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https://github.com/ethereum/solidity
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35 lines
718 B
Python
35 lines
718 B
Python
from rule import Rule
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from opcodes import *
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from util import *
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"""
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Checking conversion of exp(2, X) to shl(X, 1)
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"""
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rule = Rule()
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n_bits = 256
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# Proof of exp(2, X) = shl(X, 1) by induction:
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#
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# Base case: X = 0, exp(2, 0) = 1 = 1 = shl(0, 1)
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# Inductive step: assuming exp(2, X) = shl(X, 1) for X <= N
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# to prove: exp(2, N + 1) = shl(N + 1, 1)
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#
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# Notice that exp(2, N + 1) = 2 * exp(2, N) mod 2**256
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# since exp(2, N) = shl(N, 1), it is enough to show that
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# 2 * shl(N, 1) mod 2**256 = shl(N + 1, 1)
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#
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# Also note that N + 1 < 2**256
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N = BitVec('N', n_bits)
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inductive_step = 2 * SHL(N, 1)
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rule.check(
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inductive_step,
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If(
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N == 2**256 - 1,
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0,
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SHL(N + 1, 1)
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)
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)
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