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Proof step 161 from proof bitsand-cast (previous step)

n0 + # bits2 + # bits = # fillsgn(n0 + # bits2 + # bits, mklist(true, m - # bits2) + bits2 + bitsand(bits, true ' + false ' + bits1)), # bits = 2 + # bits1, n0 + # bits2 + # bits = # fillsgn(n0 + # bits2 + # bits, mklist(true, m) + true ' + false ' + bits1), sgn(mklist(true, m) + true ' + false ' + bits1) = true ' + false ' + bits1, bits2.first, # bits2 < m, 2 + # bits1 + m < n0 + # bits2 + # bits ⊦ mklist(true, n0) + bits2 + bitsand(bits, true ' + false ' + bits1) = fillsgn(n0 + # bits2 + # bits, mklist(true, m - # bits2) + bits2 + bitsand(bits, true ' + false ' + bits1)), bits2 = [], bits2 + bitsand(bits, true ' + false ' + bits1) = [], bits = [], bits2 + bits = []

User applied rule insert elim lemma

And obtained the following new goals

Proof step 162:
m = n + # bits2 ∧ # bits2 ≤ m, n0 + # bits2 + # bits = # fillsgn(n0 + # bits2 + # bits, mklist(true, n) + bits2 + bitsand(bits, true ' + false ' + bits1)), # bits = 2 + # bits1, n0 + # bits2 + # bits = # fillsgn(n0 + # bits2 + # bits, mklist(true, m) + true ' + false ' + bits1), sgn(mklist(true, m) + true ' + false ' + bits1) = true ' + false ' + bits1, bits2.first, # bits2 < m, 2 + # bits1 + m < n0 + # bits2 + # bits ⊦ mklist(true, n0) + bits2 + bitsand(bits, true ' + false ' + bits1) = fillsgn(n0 + # bits2 + # bits, mklist(true, n) + bits2 + bitsand(bits, true ' + false ' + bits1)), bits2 = [], bits2 + bitsand(bits, true ' + false ' + bits1) = [], bits = [], bits2 + bits = []

Rule argument

seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [m, n, n0] → [m, # bits2, n] rotate: T spec: nat inst: name: elim-sub

Additional information

Used simplifier rules: ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m < n → ¬ n < m;%n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: m, # bits2, n