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Proof step 1 from proof scolsorted-permtoArr (conclusion) (back to tree)

#r mx = #r mx0, #c mx = #c mx0, c ≤ #c mx0 ⊦ scolsorted(mx, c, mx0) → perm(mx0.toArr, mx.toArr)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: perm(mx0.toArc, mx.toArc)

Additional information

Used formulas: perm(mx0.toArc, mx.toArc)

Proof step 3 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, perm(mx0.toArc, mx.toArc), scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: rowcol-upd, name:perm-toArr-toArc, seq: ⊦ perm(mx.toArr, mx.toArc), subst:[mx] → [mx0]

Additional information

Used terms: mx0

Proof step 4 from proof scolsorted-permtoArr (previous step) (back to tree)

⊦ perm(mx.toArr, mx.toArc)

A spec-lemma

Proof step 5 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, perm(mx0.toArc, mx.toArc), perm(mx0.toArr, mx0.toArc), scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArr, mx.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-new]

Proof step 6 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, perm(mx0.toArr, mx0.toArc), perm(mx0.toArc, mx.toArc), perm(mx0.toArr, mx.toArc), scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: rowcol-upd, name:perm-toArr-toArc, seq: ⊦ perm(mx.toArr, mx.toArc), subst:[mx] → [mx]

Additional information

Used terms: mx

Proof step 7 from proof scolsorted-permtoArr (previous step) (back to tree)

⊦ perm(mx.toArr, mx.toArc)

A spec-lemma

Proof step 8 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, perm(mx0.toArr, mx.toArc), perm(mx0.toArc, mx.toArc), perm(mx0.toArr, mx0.toArc), perm(mx.toArr, mx.toArc), scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-perm[oarray-sort-new], name:sym, seq: ⊦ perm(x, y) ↔ perm(y, x), subst:[x, y] → [mx.toArr, mx.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-new] Used terms: mx.toArr, mx.toArc

Proof step 9 from proof scolsorted-permtoArr (previous step) (back to tree)

⊦ perm(x, y) ↔ perm(y, x)

A spec-lemma

Proof step 10 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(mx0.toArc, mx.toArc), ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: rowcol-upd, name:toArc-def, seq: ⊦ mx.toArc = toArc(mx, #c mx), subst:[mx] → [mx0], <all paths>

Additional information

Used terms: mx0

Proof step 11 from proof scolsorted-permtoArr (previous step) (back to tree)

⊦ mx.toArc = toArc(mx, #c mx)

A spec-lemma

Proof step 12 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(toArc(mx0, #c mx0), mx.toArc), ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: rowcol-upd, name:toArc-def, seq: ⊦ mx.toArc = toArc(mx, #c mx), subst:[mx] → [mx], <all paths>

Additional information

Used terms: mx

Proof step 13 from proof scolsorted-permtoArr (previous step) (back to tree)

⊦ mx.toArc = toArc(mx, #c mx)

A spec-lemma

Proof step 14 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(toArc(mx0, #c mx0), toArc(mx, #c mx)), ¬ perm(mx0.toArr, mx.toArr) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof scolsorted-permtoArr (previous step) (back to tree)

#c mx = #c mx0, #r mx = #r mx0, scolsorted(mx, c, mx0), c ≤ #c mx0, ¬ perm(toArc(mx0, #c mx0), toArc(mx, #c mx0)), ¬ perm(mx0.toArr, mx.toArr) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:scolsorted-permtoArc-ind, seq:#r mx = #r mx0, #c mx = #c mx0, c ≤ #c mx0, m ≤ #c mx0 ⊦ scolsorted(mx, c, mx0) → perm(toArc(mx0, m), toArc(mx, m)), subst:[mx0, m, mx, c] → [mx0, #c mx0, mx, c]

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ ¬ n < n; (s) Used terms: mx0, #c mx0, mx, c

Proof step 16 from proof scolsorted-permtoArr (previous step) (back to tree)

#r mx = #r mx0, #c mx = #c mx0, c ≤ #c mx0, m ≤ #c mx0 ⊦ scolsorted(mx, c, mx0) → perm(toArc(mx0, m), toArc(mx, m))

Lemma scolsorted-permtoArc-ind

Proof Tree for scolsorted-permtoArr (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