Proof step 1 from proof max0column-countcol (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  #c mx = #c mx0, #r mx = #r mx0, count(mx[c]c, λ j. j < i) ≠ #clean0rows(mx0, i) + count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), c0 < #c mx, c < #c mx, scolsorted(mx, #c mx, mx0), ∀ c. c < #c mx → count(mx[c]c, λ j. j < i) ≤ count(mx[c0]c, λ j. j < i) ⊦

Rule argument

Additional information

Proof step 2 from proof max0column-countcol (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 3:

  #r mx = #r mx0, #c mx = #c mx0, c < #c mx, scolsorted(mx, #c mx, mx0) ⊦ count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i)

• Proof step 4:

  #c mx = #c mx0, #r mx = #r mx0, count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx[c]c, λ j. j < i) ≠ #clean0rows(mx0, i) + count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), c < #c mx, c0 < #c mx, scolsorted(mx, #c mx, mx0), ∀ c. c < #c mx → count(mx[c]c, λ j. j < i) ≤ count(mx[c0]c, λ j. j < i) ⊦

Rule argument

Additional information

Used terms: mx, mx0, c, i

Proof step 3 from proof max0column-countcol (previous step) (back to tree)

Lemma scolsorted-samecount-above-clean0

Proof step 4 from proof max0column-countcol (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 5:

  #r mx = #r mx0, #c mx = #c mx0, c < #c mx, scolsorted(mx, #c mx, mx0) ⊦ count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i)

• Proof step 6:

  #c mx = #c mx0, #r mx = #r mx0, count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx0[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx[c]c, λ j. j < i) ≠ #clean0rows(mx0, i) + count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), c0 < #c mx, c < #c mx, scolsorted(mx, #c mx, mx0), ∀ c. c < #c mx → count(mx[c]c, λ j. j < i) ≤ count(mx[c0]c, λ j. j < i) ⊦

Rule argument

Additional information

Used terms: mx, mx0, c0, i

Proof step 5 from proof max0column-countcol (previous step) (back to tree)

Lemma scolsorted-samecount-above-clean0

Proof step 6 from proof max0column-countcol (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ count(x, p) = count(x, 0, # x, p)

• Proof step 8:

  #c mx = #c mx0, #r mx = #r mx0, count(mx0[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx[c]c, 0, # mx[c]c, λ j. j < i) ≠ #clean0rows(mx0, i) + count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), c < #c mx, c0 < #c mx, scolsorted(mx, #c mx, mx0), ∀ c. c < #c mx → count(mx[c]c, λ j. j < i) ≤ count(mx[c0]c, λ j. j < i) ⊦

Rule argument

Additional information

Used terms: mx[c]c, λ j. j < i

Proof step 7 from proof max0column-countcol (previous step) (back to tree)

A spec-lemma

Proof step 8 from proof max0column-countcol (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  m ≤ n, n ≤ n0, n0 ≤ # x ⊦ count(x, m, n, p) + count(x, n, n0, p) = count(x, m, n0, p)

• Proof step 10:

  #c mx = #c mx0, #r mx = #r mx0, count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx0[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i) = count(mx[c0]c, #clean0rows(mx0, i), #r mx, λ j. j < i), count(mx[c]c, 0, #clean0rows(mx0, i), λ j. j < i) + count(mx[c]c, #clean0rows(mx0, i), # mx[c]c, λ j. j < i) ≠ #clean0rows(mx0, i) + count(mx0[c]c, #clean0rows(mx0, i), #r mx, λ j. j < i), c0 < #c mx, c < #c mx, scolsorted(mx, #c mx, mx0), 0 ≤ #clean0rows(mx0, i), #clean0rows(mx0, i) ≤ # mx[c]c, # mx[c]c ≤ # mx[c]c, ∀ c. c < #c mx → count(mx[c]c, λ j. j < i) ≤ count(mx[c0]c, λ j. j < i) ⊦

Rule argument

Additional information

Used simplifier rules: clean0rows-bounded: ⊦ ¬ #r mx < #clean0rows(mx, i); ⊦ 0 ≤ m; (s) ⊦ # mx[c]c = #r mx; (s) ⊦ m ≤ m; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) Used terms: # mx[c]c, λ j. j < i, #clean0rows(mx0, i), 0, mx[c]c

Proof step 9 from proof max0column-countcol (previous step) (back to tree)

A spec-lemma

Proof step 10 from proof max0column-countcol (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 11:

  #r mx = #r mx0, #c mx = #c mx0, c < #c mx, scolsorted(mx, #c mx, mx0) ⊦ count(mx[c]c, 0, #clean0rows(mx0, i), λ j. j < i) = #clean0rows(mx0, i)

Rule argument

Additional information

Used simplifier rules: ⊦ # mx[c]c = #r mx; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ 0 ≤ m; (s) ⊦ m ≤ m; (s) Used terms: i, mx, c, mx0

Proof step 11 from proof max0column-countcol (previous step) (back to tree)

Lemma scolsorted-count-below-clean0rows