Re: [isabelle] questions about forall



Thanks. It seems like the simplifier can do the job when I have the right assumptions.

I have some follow up questions.

What does the "(1)" do?

I see that using drule (without the "(1)") requires me to prove the other assumption in big_step_determ, while when using "drule (1)"I am not required to do so.

In general, what does adding "(number)" to a drule do? Also can I use it with rule, erule, frule as well?

Also, what is the difference between the following?

  apply (drule (1) big_step_determ, simp)

and

  apply (drule (1)big_step_determ)
  apply (simp)

Amarin

On 10/27/2014 04:17 AM, Peter Lammich wrote:
You need to exploit determinism of your semantics, e.g.

lemma "⟦(WHILE b DO c,s) ⇒ s'';
          bval b s;
          (c,s) ⇒ s'⟧ ⟹
    (WHILE b DO c,s') ⇒ s''"
apply (erule WhileE)
apply simp
apply (drule (1) big_step_determ, simp)
done

--
   Peter

On So, 2014-10-26 at 23:55 -0500, Amarin Phaosawasdi wrote:
Hello,

This question is based on some definitions defined in the book Concrete
Semantics. Here's a high-level description of the problem.

I have to prove something in the form
"⋀ x. assumption1 ⟹ ... ⟹ assumptionN ⟹ thesis"

However, the assumptions give me the exact value of x I needed. All
other values make the assumptions false (hence making the thesis true).

How would I go about proving the theorem, removing "⋀ x" and
instantiating x to be that specific value I wanted?

In particular, given the big step semantics of while loops in a simple
programming language (see below), I'm trying to prove the following lemma.

lemma "⟦(WHILE b DO c,s) ⇒ s'';
          bval b s;
          (c,s) ⇒ s'⟧ ⟹
    (WHILE b DO c,s') ⇒ s''"

I'm stuck at the step where I need to prove.
"⋀s2. bval b s ⟹
         (c, s) ⇒ s' ⟹
         bval b s ⟹
         (c, s) ⇒ s2 ⟹
         (WHILE b DO c, s2) ⇒ s'' ⟹
   (WHILE b DO c, s') ⇒ s''"

In this case, I know that s2 has to be s'.

Below is the code.

Thank you,
Amarin

============================================
theory SemanticsQuestion imports Main begin

type_synonym vname = string
type_synonym val = int
type_synonym state = "vname ⇒ val"

datatype bexp = Bc bool
fun bval :: "bexp ⇒ state ⇒ bool" where
"bval (Bc v) s = v"

datatype
    com = SKIP |
          While bexp com ("(WHILE _/ DO _)"  [0, 61] 61)

inductive
    big_step :: "com × state ⇒ state ⇒ bool" (infix "⇒" 55)
where
WhileFalse: "⟦¬bval b s⟧ ⟹ (WHILE b DO c,s) ⇒ s" |
WhileTrue: "⟦ bval b s1;  (c,s1) ⇒ s2;  (WHILE b DO c,s2) ⇒ s3 ⟧
    ⟹ (WHILE b DO c, s1) ⇒ s3"

declare big_step.intros [intro]
lemmas big_step_induct = big_step.induct[split_format(complete)]
inductive_cases WhileE[elim]: "(WHILE b DO c,s) ⇒ t"

theorem big_step_determ: "⟦ (c,s) ⇒ t; (c,s) ⇒ u ⟧ ⟹ u = t"
    by (induction arbitrary: u as3 rule: big_step_induct) blast+

lemma "⟦(WHILE b DO c,s) ⇒ s'';
          bval b s;
          (c,s) ⇒ s'⟧ ⟹
    (WHILE b DO c,s') ⇒ s''"
apply (erule WhileE)
apply simp
sorry

end







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