Hello,
concerning the syntax errors:
1. Your _expression_ "CHOOSE k \in Nat : P(k)" appears where TLA+ expects a definition, an axiom / assumption or a theorem. Since m and k are declared as constants, I believe what you really mean is
ASSUME k \in Nat /\ (k > (1+(1+4/m)^0.5) / 2) /\ (k < (2+(1+4/m)^0.5) / 2)
Alternatively, you could remove the constant parameter k and write the definition
k == CHOOSE k \in Nat : ( (k > (1+(1+4/m)^0.5) / 2) /\ (k < (2+(1+4/m)^0.5) / 2) )
but presumably you'd like to check your spec for all possible values of k rather than a fixed one.
2. The final conjunct of your next-state relation is incomplete because the ELSE branch is missing. Perhaps you meant to write
c' >= k => E' >= r' * c'
in order to restrict transitions to those that satisfy this implication, but perhaps you rather meant to assert the invariant
c >= k => E >= r * c
and verify that this implication is true for every state along every run of the system? In that case, you should write a definition like
Inv == c >= k => E >= r * c
and use the TLA+ tools to verify that it holds (but see below).
3. [][Next] is not a well-formed formula of TLA+, you should write
Spec == Init /\ [][Next]_<<c,E,r>>
4. Although not an error, it is a bit strange that your definitions of the initial predicate and the next-state relation are written as disjunctions with a single disjunct. You may want to remove the leading "\/" (but perhaps this is a simplified version and you have other disjuncts in mind).
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More fundamentally, while TLA+ has a standard module representing the real numbers, the currently available TLA+ tools (the model checkers TLC and Apalache as well as the interactive proof system TLAPS) do not support real numbers. I am therefore afraid that TLA+ is not the most appropriate formalism for what you are trying to do.
Best regards,
Stephan