[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof abseq (conclusion) (back to tree)

⊦ abs(r) + r = 0.r ↔ r ≤ 0.r

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof abseq (previous step) (back to tree)

¬(abs(r) + r = 0.r ↔ r ≤ 0.r) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: abs-16: ⊦ 0.r < r0 → abs(r0) = r0; le-def: ⊦ r ≤ r0 ↔ ¬ r0 < r; add-comm: ⊦ r + r0 = r0 + r; from specification real-basic add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic ⊦ 0.r < r ∧ 0.r < r0 → r + r0 ≠ 0.r; (s)

Proof step 3 from proof abseq (previous step) (back to tree)

abs(r) + r ≠ 0.r, ¬ 0.r < r ⊦

Heuristic weak cut applied rule cut formula

And closed the goal!

Rule argument

fma: r < 0.r

Additional information

Used simplifier rules: Abs-neg: ⊦ r < 0.r → abs(r) = ~ r; Abs-pos: ⊦ ¬ r < 0.r → abs(r) = r; ⊦ r + r = 0.r ↔ r = 0.r; (s) ⊦ ¬ r < r0 → (r = r0 ↔ ¬ r0 < r); (ss) add-assoc: ⊦ (r + r0) + r1 = r + r0 + r1; from specification real-basic add-comm: ⊦ r + r0 = r0 + r; from specification real-basic ⊦ r = r0 → r + ~ r0 = 0.r; (s) ⊦ i.r = j.r ↔ i = j; (s) Used formulas: r < 0.r

Proof Tree for abseq (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