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Proof step 2 from proof union-diff-distr (previous step)

∀ a. count(a, bag1) ≤ count(a, bag0) ⊦ (count(a, bag0) - count(a, bag1)) + count(a, bag2) = (count(a, bag1) ≤ count(a, bag0) + count(a, bag2) ⊃ count(a, bag0) + count(a, bag2) - count(a, bag1);0)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 3:
∀ a. count(a, bag1) ≤ count(a, bag0) ⊦ count(a, bag1) ≤ count(a, bag0), (count(a, bag0) - count(a, bag1)) + count(a, bag2) = (count(a, bag1) ≤ count(a, bag0) + count(a, bag2) ⊃ count(a, bag0) + count(a, bag2) - count(a, bag1);0)
Proof step 5:
count(a, bag0) = n0 + count(a, bag1) ∧ count(a, bag1) ≤ count(a, bag0), ∀ a. count(a, bag1) ≤ count(a, bag0) ⊦ n0 + count(a, bag2) = (count(a, bag1) ≤ count(a, bag0) + count(a, bag2) ⊃ count(a, bag0) + count(a, bag2) - count(a, bag1);0)

Rule argument

spec: nat inst: name: elim-sub seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [m, n, n0] → [count(a, bag0), count(a, bag1), n] flags: []

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: count(a, bag0), count(a, bag1), n