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Proof step 8 from proof intersection-size (previous step)

# s0 + # s1 + 1 < #(s0 ∩ (s1 ++ a)), ∀ s0. ¬ # s0 + # s1 < #(s0 ∩ s1) ⊦ a ∈ s1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ a ∈ s → s ∩ (s0 ++ a) = (s ∩ s0) ++ a
Proof step 10:
# s0 + # s1 + 1 < #(s0 ∩ (s1 ++ a)), ∀ s0. ¬ # s0 + # s1 < #(s0 ∩ s1) ⊦ a ∈ s1, a ∈ s0
Proof step 12:
# s0 + # s1 + 1 < #((s0 ∩ s1) ++ a), a ∈ s0, ∀ s0. ¬ # s0 + # s1 < #(s0 ∩ s1) ⊦ a ∈ s1

Rule argument

spec: <toplevel> name: intersection-shift seq: ⊦ a ∈ s → s ∩ (s0 ++ a) = (s ∩ s0) ++ a subst: [a, s0, s] → [a, s1, s0] flags: <all paths>

Additional information

Used terms: a, s1, s0