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Proof step 7 from proof min-in (previous step)

i0 ≤ i, ∀ x0. # x0 < # (i + i0 + x) → (true → x0 = [] ∨ min(x0) ∈ x0) ⊦ min(i0, min(i ' + x)) = i, min(i0, min(i ' + x)) = i0, min(i0, min(i ' + x)) ∈ x

User applied rule apply induction

And obtained the following new goals

Proof step 6:
i ' + x = [] ∨ min(i ' + x) ∈ i ' + x, i0 ≤ i, ∀ x0. # x0 < # (i + i0 + x) → (true → x0 = [] ∨ min(x0) ∈ x0) ⊦ min(i0, min(i ' + x)) = i, min(i0, min(i ' + x)) = i0, min(i0, min(i ' + x)) ∈ x

Rule argument

[substlist-arg: [x0] → [i ' + x]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ # (x + y) = # x + # y], [CSIMPSEQ : ⊦ # d ' = 0 +1], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ d + x = d ' + x], [CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (i ' + x)]]