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Proof step 1 from proof floor-int (conclusion) (back to tree)

⊦ ⌊i.r⌋ = i

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof floor-int (previous step) (back to tree)

⌊i.r⌋ ≠ i ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:floor-le, seq: ⊦ ⌊r⌋ .r ≤ r, subst:[r] → [i.r]

Additional information

Used terms: i.r

Proof step 3 from proof floor-int (previous step) (back to tree)

⊦ ⌊r⌋ .r ≤ r

Lemma floor-le

Proof step 4 from proof floor-int (previous step) (back to tree)

⌊i.r⌋ ≠ i, ⌊i.r⌋ .r ≤ i.r ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:floor-gr-subone, seq: ⊦ r < ⌊r⌋ .r + 1.r, subst:[r] → [i.r]

Additional information

Used terms: i.r

Proof step 5 from proof floor-int (previous step) (back to tree)

⊦ r < ⌊r⌋ .r + 1.r

Lemma floor-gr-subone

Proof step 6 from proof floor-int (previous step) (back to tree)

⌊i.r⌋ ≠ i, i.r < ⌊i.r⌋ .r + 1.r, ⌊i.r⌋ .r ≤ i.r ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-basic, name: eqint-add, seq: ⊦ (* i).r + (* j).r = (i + j).r, subst:[i, j] → [⌊i.r⌋ , 1], <all paths>

Additional information

Used terms: ⌊i.r⌋ , 1

Proof step 7 from proof floor-int (previous step) (back to tree)

⊦ (* i).r + (* j).r = (i + j).r

A spec-lemma

Proof step 8 from proof floor-int (previous step) (back to tree)

⌊i.r⌋ ≠ i, i.r < (⌊i.r⌋ + 1).r, ⌊i.r⌋ .r ≤ i.r ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-basic, name: ls-num, seq: ⊦ i.r < j.r ↔ i < j, subst:[i, j] → [i, ⌊i.r⌋ + 1], <all paths>

Additional information

Used terms: i, ⌊i.r⌋ + 1

Proof step 9 from proof floor-int (previous step) (back to tree)

⊦ i.r < j.r ↔ i < j

A spec-lemma

Proof step 10 from proof floor-int (previous step) (back to tree)

⌊i.r⌋ ≠ i, i < ⌊i.r⌋ + 1, ⌊i.r⌋ .r ≤ i.r ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: real-abs, name: le-num, seq: ⊦ i.r ≤ j.r ↔ i ≤ j, subst:[i, j] → [⌊i.r⌋ , i], <all paths>

Additional information

Used simplifier rules: ⊦ i ≠ j → (i ≤ j ↔ i < j); (s) ⊦ i < j + 1 ↔ i ≤ j; (s) ⊦ i < j → (j ≤ i ↔ false); (as) add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2 add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: ⌊i.r⌋ , i

Proof step 11 from proof floor-int (previous step) (back to tree)

⊦ i.r ≤ j.r ↔ i ≤ j

A spec-lemma

Proof Tree for floor-int (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