Common use of Proof Clause in Contracts

Proof. By assumption H, at least two values v and u are written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time v is written in T [R + i], a value has already been written in T [R + i + 1]. Let p be the process that performs this first write of v in T [R + i + 1]. As v is written to T [R + K], p must exist by Lemma 1. Denote wp(R +i+1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p that returns ⊥ , and (3) in round R + i, there is a read operation from T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ for some i′′ ≥ i′, since the write of v in T [R + i] occurs in eR+i′ . Therefore, operation rp(R + i + 1) occurs after a write in T [R + i + 1], from which we conclude that rp(R + i + 1) returns a non-⊥ value. It thus follows by Xxxxx 4 that p does not write in T [R + i + 1] : a contradiction. We have shown that a write of v in T [R + i] occurs in epoch ei. A similar argument applied to value u yields that a write of u in T [R + i] occurs in ei. Since each process does not write twice in the same register, |Wi| ≥ 2.

Proof. By assumption H, at least We start by establishing that two values read operations that return v and u are respectively occur in ei. As v is written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time v is written in T [R + i], a value has already been also written in T [R + i + 1]] (Lemma 1). Let p be the process that performs this the first write of v in T [R + i + 1]. As v is written to T [R + K]By the code, p must exist by Lemma 1. Denote wp(R +i+1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing executes round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p before performing that returns ⊥ write operation, and (3) v is the estimate of p in that round. At the beginning of round R + i, there is a read operation from p either reads v in T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ for some i′′ ≥ i′]. Moreover, since the write of v in read operation on T [R + ii + 1] occurs in eR+i′ . Therefore, performed by p at the beginning of round R + i + 1 returns ⊥ (Otherwise p does not perform a write operation rp(R on T [R + i + 1) ]). Therefore, every operation performed by p while it is executing round R + i occurs in epoch eR+i. K In particular, the read of T [R + j] performed by p at line 10 occurs in eR+i. This read must return v. Otherwise, p writes true in C[R + i], and this operation occurs in eR+i. As no process ever writes false in C[R + i], every read operation performed on C[R + i] that occurs in later epochs return true. Consider a process p′ executing round R + K. p′ reads C[R + i] at line 15. This read operation occurs after a write operation has been performed on T [R + ], so it occurs after the end of epoch eR+i. Hence, that operation returns true and thus p′ cannot write in D in that round. Therefore, no value is committed in round R + K, contradicting assumption H. Similarly, by considering the process that performs the first write of u in T [R + i + 1], from which we conclude get that rp(R + i + 1) returns a non-⊥ value. It thus follows by Xxxxx 4 that p does not write in read operation of T [R + i + 1j] : a contradictionthat returns u occurs in eR+i. We have shown that a write j Finally, as there are two read operations of v in T [R + ij] occurs returning two different values occur in epoch ei. A similar argument applied to value u yields that , there must exist a write of u in operation on T [R + iR+j] that occurs in ei. Since each process does not write twice in the same register, |Wi| ≥ 2.We thus conclude that Wi /= ∅.

Proof. By assumption H, at least We start by establishing that two values read operations that return v and u are respectively occur in ei. As v is written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time v is written in T [R + i], a value has already been also written in T [R + i + 1]] (Lemma 1). Let p be the process that performs this the first write of v in T [R + i + 1]. As v is written to T [R + K]By the code, p must exist by Lemma 1. Denote wp(R +i+1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing executes round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p before performing that returns ⊥ write operation, and (3) v is the estimate of p in that round. At the beginning of round R + i, there is a read operation from p either reads v in T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ for some i′′ ≥ i′]. Moreover, since the write of v in read operation on T [R + ii + 1] occurs in eR+i′ . Therefore, performed by p at the beginning of round R + i + 1 returns ⊥ (Otherwise p does not perform a write operation rp(R on T [R + i + 1) ]). Therefore, every operation performed by p while it is executing round R + i occurs in epoch eR+i. K In particular, the read of T [R + j] performed by p at line 10 occurs in eR+i. This read must return v. Otherwise, p writes true in C[R + i], and this operation occurs in eR+i. As no process ever writes false in C[R + i], every read operation performed on C[R + i] that occurs in later epochs return true. Consider a process p′ executing round R + K. p′ reads C[R + i] at line 15. This read operation occurs after a write operation has been performed on T [R + ], so it occurs after the end of epoch eR+i. Hence, that operation returns true and thus p′ cannot write in D in that round. Therefore, no value is committed in round R + K, contradicting assumption H. Similarly, by considering the process that performs the first write of u in T [R + i + 1], from which we conclude get that rp(R + i + 1) returns a non-⊥ value. It thus follows by Xxxxx 4 that p does not write in read operation of T [R + i + 1j] : a contradictionthat returns u occurs in eR+i. We have shown that a write j Finally, as there are two read operations of v in T [R + ij] occurs returning two different values occur in epoch ei. A similar argument applied to value u yields that , there must exist a write of u in operation on T [R + iR+j] that occurs in ei. Since each process does not write twice in the same register, |Wi| ≥ 2.We thus conclude that Wi ƒ= ∅.

Proof. By assumption H, at least two values v and u are written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. ≥ K Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ eR+i' , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time v is written in T [R + i], a value has already been written in T [R + i + 1R+i+1]. Let p be the process that performs this first write of v in T [R + i + 1R+i+1]. As v is written to T [R + K], p must exist by Lemma 1. Denote wp(R +i+1+ i + 1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p that returns ⊥ , and (3) in round R + i, there is a read operation from T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last ⊥ operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ eR+i'' for some i′′ ≥ i′, since the write of v in T [R + i] occurs in eR+i′ eR+i' . Therefore, operation rp(R + i + 1+1) occurs after a write in T [R + i + 1+1], from which we conclude that rp(R + i + 1) returns a non-⊥ non- value. It thus follows by Xxxxx 4 that p does not write in T [R + i + 1] : a contradiction. We have shown that a write of v in T [R + i] occurs in epoch ei. A similar argument applied to value u yields that a write of u in T [R + i] occurs in ei. Since each process does not write twice in the same register, |Wi| |W i| ≥ 2.

Proof. By assumption H, at least two values v and u are written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. ≥ K Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ eR+it , for some i′ ≥ i. xx x. Suppose for the sake of contradiction that i′ ij > i. Hence, the first time v is written in T [R + i], a value has already been written in T [R + i + 1R+i+1]. Let p be the process that performs this first write of v in T [R + i + 1R+i+1]. As v is written to T [R + K], p must exist by Lemma 1. Denote wp(R +i+1+ i + 1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p that returns ⊥ , and (3) in round R + i, there is a read operation from T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last ⊥ operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ eR+itt for some i′′ ijj ≥ i′ij, since the write of v in T [R + i] occurs in eR+i′ eR+it . Therefore, operation rp(R + i + 1) occurs after a write in T [R + i + 1], from which we conclude that rp(R + i + 1) returns a non-⊥ non- value. It thus follows by Xxxxx Lemma 4 that p does not write in T [R + i + 1] : a contradiction. We have shown that a write of v in T [R + i] occurs in epoch ei. A similar argument applied to value u yields that a write of u in T [R + i] occurs in ei. Since each process does not write twice in the same register, |Wi| ≥ 2.

Proof. By assumption H, at least We start by establishing that two values read operations that return v and u are written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs respectively occur in ei. Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time K As v is written in T [R + i], a value has already been v is also written in T [R + i + 1]] (Lemma 1). Let p be the process that performs this the first write of v in T [R + i + 1]. As v is written to T [R + K]By the code, p must exist by Lemma 1. Denote wp(R +i+1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing executes round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p before performing that returns ⊥ write operation, and (3) v is the ⊥ estimate of p in that round. At the beginning of round R + i, there is a read operation from p either reads v in T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ for some i′′ ≥ i′]. Moreover, since the write of v in read operation on T [R + ii + 1] occurs in eR+i′ . Therefore, performed by p at the beginning of round R + i + 1 returns (Otherwise p does not perform a write operation rp(R on T [R + i + 1) ]). Therefore, every operation performed by p while it is executing round R + i occurs in epoch eR+i. X X K In particular, the read of T [R + j] performed by p at line 10 occurs in eR+i. This read must return v. Otherwise, p writes true in C[R + i], and this operation occurs in eR+i. As no process ever writes false in C[R + i], every read operation performed on C[R+i] that occurs in later epochs return true. Consider a process p′ executing round R+ . p′ reads C[R+i] at line 15. This read operation occurs after a write operation has been performed on T [R+ ], so it occurs after the end of epoch eR+i. Hence, that operation returns true and thus p′ cannot write in D in that round. Therefore, no value is committed in round R + , contradicting assumption H. Similarly, by considering the process that performs the first write of u in T [R + i + 1], from which we conclude get that rp(R + i + 1) returns a non-⊥ value. It thus follows by Xxxxx 4 that p does not write in read operation of T [R + i + 1j] : a contradictionthat returns u occurs in eR+i. We have shown that a write Finally, as there are two read operations of v in T [R + ij] occurs returning two different values occur in epoch ei. A similar argument applied to value u yields that , there must exist a write of u in operation on T [R + ij] that occurs in j ei. Since each process does not write twice in the same register, |Wi| ≥ 2.We thus conclude that W i /= ∅.

Proof. By assumption H, at least two values v and u are written in T [R + K]. It follows from Lemma 1 that v and u must have been written in T [R + i] for each i such that 1 ≤ i < K. It remains to show that such a write operation with parameter v (resp. u) occurs in ei. Let us consider the first write of v in T [R + i]. Clearly, this operation occurs in epoch eR+i′ , for some i′ ≥ i. Suppose for the sake of contradiction that i′ > i. Hence, the first time v is written in T [R + i], a value has already been written in T [R + i + 1]. Let p be the process that performs this first write of v in T [R + i + 1]. As v is written to T [R + K], p must exist by Lemma 1. Denote wp(R +i+1) the write operation of p. According to the code of Janus we know that: (1) p performs that operation while it is executing round R + i + 1 (line 7), (2) wp(R + i + 1) is preceded by a read operation of T [R + i + 1] (denoted rp(R + i + 1)) by p that returns ⊥ , and (3) in round R + i, there is a read operation from T [R + i] that returns v or a write of v by p to T [R + i]. Denote by opp(R + i) this last operation, and opp(R + i), rp(R + i + 1), wp(R + i + 1) the operations that occur in this order. Moreover, opp(R + i), which reads or writes v in T [R + i] occurs in epoch eR+i′′ for some i′′ ≥ i′, since the write of v in T [R + i] occurs in eR+i′ . Therefore, operation rp(R + i + 1) occurs after a write in T [R + i + 1], from which we conclude that rp(R + i + 1) returns a non-⊥ value. It thus follows by Xxxxx Lemma 4 that p does not write in T [R + i + 1] : a contradiction. We have shown that a write of v in T [R + i] occurs in epoch ei. A similar argument applied to value u yields that a write of u in T [R + i] occurs in ei. Since each process does not write twice in the same register, |Wi| ≥ 2.