Proof step 1 from proof butmin-rewput (conclusion) (back to tree)

⊦ a ≠ (st[a := d]).min → (st[a := d]).butmin = st.butmin[a := d]

System applied rule normalize

And obtained the following new goals

Proof step 2:
(st[a := d]).butmin ≠ st.butmin[a := d], a ≠ (st[a := d]).min ⊦

Rule argument

Additional information

Proof step 2 from proof butmin-rewput (previous step) (back to tree)

(st[a := d]).butmin ≠ st.butmin[a := d], a ≠ (st[a := d]).min ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
st[a := d] -- (st[a := d]).min ≠ (st -- st.min)[a := d], a ≠ (st[a := d]).min ⊦

Rule argument

Additional information

Used simplifier rules: Butmin: ⊦ st.butmin = st -- st.min;

Proof step 3 from proof butmin-rewput (previous step) (back to tree)

st[a := d] -- (st[a := d]).min ≠ (st -- st.min)[a := d], a ≠ (st[a := d]).min ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ st1 = st2 ↔ (∀ a. (a ∈ st1 ↔ a ∈ st2) ∧ st1[a] = st2[a])
Proof step 5:
a ≠ (st[a := d]).min, ∃ a0. ¬((a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]) ⊦

Rule argument

spec: store, name:Extension, seq: ⊦ st1 = st2 ↔ (∀ a. (a ∈ st1 ↔ a ∈ st2) ∧ st1[a] = st2[a]), subst:[st1, st2] → [st[a := d] -- (st[a := d]).min, (st -- st.min)[a := d]]; ['a] → [tord], <all paths>

Additional information

Used terms: st[a := d] -- (st[a := d]).min, (st -- st.min)[a := d]

Proof step 4 from proof butmin-rewput (previous step) (back to tree)

⊦ st1 = st2 ↔ (∀ a. (a ∈ st1 ↔ a ∈ st2) ∧ st1[a] = st2[a])

A spec-lemma

Proof step 5 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min, ∃ a0. ¬((a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
a ≠ (st[a := d]).min ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]

Rule argument

Additional information

Proof step 6 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 7:
a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]

Rule argument

fma: a0 = a

Additional information

Used simplifier rules: ⊦ a ∈ st -- b ↔ a ≠ b ∧ a ∈ st; (ws) ⊦ a ∈ st[a := d]; (s) ⊦ a ≠ b → (st -- b)[a] = st[a]; (s) ⊦ (st[a := d])[a] = d; (s) Used formulas: a0 = a

Proof step 7 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ (st -- st.min)[a := d]) ∧ (st[a := d] -- (st[a := d]).min)[a0] = ((st -- st.min)[a := d])[a0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d] -- (st[a := d]).min)[a0] = (st -- st.min)[a0]

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (st[b := d])[a] = st[a]; (s) ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s)

Proof step 8 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d] -- (st[a := d]).min)[a0] = (st -- st.min)[a0]

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ a ≠ b → (st -- b)[a] = st[a]
Proof step 10:
a0 = (st[a := d]).min, a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d] -- (st[a := d]).min)[a0] = (st -- st.min)[a0]
Proof step 15:
a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d])[a0] = (st -- st.min)[a0]

Rule argument

spec: store, name:Del-at, seq: ⊦ a ≠ b → (st -- b)[a] = st[a], subst:[a, b, st] → [a0, (st[a := d]).min, st[a := d]]; ['a] → [tord], <all paths>

Additional information

Used terms: a0, (st[a := d]).min, st[a := d]

Proof step 9 from proof butmin-rewput (previous step) (back to tree)

⊦ a ≠ b → (st -- b)[a] = st[a]

A spec-lemma

Proof step 10 from proof butmin-rewput (previous step) (back to tree)

a0 = (st[a := d]).min, a ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d] -- (st[a := d]).min)[a0] = (st -- st.min)[a0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
(st[a := d]).min ≠ st.min, (st[a := d]).min ≠ a, st.min < (st[a := d]).min, (st[a := d]).min ∈ st ⊦

Rule argument

Additional information

Used simplifier rules: Min-min: ⊦ a ∈ st → ¬ a < st.min; inf: ⊦ a ∈ st → a ∈ st.keys; Keys: ⊦ a ∈ st.keys ↔ a ∈ st; ⊦ a ≠ b → (a < b ↔ ¬ b < a); (ss) ⊦ a ∈ st -- b ↔ a ≠ b ∧ a ∈ st; (ws) ⊦ ¬ a ∈ st → st[a] = ∅[a]; (s) ⊦ (st -- a)[a] = ∅[a]; (s) ⊦ ¬ a ∈ st -- a; (s)

Proof step 11 from proof butmin-rewput (previous step) (back to tree)

(st[a := d]).min ≠ st.min, (st[a := d]).min ≠ a, st.min < (st[a := d]).min, (st[a := d]).min ∈ st ⊦

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 12:
(st[a := d]).min ≠ st.min, (st[a := d]).min ≠ a, st ≠ ∅, st.min < (st[a := d]).min, (st[a := d]).min ∈ st ⊦

Rule argument

fma: st = ∅

Additional information

Used simplifier rules: min: ⊦ (∅[a := d]).min = a; Used formulas: st = ∅

Proof step 12 from proof butmin-rewput (previous step) (back to tree)

(st[a := d]).min ≠ st.min, (st[a := d]).min ≠ a, st ≠ ∅, st.min < (st[a := d]).min, (st[a := d]).min ∈ st ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
(st[a := d]).min ≠ a, (st[a := d]).min ≠ st.keys.min, st ≠ ∅, st.keys.min < (st[a := d]).min, (st[a := d]).min ∈ st, st.keys.min ∈ st ⊦

Rule argument

Additional information

Used simplifier rules: Min-in: ⊦ st ≠ ∅ → st.min ∈ st; Min-min: ⊦ a ∈ st → ¬ a < st.min; inf: ⊦ a ∈ st → a ∈ st.keys; Keys: ⊦ a ∈ st.keys ↔ a ∈ st; min-def: ⊦ st ≠ ∅ → st.min = st.keys.min; ⊦ ¬ a < a; (s)

Proof step 13 from proof butmin-rewput (previous step) (back to tree)

(st[a := d]).min ≠ a, (st[a := d]).min ≠ st.keys.min, st ≠ ∅, st.keys.min < (st[a := d]).min, (st[a := d]).min ∈ st, st.keys.min ∈ st ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min

Rule argument

spec: <toplevel>, name:min-put, seq: ⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min, subst:[d, a, st] → [d, a, st], <all paths>

Additional information

Used simplifier rules: min-def: ⊦ st ≠ ∅ → st.min = st.keys.min; Used terms: d, a, st

Proof step 14 from proof butmin-rewput (previous step) (back to tree)

⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min

Lemma min-put

Proof step 15 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st[a := d] -- (st[a := d]).min ↔ a0 ∈ st -- st.min) ∧ (st[a := d])[a0] = (st -- st.min)[a0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st ↔ a0 ∈ st -- st.min) ∧ st[a0] = (st -- st.min)[a0]

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (st[b := d])[a] = st[a]; (s) ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s) ⊦ a ∈ st -- b ↔ a ≠ b ∧ a ∈ st; (ws)

Proof step 16 from proof butmin-rewput (previous step) (back to tree)

a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st ↔ a0 ∈ st -- st.min) ∧ st[a0] = (st -- st.min)[a0]

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ a ≠ b → (st -- b)[a] = st[a]
Proof step 18:
a0 = st.min, a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st ↔ a0 ∈ st -- st.min) ∧ st[a0] = (st -- st.min)[a0]

Rule argument

spec: store, name:Del-at, seq: ⊦ a ≠ b → (st -- b)[a] = st[a], subst:[a, b, st] → [a0, st.min, st]; ['a] → [tord], <all paths>

Additional information

Used simplifier rules: ⊦ a ∈ st -- b ↔ a ≠ b ∧ a ∈ st; (ws) Used terms: a0, st.min, st

Proof step 17 from proof butmin-rewput (previous step) (back to tree)

⊦ a ≠ b → (st -- b)[a] = st[a]

A spec-lemma

Proof step 18 from proof butmin-rewput (previous step) (back to tree)

a0 = st.min, a ≠ (st[a := d]).min, a0 ≠ (st[a := d]).min, a0 ≠ a ⊦ (a0 ∈ st ↔ a0 ∈ st -- st.min) ∧ st[a0] = (st -- st.min)[a0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
st.min ≠ a, st.min ≠ (st[a := d]).min, a ≠ (st[a := d]).min, st.min ∈ st ⊦

Rule argument

Additional information

Used simplifier rules: Min-min: ⊦ a ∈ st → ¬ a < st.min; inf: ⊦ a ∈ st → a ∈ st.keys; Keys: ⊦ a ∈ st.keys ↔ a ∈ st; ⊦ ¬ a < a; (s) ⊦ ¬ a ∈ st → st[a] = ∅[a]; (s) ⊦ ¬ a ∈ st → st -- a = st; (s) ⊦ a ∈ st -- b ↔ a ≠ b ∧ a ∈ st; (ws)

Proof step 19 from proof butmin-rewput (previous step) (back to tree)

st.min ≠ a, st.min ≠ (st[a := d]).min, a ≠ (st[a := d]).min, st.min ∈ st ⊦

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 20:
st.min ≠ a, st.min ≠ (st[a := d]).min, a ≠ (st[a := d]).min, st ≠ ∅, st.min ∈ st ⊦

Rule argument

fma: st = ∅

Additional information

Used simplifier rules: min: ⊦ (∅[a := d]).min = a; Used formulas: st = ∅

Proof step 20 from proof butmin-rewput (previous step) (back to tree)

st.min ≠ a, st.min ≠ (st[a := d]).min, a ≠ (st[a := d]).min, st ≠ ∅, st.min ∈ st ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
st.keys.min ≠ (st[a := d]).min, st.keys.min ≠ a, a ≠ (st[a := d]).min, st ≠ ∅, st.keys.min ∈ st ⊦

Rule argument

Additional information

Used simplifier rules: Min-min: ⊦ a ∈ st → ¬ a < st.min; inf: ⊦ a ∈ st → a ∈ st.keys; Min-in: ⊦ st ≠ ∅ → st.min ∈ st; Keys: ⊦ a ∈ st.keys ↔ a ∈ st; min-def: ⊦ st ≠ ∅ → st.min = st.keys.min; ⊦ ¬ a < a; (s)

Proof step 21 from proof butmin-rewput (previous step) (back to tree)

st.keys.min ≠ (st[a := d]).min, st.keys.min ≠ a, a ≠ (st[a := d]).min, st ≠ ∅, st.keys.min ∈ st ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min

Rule argument

spec: <toplevel>, name:min-put, seq: ⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min, subst:[d, a, st] → [d, a, st], <all paths>

Additional information

Used simplifier rules: min-def: ⊦ st ≠ ∅ → st.min = st.keys.min; Used terms: d, a, st

Proof step 22 from proof butmin-rewput (previous step) (back to tree)

⊦ a ≠ (st[a := d]).min → (st[a := d]).min = st.min

Lemma min-put

Proof Tree for butmin-rewput (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.