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Proof step 3 from proof index-update-range (previous step)

# kx = # ax, ax.toSet ∩ range ri = ∅, ri-inj(ri), kx.toSet ⊆ dom ri ⊦ range ri[kx, ax] = (range ri \ ri[kx].toSet) ∪ ax.toSet, dups(kx), dups(ax)

User applied rule induction

And obtained the following new goals

Proof step 4:
# kx = # ax, ax.toSet ∩ range ri = ∅, ri-inj(ri), kx.toSet ⊆ dom ri, ∀ kx0, ax, ri. # kx0 < # kx → # kx0 = # ax ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ kx0.toSet ⊆ dom ri → range ri[kx0, ax] = (range ri \ ri[kx0].toSet) ∪ ax.toSet ∨ dups(kx0) ∨ dups(ax) ⊦ range ri[kx, ax] = (range ri \ ri[kx].toSet) ∪ ax.toSet, dups(kx), dups(ax)

Rule argument

indvar: # kx wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: # kx