31. A formal proof of the lemma relies on taking the limit of a sequence of random variables. 32. The realizer, however, usually gives more information about the formula than a formal proof would directly provide. 33. John Cowles developed a formal proof that Knuth's generalized function was total, using the ACL2 theorem prover. 34. The FBI estimated the cost at dlrs 250, 000, although no formal proof was provided in court. 35. In other formal proof systems, the deduction theorem is sometimes taken as a primitive rule of inference. 36. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use. 37. The military frustrated from not having received formal proof of Chavez's resignation withdrew their support of Carmona's presidency. 38. We will present the applicable result from ( Wald, 1984 eq 11.2 . 10 ) without a formal proof . 39. For the formal proof , algorithms are presumed to define partial functions over strings and are themselves represented by strings. 40. On 10 August 2014 Hales announced the completion of a formal proof using automated proof checking, removing any doubt.
