# Theory Examples

theory Examples
imports Predicate_Compile_Alternative_Defs
```theory Examples
imports Main "HOL-Library.Predicate_Compile_Alternative_Defs"
begin

declare [[values_timeout = 480.0]]

section ‹Formal Languages›

subsection ‹General Context Free Grammars›

text ‹a contribution by Aditi Barthwal›

datatype ('nts,'ts) symbol = NTS 'nts
| TS 'ts

datatype ('nts,'ts) rule = rule 'nts "('nts,'ts) symbol list"

type_synonym ('nts,'ts) grammar = "('nts,'ts) rule set * 'nts"

fun rules :: "('nts,'ts) grammar => ('nts,'ts) rule set"
where
"rules (r, s) = r"

definition derives
where
"derives g = { (lsl,rsl). ∃s1 s2 lhs rhs.
(s1 @ [NTS lhs] @ s2 = lsl) ∧
(s1 @ rhs @ s2) = rsl ∧
(rule lhs rhs) ∈ fst g }"

definition derivesp ::
"(('nts, 'ts) rule => bool) * 'nts => ('nts, 'ts) symbol list => ('nts, 'ts) symbol list => bool"
where
"derivesp g = (λ lhs rhs. (lhs, rhs) ∈ derives (Collect (fst g), snd g))"

lemma [code_pred_def]:
"derivesp g = (λ lsl rsl. ∃s1 s2 lhs rhs.
(s1 @ [NTS lhs] @ s2 = lsl) ∧
(s1 @ rhs @ s2) = rsl ∧
(fst g) (rule lhs rhs))"
unfolding derivesp_def derives_def by auto

abbreviation "example_grammar ==
({ rule ''S'' [NTS ''A'', NTS ''B''],
rule ''S'' [TS ''a''],
rule ''A'' [TS ''b'']}, ''S'')"

definition "example_rules ==
(%x. x = rule ''S'' [NTS ''A'', NTS ''B''] ∨
x = rule ''S'' [TS ''a''] ∨
x = rule ''A'' [TS ''b''])"

code_pred [inductify, skip_proof] derivesp .

thm derivesp.equation

definition "testp = (% rhs. derivesp (example_rules, ''S'') [NTS ''S''] rhs)"

code_pred (modes: o ⇒ bool) [inductify] testp .
thm testp.equation

values "{rhs. testp rhs}"

declare rtranclp.intros(1)[code_pred_def] converse_rtranclp_into_rtranclp[code_pred_def]

code_pred [inductify] rtranclp .

definition "test2 = (λ rhs. rtranclp (derivesp (example_rules, ''S'')) [NTS ''S''] rhs)"

code_pred [inductify, skip_proof] test2 .

values "{rhs. test2 rhs}"

subsection ‹Some concrete Context Free Grammars›

datatype alphabet = a | b

inductive_set S⇩1 and A⇩1 and B⇩1 where
"[] ∈ S⇩1"
| "w ∈ A⇩1 ⟹ b # w ∈ S⇩1"
| "w ∈ B⇩1 ⟹ a # w ∈ S⇩1"
| "w ∈ S⇩1 ⟹ a # w ∈ A⇩1"
| "w ∈ S⇩1 ⟹ b # w ∈ S⇩1"
| "⟦v ∈ B⇩1; v ∈ B⇩1⟧ ⟹ a # v @ w ∈ B⇩1"

code_pred [inductify] S⇩1p .
code_pred [random_dseq inductify] S⇩1p .
thm S⇩1p.equation
thm S⇩1p.random_dseq_equation

values [random_dseq 5, 5, 5] 5 "{x. S⇩1p x}"

inductive_set S⇩2 and A⇩2 and B⇩2 where
"[] ∈ S⇩2"
| "w ∈ A⇩2 ⟹ b # w ∈ S⇩2"
| "w ∈ B⇩2 ⟹ a # w ∈ S⇩2"
| "w ∈ S⇩2 ⟹ a # w ∈ A⇩2"
| "w ∈ S⇩2 ⟹ b # w ∈ B⇩2"
| "⟦v ∈ B⇩2; v ∈ B⇩2⟧ ⟹ a # v @ w ∈ B⇩2"

code_pred [random_dseq inductify] S⇩2p .
thm S⇩2p.random_dseq_equation
thm A⇩2p.random_dseq_equation
thm B⇩2p.random_dseq_equation

values [random_dseq 5, 5, 5] 10 "{x. S⇩2p x}"

inductive_set S⇩3 and A⇩3 and B⇩3 where
"[] ∈ S⇩3"
| "w ∈ A⇩3 ⟹ b # w ∈ S⇩3"
| "w ∈ B⇩3 ⟹ a # w ∈ S⇩3"
| "w ∈ S⇩3 ⟹ a # w ∈ A⇩3"
| "w ∈ S⇩3 ⟹ b # w ∈ B⇩3"
| "⟦v ∈ B⇩3; w ∈ B⇩3⟧ ⟹ a # v @ w ∈ B⇩3"

code_pred [inductify, skip_proof] S⇩3p .
thm S⇩3p.equation

values 10 "{x. S⇩3p x}"

inductive_set S⇩4 and A⇩4 and B⇩4 where
"[] ∈ S⇩4"
| "w ∈ A⇩4 ⟹ b # w ∈ S⇩4"
| "w ∈ B⇩4 ⟹ a # w ∈ S⇩4"
| "w ∈ S⇩4 ⟹ a # w ∈ A⇩4"
| "⟦v ∈ A⇩4; w ∈ A⇩4⟧ ⟹ b # v @ w ∈ A⇩4"
| "w ∈ S⇩4 ⟹ b # w ∈ B⇩4"
| "⟦v ∈ B⇩4; w ∈ B⇩4⟧ ⟹ a # v @ w ∈ B⇩4"

code_pred (expected_modes: o => bool, i => bool) S⇩4p .

hide_const a b

section ‹Semantics of programming languages›

subsection ‹IMP›

type_synonym var = nat
type_synonym state = "int list"

datatype com =
Skip |
Ass var "state => int" |
Seq com com |
IF "state => bool" com com |
While "state => bool" com

inductive exec :: "com => state => state => bool" where
"exec Skip s s" |
"exec (Ass x e) s (s[x := e(s)])" |
"exec c1 s1 s2 ==> exec c2 s2 s3 ==> exec (Seq c1 c2) s1 s3" |
"b s ==> exec c1 s t ==> exec (IF b c1 c2) s t" |
"~b s ==> exec c2 s t ==> exec (IF b c1 c2) s t" |
"~b s ==> exec (While b c) s s" |
"b s1 ==> exec c s1 s2 ==> exec (While b c) s2 s3 ==> exec (While b c) s1 s3"

code_pred exec .

values "{t. exec
(While (%s. s!0 > 0) (Seq (Ass 0 (%s. s!0 - 1)) (Ass 1 (%s. s!1 + 1))))
[3,5] t}"

subsection ‹Lambda›

datatype type =
Atom nat
| Fun type type    (infixr "⇒" 200)

datatype dB =
Var nat
| App dB dB (infixl "°" 200)
| Abs type dB

primrec
nth_el :: "'a list ⇒ nat ⇒ 'a option" ("_⟨_⟩" [90, 0] 91)
where
"[]⟨i⟩ = None"
| "(x # xs)⟨i⟩ = (case i of 0 ⇒ Some x | Suc j ⇒ xs ⟨j⟩)"

inductive nth_el' :: "'a list ⇒ nat ⇒ 'a ⇒ bool"
where
"nth_el' (x # xs) 0 x"
| "nth_el' xs i y ⟹ nth_el' (x # xs) (Suc i) y"

inductive typing :: "type list ⇒ dB ⇒ type ⇒ bool"  ("_ ⊢ _ : _" [50, 50, 50] 50)
where
Var [intro!]: "nth_el' env x T ⟹ env ⊢ Var x : T"
| Abs [intro!]: "T # env ⊢ t : U ⟹ env ⊢ Abs T t : (T ⇒ U)"
| App [intro!]: "env ⊢ s : T ⇒ U ⟹ env ⊢ t : T ⟹ env ⊢ (s ° t) : U"

primrec
lift :: "[dB, nat] => dB"
where
"lift (Var i) k = (if i < k then Var i else Var (i + 1))"
| "lift (s ° t) k = lift s k ° lift t k"
| "lift (Abs T s) k = Abs T (lift s (k + 1))"

primrec
subst :: "[dB, dB, nat] => dB"  ("_[_'/_]" [300, 0, 0] 300)
where
subst_Var: "(Var i)[s/k] =
(if k < i then Var (i - 1) else if i = k then s else Var i)"
| subst_App: "(t ° u)[s/k] = t[s/k] ° u[s/k]"
| subst_Abs: "(Abs T t)[s/k] = Abs T (t[lift s 0 / k+1])"

inductive beta :: "[dB, dB] => bool"  (infixl "→⇩β" 50)
where
beta [simp, intro!]: "Abs T s ° t →⇩β s[t/0]"
| appL [simp, intro!]: "s →⇩β t ==> s ° u →⇩β t ° u"
| appR [simp, intro!]: "s →⇩β t ==> u ° s →⇩β u ° t"
| abs [simp, intro!]: "s →⇩β t ==> Abs T s →⇩β Abs T t"

code_pred (expected_modes: i => i => o => bool, i => i => i => bool) typing .
thm typing.equation

code_pred (modes: i => i => bool,  i => o => bool as reduce') beta .
thm beta.equation

values "{x. App (Abs (Atom 0) (Var 0)) (Var 1) →⇩β x}"

definition "reduce t = Predicate.the (reduce' t)"

value "reduce (App (Abs (Atom 0) (Var 0)) (Var 1))"

code_pred [dseq] typing .
code_pred [random_dseq] typing .

values [random_dseq 1,1,5] 10 "{(Γ, t, T). Γ ⊢ t : T}"

subsection ‹A minimal example of yet another semantics›

text ‹thanks to Elke Salecker›

type_synonym vname = nat
type_synonym vvalue = int
type_synonym var_assign = "vname ⇒ vvalue"  ― ‹variable assignment›

datatype ir_expr =
IrConst vvalue
| Add ir_expr ir_expr

datatype val =
IntVal  vvalue

record  configuration =
Env :: var_assign

inductive eval_var ::
"ir_expr ⇒ configuration ⇒ val ⇒ bool"
where
irconst: "eval_var (IrConst i) conf (IntVal i)"
| objaddr: "⟦ Env conf n = i ⟧ ⟹ eval_var (ObjAddr n) conf (IntVal i)"
| plus: "⟦ eval_var l conf (IntVal vl); eval_var r conf (IntVal vr) ⟧ ⟹
eval_var (Add l r) conf (IntVal (vl+vr))"

code_pred eval_var .
thm eval_var.equation

values "{val. eval_var (Add (IrConst 1) (IrConst 2)) (| Env = (λx. 0)|) val}"

subsection ‹Another semantics›

type_synonym name = nat ― ‹For simplicity in examples›
type_synonym state' = "name ⇒ nat"

datatype aexp = N nat | V name | Plus aexp aexp

fun aval :: "aexp ⇒ state' ⇒ nat" where
"aval (N n) _ = n" |
"aval (V x) st = st x" |
"aval (Plus e⇩1 e⇩2) st = aval e⇩1 st + aval e⇩2 st"

datatype bexp = B bool | Not bexp | And bexp bexp | Less aexp aexp

primrec bval :: "bexp ⇒ state' ⇒ bool" where
"bval (B b) _ = b" |
"bval (Not b) st = (¬ bval b st)" |
"bval (And b1 b2) st = (bval b1 st ∧ bval b2 st)" |
"bval (Less a⇩1 a⇩2) st = (aval a⇩1 st < aval a⇩2 st)"

datatype
com' = SKIP
| Assign name aexp         ("_ ::= _" [1000, 61] 61)
| Semi   com'  com'          ("_; _"  [60, 61] 60)
| If     bexp com' com'     ("IF _ THEN _ ELSE _"  [0, 0, 61] 61)
| While  bexp com'         ("WHILE _ DO _"  [0, 61] 61)

inductive
big_step :: "com' * state' ⇒ state' ⇒ bool" (infix "⇒" 55)
where
Skip:    "(SKIP,s) ⇒ s"
| Assign:  "(x ::= a,s) ⇒ s(x := aval a s)"

| Semi:    "(c⇩1,s⇩1) ⇒ s⇩2  ⟹  (c⇩2,s⇩2) ⇒ s⇩3  ⟹ (c⇩1;c⇩2, s⇩1) ⇒ s⇩3"

| IfTrue:  "bval b s  ⟹  (c⇩1,s) ⇒ t  ⟹  (IF b THEN c⇩1 ELSE c⇩2, s) ⇒ t"
| IfFalse: "¬bval b s  ⟹  (c⇩2,s) ⇒ t  ⟹  (IF b THEN c⇩1 ELSE c⇩2, s) ⇒ t"

| WhileFalse: "¬bval b s ⟹ (WHILE b DO c,s) ⇒ s"
| WhileTrue:  "bval b s⇩1  ⟹  (c,s⇩1) ⇒ s⇩2  ⟹  (WHILE b DO c, s⇩2) ⇒ s⇩3
⟹ (WHILE b DO c, s⇩1) ⇒ s⇩3"

code_pred big_step .

thm big_step.equation

definition list :: "(nat ⇒ 'a) ⇒ nat ⇒ 'a list" where
"list s n = map s [0 ..< n]"

values [expected "{[42::nat, 43]}"] "{list s 2|s. (SKIP, nth [42, 43]) ⇒ s}"

subsection ‹CCS›

text‹This example formalizes finite CCS processes without communication or
recursion. For simplicity, labels are natural numbers.›

datatype proc = nil | pre nat proc | or proc proc | par proc proc

inductive step :: "proc ⇒ nat ⇒ proc ⇒ bool" where
"step (pre n p) n p" |
"step p1 a q ⟹ step (or p1 p2) a q" |
"step p2 a q ⟹ step (or p1 p2) a q" |
"step p1 a q ⟹ step (par p1 p2) a (par q p2)" |
"step p2 a q ⟹ step (par p1 p2) a (par p1 q)"

code_pred step .

inductive steps where
"steps p [] p" |
"step p a q ⟹ steps q as r ⟹ steps p (a#as) r"

code_pred steps .

values 3
"{as . steps (par (or (pre 0 nil) (pre 1 nil)) (pre 2 nil)) as (par nil nil)}"

values 5
"{as . steps (par (or (pre 0 nil) (pre 1 nil)) (pre 2 nil)) as (par nil nil)}"

values 3 "{(a,q). step (par nil nil) a q}"

end

```