[isabelle] questions about forall


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,

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"

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

  big_step :: "com × state ⇒ state ⇒ bool" (infix "⇒" 55)
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


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