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

⊦ #r mx = # a ∧ c < #c mx ∧ perm(mx[c]c, a) → perm(mx.toArr, mx[c, a]c.toArr)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

#r mx = # a, c < #c mx, perm(mx[c]c, a), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:perm-updcol-aux, seq:⊦ #r mx = # a ∧ c < #c mx ∧ perm(mx[c]c, a) → perm(mx.toArc, mx[c, a]c.toArc), subst:[mx, a, c] → [mx, a, c]

Additional information

Used terms: mx, a, c

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

⊦ #r mx = # a ∧ c < #c mx ∧ perm(mx[c]c, a) → perm(mx.toArc, mx[c, a]c.toArc)

Lemma perm-updcol-aux

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

#r mx = # a, c < #c mx, perm(mx[c]c, a), ¬ perm(mx.toArr, mx[c, a]c.toArr), #r mx = # a ∧ c < #c mx ∧ perm(mx[c]c, a) → perm(mx.toArc, mx[c, a]c.toArc) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

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

#r mx = # a, c < #c mx, perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c]c, a), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:perm-toArr-toArc, seq: ⊦ perm(mx.toArr, mx.toArc), subst:[mx] → [mx[c, a]c]

Additional information

Used terms: mx[c, a]c

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

⊦ perm(mx.toArr, mx.toArc)

Lemma perm-toArr-toArc

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

#r mx = # a, c < #c mx, perm(mx[c]c, a), perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c, a]c.toArr, mx[c, a]c.toArc), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:perm-toArr-toArc, seq: ⊦ perm(mx.toArr, mx.toArc), subst:[mx] → [mx]

Additional information

Used terms: mx

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

⊦ perm(mx.toArr, mx.toArc)

Lemma perm-toArr-toArc

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

#r mx = # a, c < #c mx, perm(mx[c, a]c.toArr, mx[c, a]c.toArc), perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c]c, a), perm(mx.toArr, mx.toArc), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: trans: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z); (forward) from specification list-perm[oarray-sort]

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

#r mx = # a, c < #c mx, perm(mx.toArr, mx.toArc), perm(mx[c]c, a), perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c, a]c.toArr, mx[c, a]c.toArc), perm(mx.toArr, mx[c, a]c.toArc), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-perm[oarray-sort], name:trans, seq: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z), subst:[y, x, z] → [mx.toArc, mx.toArr, mx[c, a]c.toArr], <all paths>

Additional information

Used terms: mx.toArc, mx.toArr, mx[c, a]c.toArr

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

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

A spec-lemma

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

#r mx = # a, c < #c mx, perm(mx.toArr, mx[c, a]c.toArc), perm(mx[c, a]c.toArr, mx[c, a]c.toArc), perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c]c, a), perm(mx.toArr, mx.toArc), ¬ perm(mx.toArr, mx[c, a]c.toArr) ⊦ perm(mx.toArr, mx.toArc) ∧ perm(mx.toArc, mx[c, a]c.toArr)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

#r mx = # a, c < #c mx, perm(mx.toArr, mx.toArc), perm(mx[c]c, a), perm(mx.toArc, mx[c, a]c.toArc), perm(mx[c, a]c.toArr, mx[c, a]c.toArc), perm(mx.toArr, mx[c, a]c.toArc), ¬ perm(mx.toArr, mx[c, a]c.toArr), ¬ perm(mx.toArc, mx[c, a]c.toArr) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-perm[oarray-sort], name:sym, seq: ⊦ perm(x, y) ↔ perm(y, x), subst:[x, y] → [mx[c, a]c.toArr, mx[c, a]c.toArc], <all paths>

Additional information

Used simplifier rules: trans: ⊦ perm(x, y) ∧ perm(y, z) → perm(x, z); (forward) from specification list-perm[oarray-sort] Used terms: mx[c, a]c.toArr, mx[c, a]c.toArc

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

⊦ perm(x, y) ↔ perm(y, x)

A spec-lemma

Proof Tree for perm-updcol (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