[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 8 from proof no-dups-keyset-size (previous step)

(∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1]) ↔ # keyset(kar, m0) = m0, m0 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1] ⊦ #(keyset(kar, m0) ++ kar[m0]) = m0 + 1

User applied rule case distinction

And obtained the following new goals

Proof step 9:
m0 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1] ⊦ #(keyset(kar, m0) ++ kar[m0]) = m0 + 1, # keyset(kar, m0) = m0, ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1]
Proof step 11:
# keyset(kar, m0) = m0, m0 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1], ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1] ⊦ #(keyset(kar, m0) ++ kar[m0]) = m0 + 1

Rule argument

pos: left 1

Proof step 8 from proof no-dups-keyset-size (previous step)

User applied rule case distinction

And obtained the following new goals

Rule argument

Additional information