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Proof step 1 from proof next-perm-smallerPerms (conclusion) (back to tree)

⊦ nextperm(ar1, ar2) → smallerPerms(ar2) = smallerPerms(ar1) ++ ar1

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof next-perm-smallerPerms (previous step) (back to tree)

smallerPerms(ar2) ≠ smallerPerms(ar1) ++ ar1, nextperm(ar1, ar2) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:next-permutation, seq:⊦ nextperm(ar1, ar2) ↔ perm(ar1, ar2) ∧ ar1 ≠ ar2 ∧ ar1 ≤ ar2 ∧ (∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬(ar1 ≤ ar3 ∧ ar3 ≤ ar2)), subst:[ar1, ar2] → [ar1, ar2], <all paths>

Additional information

Used terms: ar1, ar2

Proof step 3 from proof next-perm-smallerPerms (previous step) (back to tree)

⊦ nextperm(ar1, ar2) ↔ perm(ar1, ar2) ∧ ar1 ≠ ar2 ∧ ar1 ≤ ar2 ∧ (∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬(ar1 ≤ ar3 ∧ ar3 ≤ ar2))

Lemma next-permutation

Proof step 4 from proof next-perm-smallerPerms (previous step) (back to tree)

ar1 ≠ ar2, smallerPerms(ar2) ≠ smallerPerms(ar1) ++ ar1, perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬(ar1 ≤ ar3 ∧ ar3 ≤ ar2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof next-perm-smallerPerms (previous step) (back to tree)

ar1 ≠ ar2, smallerPerms(ar2) ≠ smallerPerms(ar1) ++ ar1, perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: set-basic[array-set], name:Extension, seq: ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2), subst:[s1, s2] → [smallerPerms(ar2), smallerPerms(ar1) ++ ar1], <all paths>

Additional information

Used terms: smallerPerms(ar2), smallerPerms(ar1) ++ ar1

Proof step 6 from proof next-perm-smallerPerms (previous step) (back to tree)

⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)

A spec-lemma

Proof step 7 from proof next-perm-smallerPerms (previous step) (back to tree)

ar1 ≠ ar2, perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2, ∃ a. ¬(a ∈ smallerPerms(ar2) ↔ a ∈ smallerPerms(ar1) ++ ar1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: smaller-perms: ⊦ ar0 ∈ smallerPerms(ar) ↔ ar ≠ ar0 ∧ perm(ar, ar0) ∧ ar0 ≤ ar; ⊦ a ∈ s ++ b ↔ a = b ∨ a ∈ s; (s)

Proof step 8 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(ar2 ≠ ar ∧ perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ perm(ar1, ar) ∧ ar ≤ ar1), ar1 ≠ ar2, perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: perm(ar1, ar)

Additional information

Used simplifier rules: ⊦ perm(x, x); (s) ⊦ perm(x, y) ∧ ¬ perm(x0, y) → x ≠ x0; (s) trans: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z); (forward) from specification list-perm[natarray-oarray-sort] Used formulas: perm(ar1, ar)

Proof step 9 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(ar2 ≠ ar ∧ perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ perm(ar1, ar) ∧ ar ≤ ar1), ar1 ≠ ar2, perm(ar1, ar2), perm(ar1, ar), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(ar2 ≠ ar ∧ perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar1 ≠ ar2, perm(ar1, ar), perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: ar = ar2

Additional information

Used simplifier rules: array-neg-le-eq-2: ⊦ ar0 ≠ ar1 ∧ ar0 ≤ ar1 → ¬ ar1 ≤ ar0; Used formulas: ar = ar2

Proof step 11 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(ar2 ≠ ar ∧ perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar ≠ ar2, ar1 ≠ ar2, perm(ar1, ar2), perm(ar1, ar), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar ≠ ar2, ar1 ≠ ar2, perm(ar1, ar), perm(ar1, ar2), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-perm[natarray-oarray-sort], name:trans, seq: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z), subst:[y, x, z] → [ar1, ar2, ar], <all paths>

Additional information

Used terms: ar1, ar2, ar

Proof step 13 from proof next-perm-smallerPerms (previous step) (back to tree)

⊦ perm(x, y) ∧ perm(y, z) → perm(x, z)

A spec-lemma

Proof step 14 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar ≠ ar2, ar1 ≠ ar2, perm(ar1, ar2), perm(ar1, ar), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦ perm(ar2, ar1) ∧ perm(ar1, ar)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(perm(ar2, ar) ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar ≠ ar2, ar1 ≠ ar2, perm(ar1, ar), perm(ar1, ar2), ar1 ≤ ar2, ¬ perm(ar2, ar1), ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-perm[natarray-oarray-sort], name:sym, seq: ⊦ perm(x, y) ↔ perm(y, x), subst:[x, y] → [ar1, ar2], <all paths>

Additional information

Used terms: ar1, ar2

Proof step 16 from proof next-perm-smallerPerms (previous step) (back to tree)

⊦ perm(x, y) ↔ perm(y, x)

A spec-lemma

Proof step 17 from proof next-perm-smallerPerms (previous step) (back to tree)

¬(true ∧ ar ≤ ar2 ↔ ar = ar1 ∨ ar ≤ ar1), ar ≠ ar2, ar1 ≠ ar2, perm(ar1, ar2), perm(ar2, ar1), perm(ar1, ar), ar1 ≤ ar2, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: array-le-trans: ⊦ ar0 ≤ ar1 ∧ ar1 ≤ ar2 → ar0 ≤ ar2; trans: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z); (forward) from specification list-perm[natarray-oarray-sort] ⊦ perm(x, x); (s)

Proof step 18 from proof next-perm-smallerPerms (previous step) (back to tree)

ar ≠ ar1, ar ≠ ar2, ar1 ≠ ar2, perm(ar2, ar), perm(ar1, ar2), perm(ar2, ar1), perm(ar1, ar), ar ≤ ar2, ar1 ≤ ar2, ¬ ar ≤ ar1, ∀ ar3. perm(ar1, ar3) ∧ ar3 ≠ ar1 ∧ ar3 ≠ ar2 → ¬ ar1 ≤ ar3 ∨ ¬ ar3 ≤ ar2 ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: ar abort: F discard: F computed: T

Additional information

Used simplifier rules: array-neg-le-eq: ⊦ ar0 ≠ ar1 ∧ ¬ ar1 ≤ ar0 → ar0 ≤ ar1; Used terms: ar

Proof Tree for next-perm-smallerPerms (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree