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Proof step 1 from proof uni-ass (conclusion) (back to tree)

⊦ h0 ⊥ h1 ∧ h1 ⊥ h ∧ h ⊥ h0 → (h0 ∪ h1) ∪ h = h0 ∪ h1 ∪ h

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof uni-ass (previous step) (back to tree)

(h0 ∪ h1) ∪ h ≠ h0 ∪ h1 ∪ h, h ⊥ h0, h0 ⊥ h1, h1 ⊥ h ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof uni-ass (previous step) (back to tree)

(h0 ∪ h1) ∪ h ≠ h0 ∪ h1 ∪ h, h ⊥ h0, h0 ⊥ h1, h1 ⊥ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: ext, seq: ⊦ h0 = h1 ↔ (∀ r. (r ∈ h0 ↔ r ∈ h1) ∧ (r ∈ h0 → h0(r) = h1(r))), subst:[h1, h0] → [h0 ∪ h1 ∪ h, (h0 ∪ h1) ∪ h], <all paths>

Additional information

Used terms: h0 ∪ h1 ∪ h, (h0 ∪ h1) ∪ h

Proof step 4 from proof uni-ass (previous step) (back to tree)

⊦ h0 = h1 ↔ (∀ r. (r ∈ h0 ↔ r ∈ h1) ∧ (r ∈ h0 → h0(r) = h1(r)))

Lemma ext

Proof step 5 from proof uni-ass (previous step) (back to tree)

h ⊥ h0, h0 ⊥ h1, h1 ⊥ h, ∃ r. ¬((r ∈ (h0 ∪ h1) ∪ h ↔ r ∈ h0 ∪ h1 ∪ h) ∧ (r ∈ (h0 ∪ h1) ∪ h → ((h0 ∪ h1) ∪ h)(r) = (h0 ∪ h1 ∪ h)(r))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof uni-ass (previous step) (back to tree)

h ⊥ h0, h0 ⊥ h1, h1 ⊥ h ⊦ (r ∈ (h0 ∪ h1) ∪ h ↔ r ∈ h0 ∪ h1 ∪ h) ∧ (r ∈ (h0 ∪ h1) ∪ h → ((h0 ∪ h1) ∪ h)(r) = (h0 ∪ h1 ∪ h)(r))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: in-uni, seq: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1), subst:[h1, h0, r] → [h, h0 ∪ h1, r], <all paths>

Additional information

Used simplifier rules: disj-distr-uni-com: ⊦ h1 ⊥ h2 → (h1 ∪ h2 ⊥ h0 ↔ h0 ⊥ h1 ∧ h0 ⊥ h2); disj-com-rw: ⊦ h0 ⊥ h1 → h1 ⊥ h0; Used terms: h, h0 ∪ h1, r

Proof step 7 from proof uni-ass (previous step) (back to tree)

⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1)

Lemma in-uni

Proof step 8 from proof uni-ass (previous step) (back to tree)

h ⊥ h0, h0 ⊥ h1, h0 ∪ h1 ⊥ h, h1 ⊥ h ⊦ (r ∈ h0 ∪ h1 ∨ r ∈ h ↔ r ∈ h0 ∪ h1 ∪ h) ∧ (r ∈ h0 ∪ h1 ∨ r ∈ h → ((h0 ∪ h1) ∪ h)(r) = (h0 ∪ h1 ∪ h)(r))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: disj-distr-uni-com: ⊦ h1 ⊥ h2 → (h1 ∪ h2 ⊥ h0 ↔ h0 ⊥ h1 ∧ h0 ⊥ h2); in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1);

Proof step 9 from proof uni-ass (previous step) (back to tree)

h ⊥ h0, h ⊥ h1, h0 ⊥ h1, h1 ⊥ h ⊦ (r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ↔ r ∈ h0 ∪ h1 ∪ h) ∧ (r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h → ((h0 ∪ h1) ∪ h)(r) = (h0 ∪ h1 ∪ h)(r))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: in-uni, seq: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1), subst:[h1, h0, r] → [h1 ∪ h, h0, r], <all paths>

Additional information

Used simplifier rules: disj-distr-uni: ⊦ h1 ⊥ h → (h0 ⊥ h1 ∪ h ↔ h0 ⊥ h1 ∧ h0 ⊥ h); disj-com-rw: ⊦ h0 ⊥ h1 → h1 ⊥ h0; Used terms: h1 ∪ h, h0, r

Proof step 10 from proof uni-ass (previous step) (back to tree)

⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1)

Lemma in-uni

Proof step 11 from proof uni-ass (previous step) (back to tree)

h ⊥ h1, h ⊥ h0, h0 ⊥ h1, h0 ⊥ h1 ∪ h, h1 ⊥ h ⊦ (r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ↔ r ∈ h0 ∨ r ∈ h1 ∪ h) ∧ (r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h → ((h0 ∪ h1) ∪ h)(r) = (h0 ∪ h1 ∪ h)(r))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: disj-distr-uni: ⊦ h1 ⊥ h → (h0 ⊥ h1 ∪ h ↔ h0 ⊥ h1 ∧ h0 ⊥ h); in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1);

Proof step 12 from proof uni-ass (previous step) (back to tree)

((h0 ∪ h1) ∪ h)(r) ≠ (h0 ∪ h1 ∪ h)(r), h ⊥ h0, h ⊥ h1, h0 ⊥ h1, h0 ⊥ h, h1 ⊥ h, r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: at-uni, seq:r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)), subst:[r, h1, h0] → [r, h, h0 ∪ h1], <all paths>

Additional information

Used simplifier rules: in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); disj-distr-uni-com: ⊦ h1 ⊥ h2 → (h1 ∪ h2 ⊥ h0 ↔ h0 ⊥ h1 ∧ h0 ⊥ h2); disj-com-rw: ⊦ h0 ⊥ h1 → h1 ⊥ h0; Used terms: r, h, h0 ∪ h1

Proof step 13 from proof uni-ass (previous step) (back to tree)

r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r))

Lemma at-uni

Proof step 14 from proof uni-ass (previous step) (back to tree)

(r ∈ h0 ∪ h1 ⊃ (h0 ∪ h1)(r);h(r)) ≠ (h0 ∪ h1 ∪ h)(r), h ⊥ h1, h ⊥ h0, h0 ⊥ h, h0 ⊥ h1, h0 ∪ h1 ⊥ h, h1 ⊥ h, r ∈ (h0 ∪ h1) ∪ h, r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: at-uni: r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)); in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); disj-distr-uni-com: ⊦ h1 ⊥ h2 → (h1 ∪ h2 ⊥ h0 ↔ h0 ⊥ h1 ∧ h0 ⊥ h2);

Proof step 15 from proof uni-ass (previous step) (back to tree)

(r ∈ h0 ∨ r ∈ h1 ⊃ (r ∈ h0 ⊃ h0(r);h1(r));h(r)) ≠ (h0 ∪ h1 ∪ h)(r), h ⊥ h0, h ⊥ h1, h0 ⊥ h1, h0 ⊥ h, h1 ⊥ h, r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: at-uni, seq:r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)), subst:[r, h1, h0] → [r, h1 ∪ h, h0], <all paths>

Additional information

Used simplifier rules: in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); disj-distr-uni: ⊦ h1 ⊥ h → (h0 ⊥ h1 ∪ h ↔ h0 ⊥ h1 ∧ h0 ⊥ h); disj-com-rw: ⊦ h0 ⊥ h1 → h1 ⊥ h0; Used terms: r, h1 ∪ h, h0

Proof step 16 from proof uni-ass (previous step) (back to tree)

r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r))

Lemma at-uni

Proof step 17 from proof uni-ass (previous step) (back to tree)

(r ∈ h0 ∨ r ∈ h1 ⊃ (r ∈ h0 ⊃ h0(r);h1(r));h(r)) ≠ (r ∈ h0 ⊃ h0(r);(h1 ∪ h)(r)), h ⊥ h1, h ⊥ h0, h0 ⊥ h, h0 ⊥ h1, h0 ⊥ h1 ∪ h, h1 ⊥ h, r ∈ h0 ∪ h1 ∪ h, r ∈ h0 ∨ r ∈ h1 ∨ r ∈ h ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: disj-distr-uni: ⊦ h1 ⊥ h → (h0 ⊥ h1 ∪ h ↔ h0 ⊥ h1 ∧ h0 ⊥ h); disj-r: ⊦ r ∈ h1 ∧ h0 ⊥ h1 → (r ∈ h0 ↔ false);

Proof step 18 from proof uni-ass (previous step) (back to tree)

(r ∈ h1 ⊃ h1(r);h(r)) ≠ (h1 ∪ h)(r), h ⊥ h1, h ⊥ h0, h0 ⊥ h, h0 ⊥ h1, h1 ⊥ h, r ∈ h0 ∪ h1 ∪ h, ¬ r ∈ h0, r ∈ h ∨ r ∈ h1 ⊦

Heuristic cut applied rule cut formula

And closed the goal!

Rule argument

fma: r ∈ h1

Additional information

Used simplifier rules: in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); at-uni: r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)); in-uni: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); at-uni: r ∈ h0 ∪ h1 ⊦ h0 ⊥ h1 → (h0 ∪ h1)(r) = (r ∈ h0 ⊃ h0(r);h1(r)); disj-l: ⊦ r ∈ h0 ∧ h0 ⊥ h1 → (r ∈ h1 ↔ false); Used formulas: r ∈ h1

Proof Tree for uni-ass (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