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Proof step 9 from proof linearity-path (previous step)

path(H[r].nxt ' + x0, H), path(r ' + y, H), r ∈ H, ∀ r, H, y. path(r ' + x0, H) ∧ path(r ' + y, H) → x0 = y ∨ (∃ r0, x1. y = x0 + r0 ' + x1) ∨ (∃ r0, y0. x0 = y + r0 ' + y0) ⊦ H[r].nxt ' + x0 = y, r = null, ∃ r1, y0. H[r].nxt ' + x0 = y + r1 ' + y0, ∃ r1, x. y = (H[r].nxt ' + x0) + r1 ' + x

User applied rule constructor cut

And obtained the following new goals

Proof step 10:
y = [], path(r ' + y, H), path(H[r].nxt ' + x0, H), r ∈ H, ∀ r, H, y. path(r ' + x0, H) ∧ path(r ' + y, H) → x0 = y ∨ (∃ r0, x1. y = x0 + r0 ' + x1) ∨ (∃ r0, y0. x0 = y + r0 ' + y0) ⊦ H[r].nxt ' + x0 = y, r = null, ∃ r1, x. y = (H[r].nxt ' + x0) + r1 ' + x, ∃ r1, y0. H[r].nxt ' + x0 = y + r1 ' + y0
Proof step 12:
y = r2 + y1, path(r ' + y, H), path(H[r].nxt ' + x0, H), r ∈ H, ∀ r, H, y. path(r ' + x0, H) ∧ path(r ' + y, H) → x0 = y ∨ (∃ r0, x1. y = x0 + r0 ' + x1) ∨ (∃ r0, y0. x0 = y + r0 ' + y0) ⊦ H[r].nxt ' + x0 = y, r = null, ∃ r1, x. y = (H[r].nxt ' + x0) + r1 ' + x, ∃ r1, y0. H[r].nxt ' + x0 = y + r1 ' + y0

Rule argument

var: y

Additional information

Used simplifier rules: reflist freely generated by [], +;;