto
+ moreover
+ have "<t1, CAppL \<box> t2 # Es> \<mapsto>* <Lam [x].t, CAppL \<box> t2 # Es>" using ih1 by auto
+ moreover
+ have "<Lam [x].t, CAppL \<box> t2 # Es> \<mapsto>* <t2, CAppR (Lam [x].t) \<box> # Es>" by auto
+ moreover
+ have "<t2, CAppR (Lam [x].t) \<box> # Es> \<mapsto>* <v', CAppR (Lam [x].t) \<box> # Es>"
+ using ih2 by auto
+ moreover
+ have "val v'" using a2 eval_to_val by auto
+ then have "<v', CAppR (Lam [x].t) \<box> # Es> \<mapsto>* <t[x::=v'], Es>" by auto
+ moreover
+ have "<t[x::=v'], Es> \<mapsto>* <v, Es>" using ih3 by auto
+ ultimately show "<App t1 t2, Es> \<mapsto>* <v, Es>" by blast
+qed
+
+
+text {*
+ Again the automatic tools in Isabelle can discharge automatically
+ of the routine work in these proofs. We can write:
+*}
+
+theorem eval_implies_machines_ctx:
+ assumes a: "t \<Down> t'"
+ shows "<t, Es> \<mapsto>* <t', Es>"
+using a
+by (induct arbitrary: Es)
+ (metis eval_to_val machine.intros ms1 ms2 ms3 v_Lam)+
+
+corollary eval_implies_machines:
+ assumes a: "t \<Down> t'"
+ shows "<t, []> \<mapsto>* <t', []>"
+using a eval_implies_machines_ctx by simp
+
+
+section {* Function Definitions: Filling a Lambda-Term into a Context *}
+
+text {*
+ Many functions over datatypes can be defined by recursion on the
+ structure. For this purpose, Isabelle provides "fun". To use it one needs
+ to give a name for the function, its type, optionally some pretty-syntax
+ and then some equations defining the function. Like in "inductive",
+ "fun" expects that more than one such equation is separated by "|".
+*}
+
+fun
+ filling :: "ctx \<Rightarrow> lam \<Rightarrow> lam" ("_\<lbrakk>_\<rbrakk>" [100, 100] 100)
+where
+ "\<box>\<lbrakk>t\<rbrakk> = t"
+| "(CAppL E t')\<lbrakk>t\<rbrakk> = App (E\<lbrakk>t\<rbrakk>) t'"
+| "(CAppR t' E)\<lbrakk>t\<rbrakk> = App t' (E\<lbrakk>t\<rbrakk>)"
+
+
+text {*
+ After this definition Isabelle will be able to simplify
+ statements like:
+*}
+
+lemma
+ shows "(CAppL \<box> (Var x))\<lbrakk>Var y\<rbrakk> = App (Var y) (Var x)"
+ by simp
+
+fun
+ ctx_compose :: "ctx \<Rightarrow> ctx \<Rightarrow> ctx" (infixr "\<odot>" 99)
+where
+ "\<box> \<odot> E' = E'"
+| "(CAppL E t') \<odot> E' = CAppL (E \<odot> E') t'"
+| "(CAppR t' E) \<odot> E' = CAppR t' (E \<odot> E')"
+
+fun
+ ctx_composes :: "ctxs \<Rightarrow> ctx" ("_\<down>" [110] 110)
+where
+ "[]\<down> = \<box>"
+ | "(E # Es)\<down> = (Es\<down>) \<odot> E"
+
+text {*
+ Notice that we not just have given a pretty syntax for the functions, but
+ also some precedences. The numbers inside the [\<dots>] stand for the precedences
+ of the arguments; the one next to it the precedence of the whole term.
+
+ This means we have to write (Es1 \<odot> Es2) \<odot> Es3 otherwise Es1 \<odot> Es2 \<odot> Es3 is
+ interpreted as Es1 \<odot> (Es2 \<odot> Es3).
+*}
+
+section {* Structural Inductions over Contexts *}
+
+text {*
+ So far we have had a look at an induction over an inductive predicate.
+ Another important induction principle is structural inductions for
+ datatypes. To illustrate structural inductions we prove some facts
+ about context composition, some of which we will need later on.
+*}
+
+subsection {* EXERCISE 5 *}
+
+text {* Complete the proof and remove the sorries. *}
+
+lemma
+ shows "(E1 \<odot> E2)\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>"
+proof (induct E1)
+ case Hole
+ show "(\<box> \<odot> E2)\<lbrakk>t\<rbrakk> = \<box>\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" by simp
+next
+ case (CAppL E1 t')
+ have ih: "(E1 \<odot> E2)\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" by fact
+ show "((CAppL E1 t') \<odot> E2)\<lbrakk>t\<rbrakk> = (CAppL E1 t')\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" using ih by simp
+next
+ case (CAppR t' E1)
+ have ih: "(E1 \<odot> E2)\<lbrakk>t\<rbrakk> = E1\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" by fact
+ show "((CAppR t' E1) \<odot> E2)\<lbrakk>t\<rbrakk> = (CAppR t' E1)\<lbrakk>E2\<lbrakk>t\<rbrakk>\<rbrakk>" using ih by simp
+qed
+
+subsection {* EXERCISE 6 *}
+
+text {*
+ Remove the sorries in the ctx_append proof below. You can make
+ use of the following two properties.
+*}
+
+lemma neut_hole:
+ shows "E \<odot> \<box> = E"
+by (induct E) (simp_all)
+
+lemma compose_assoc:
+ shows "(E1 \<odot> E2) \<odot> E3 = E1 \<odot> (E2 \<odot> E3)"
+by (induct E1) (simp_all)
+
+lemma
+ shows "(Es1 @ Es2)\<down> = (Es2\<down>) \<odot> (Es1\<down>)"
+proof (induct Es1)
+ case Nil
+ show "([] @ Es2)\<down> = Es2\<down> \<odot> []\<down>" using neut_hole by simp
+next
+ case (Cons E Es1)
+ have ih: "(Es1 @ Es2)\<down> = Es2\<down> \<odot> Es1\<down>" by fact
+ have eq1: "((E # Es1) @ Es2)\<down> = (E # (Es1 @ Es2))\<down>" by simp
+ have eq2: "(E # (Es1 @ Es2))\<down> = (Es1 @ Es2)\<down> \<odot> E" by simp
+ have eq3: "(Es1 @ Es2)\<down> \<odot> E = (Es2\<down> \<odot> Es1\<down>) \<odot> E" using ih by simp
+ have eq4: "(Es2\<down> \<odot> Es1\<down>) \<odot> E = Es2\<down> \<odot> (Es1\<down> \<odot> E)" using compose_assoc by simp
+ have eq5: "Es2\<down> \<odot> (Es1\<down> \<odot> E) = Es2\<down> \<odot> (E # Es1)\<down>" by simp
+ show "((E # Es1) @ Es2)\<down> = Es2\<down> \<odot> (E # Es1)\<down>" using eq1 eq2 eq3 eq4 eq5 by (simp only:)
+qed
+
+text {*
+ The last proof involves several steps of equational reasoning.
+ Isar provides some convenient means to express such equational
+ reasoning in a much cleaner fashion using the "also have"
+ construction.
+*}
+
+lemma
+ shows "(Es1 @ Es2)\<down> = (Es2\<down>) \<odot> (Es1\<down>)"
+proof (induct Es1)
+ case Nil
+ show "([] @ Es2)\<down> = Es2\<down> \<odot> []\<down>" using neut_hole by simp
+next
+ case (Cons E Es1)
+ have ih: "(Es1 @ Es2)\<down> = Es2\<down> \<odot> Es1\<down>" by fact
+ have "((E # Es1) @ Es2)\<down> = (E # (Es1 @ Es2))\<down>" by simp
+ also have "\<dots> = (Es1 @ Es2)\<down> \<odot> E" by simp
+ also have "\<dots> = (Es2\<down> \<odot> Es1\<down>) \<odot> E" using ih by simp
+ also have "\<dots> = Es2\<down> \<odot> (Es1\<down> \<odot> E)" using compose_assoc by simp
+ also have "\<dots> = Es2\<down> \<odot> (E # Es1)\<down>" by simp
+ finally show "((E # Es1) @ Es2)\<down> = Es2\<down> \<odot> (E # Es1)\<down>" by simp
+qed
+
+
+end
+