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Proof step 1 from proof mk-exists (conclusion) (back to tree)

⊦ ∃ rs, f, g. ha = mk(rs, f, g)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: mk-def: ⊦ mk(rs, f, g) = mk(rs, λ r. (f(r), g(r)));

Proof step 2 from proof mk-exists (previous step) (back to tree)

⊦ ∃ rs, f, g. ha = mk(rs, λ r. (f(r), g(r)))

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: heap-fun, name:mk-exists, seq: ⊦ ∃ rs, f. h = mk(rs, f), subst:[h] → [ha]; [A] → [(A, B)]

Additional information

Used terms: ha

Proof step 3 from proof mk-exists (previous step) (back to tree)

⊦ ∃ rs, f. h = mk(rs, f)

A spec-lemma

Proof step 4 from proof mk-exists (previous step) (back to tree)

∃ x_rs, x_f. ha = mk(x_rs, x_f) ⊦ ∃ rs, f, g. ha = mk(rs, λ r. (f(r), g(r)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof mk-exists (previous step) (back to tree)

⊦ ∃ rs0, f, g. mk(rs, x_f) = mk(rs0, λ r. (f(r), g(r)))

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: rs, λ r. x_f(r)._1, λ r. x_f(r)._2 abort: F discard: T computed: F

Additional information

Used terms: rs, λ r. x_f(r)._1, λ r. x_f(r)._2

Proof step 6 from proof mk-exists (previous step) (back to tree)

mk(rs, x_f) ≠ mk(rs, λ r. (x_f(r)._1, x_f(r)._2)) ⊦

User applied rule reduce equation

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 7 from proof mk-exists (previous step) (back to tree)

x_f ≠ (λ r. (x_f(r)._1, x_f(r)._2)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof mk-exists (previous step) (back to tree)

x_f(r) ≠ (x_f(r)._1, x_f(r)._2) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:elim-objaux, seq: ⊦ o = oa._1 ∧ a = oa._2 ↔ oa = (o, a), subst:[oa, o, a] → [x_f(r), o, a]

Additional information

Used terms: x_f(r), o, a

Proof step 9 from proof mk-exists (previous step) (back to tree)

⊦ o = oa._1 ∧ a = oa._2 ↔ oa = (o, a)

Lemma elim-objaux

Proof Tree for mk-exists (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