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Proof step 18 from proof reach-n-path-n (previous step)

reachable-1(H, r, r1), reachable-1(H, r1, r2), reachable-n(H, n, r2, r0), ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n) ⊦ ∃ x. path(x, H) ∧ hd x = r ∧ x.last = r0 ∧ # x = 3 + n

User applied rule constructor cut

And obtained the following new goals

Proof step 19:
n = 0, reachable-1(H, r, r1), reachable-1(H, r1, r2), reachable-n(H, n, r2, r0), ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n) ⊦ ∃ x. path(x, H) ∧ hd x = r ∧ x.last = r0 ∧ # x = 3 + n
Proof step 27:
n = n0 +1, reachable-1(H, r, r1), reachable-1(H, r1, r2), reachable-n(H, n, r2, r0), ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n) ⊦ ∃ x. path(x, H) ∧ hd x = r ∧ x.last = r0 ∧ # x = 3 + n

Rule argument

var: n

Additional information

Used simplifier rules: nat freely generated by 0, +1;;