This is a listing of all goals defined over the current specification. Axioms are marked yellow, proved theorems green and unproved theorems blue. Click on individual names to get more information on the proof.
| Name: | Sequent: | Type: |
|---|---|---|
| One | ⊦ {a} = ∅ ++ a | An Axiom |
| Intersect | ⊦ a ∈ s1 ∩ s2 ↔ a ∈ s1 ∧ a ∈ s2 | An Axiom |
| map-base | ⊦ ∅ map f = ∅ | An Axiom |
| Difference | ⊦ a ∈ s1 \ s2 ↔ a ∈ s1 ∧ ¬ a ∈ s2 | An Axiom |
| map-rec | ⊦ ¬ a ∈ s → (s ++ a) map f = (s map f) ++ f(a) | An Axiom |
| Disjoint | ⊦ s1 ∩ s2 = ∅ ↔ disjoint(s1, s2) | An Axiom |
| Union | ⊦ a ∈ s1 ∪ s2 ↔ a ∈ s1 ∨ a ∈ s2 | An Axiom |
| Delete | ⊦ a ∈ s -- a0 ↔ a ≠ a0 ∧ a ∈ s | An Axiom |
| filter-in | ⊦ a ∈ filter(s, p) ↔ a ∈ s ∧ p(a) | An Axiom |
| add-inter-rem | ⊦ ((s0 ++ a) ∩ s1) -- a = (s0 ∩ s1) -- a | A User Theorem |
| add-rem-sub | ⊦ (s0 ++ a) \ (s1 -- a) = (s0 \ s1) ++ a | A User Theorem |
| del-addeq | ⊦ a ≠ a0 ∧ ¬ a ∈ s ∧ ¬ a ∈ s0 → ((s ++ a) -- a0 = s0 ++ a ↔ s -- a0 = s0) | A User Theorem |
| del-addeq-01 | ⊦ (s ++ a) -- a0 -- a = s -- a0 -- a | A User Theorem |
| del-addeq-02 | ⊦ a ∈ s → (s -- a) ++ a = s | A User Theorem |
| del-addeq-03 | ⊦ a ∈ s ∧ a ≠ a0 → (s -- a0) ++ a = s -- a0 | A User Theorem |
| del-addeq-04 | ⊦ ¬ a ∈ s → (s ++ a) -- a = s | A User Theorem |
| del-addeq-05 | ⊦ (s ++ a) -- a = s -- a | A User Theorem |
| del-addeq-06 | ⊦ (s ++ a ++ a0) -- a = (s ++ a0) -- a | A User Theorem |
| del-addin | ⊦ ¬ a ∈ s → (a ∈ (s -- a0) ++ a1 ++ a2 ↔ a = a1 ∨ a = a2) | A User Theorem |
| del-addin-01 | ⊦ ¬ a ∈ s → (a ∈ (s -- a0) ++ a1 ↔ a = a1) | A User Theorem |
| del-d | ⊦ s -- a -- a = s -- a | A User Theorem |
| del-d-01 | ⊦ s -- a -- a0 -- a = s -- a0 -- a | A User Theorem |
| del-del-one | ⊦ a ≠ a0 → (∅ ++ a) -- a0 = ∅ ++ a | A User Theorem |
| del-e | ⊦ ¬ a ∈ s → ((s ++ a) -- a0 = ∅ ↔ s = ∅ ∧ a = a0) | A User Theorem |
| del-e-01 | ⊦ ∅ -- a = ∅ | A User Theorem |
| del-e-02 | ⊦ s -- a = ∅ ↔ s = ∅ ∨ s = ∅ ++ a | A User Theorem |
| del-eq | ⊦ a ∈ s ∧ a ∈ s0 → (s -- a = s0 -- a ↔ s = s0) | A User Theorem |
| del-feq | ⊦ s -- a = s0 ++ a0 → a ≠ a0 ∧ a0 ∈ s ∧ ¬ a ∈ s0 | A User Theorem |
| del-feq-01 | ⊦ (s ++ a) -- a0 = s0 ++ a1 ∧ ¬ a1 ∈ s → a = a1 ∧ a0 ≠ a1 | A User Theorem |
| del-feq-02 | ⊦ (s ++ a) -- a0 = s0 ++ a1 ∧ a ≠ a0 ∧ ¬ a ∈ s0 → a = a1 | A User Theorem |
| del-in | ⊦ ¬ a ∈ s → s -- a = s | A User Theorem |
| del-in-01 | ⊦ a ≠ a0 → (a ∈ s -- a0 ↔ a ∈ s) | A User Theorem |
| del-in-02 | ⊦ ¬ a ∈ s -- a | A User Theorem |
| del-in-03 | ⊦ s -- a = s ↔ ¬ a ∈ s | A User Theorem |
| del-ne | ⊦ 1 < # s → s -- a ≠ ∅ | A User Theorem |
| del-neq | ⊦ s -- a ≠ s0 -- a → s ≠ s0 | A User Theorem |
| del-neq-01 | ⊦ a ≠ a0 → (s ++ a) -- a0 ≠ ∅ | A User Theorem |
| del-neq-02 | ⊦ s -- a ≠ s0 ++ a | A User Theorem |
| del-notin | ⊦ s0 = s -- a → ¬ a ∈ s0 | A User Theorem |
| del-size | ⊦ a ∈ s → #(s -- a) = # s - 1 | A User Theorem |
| del-size-01 | ⊦ a ∈ s → (# s = n + 1 ↔ #(s -- a) = n) | A User Theorem |
| del-size-02 | ⊦ a ∈ s ∧ a ≠ a0 → #(s -- a0 -- a) = #(s -- a0) - 1 | A User Theorem |
| del-size-le | ⊦ ¬ # s < #(s -- a) | A User Theorem |
| del-three | ⊦ 2 < # s → #(s -- a -- a0) ≠ 0 | A User Theorem |
| del-three-01 | ⊦ 2 < # s ∧ a0 ∈ s ∧ a ≠ a0 → #(s -- a -- a0) = #(s -- a) - 1 | A User Theorem |
| del-two | s -- a -- a0 = ∅ ⊦ s = ∅ ∨ s = ∅ ++ a ∨ s = ∅ ++ a0 ∨ s = ∅ ++ a ++ a0 | A User Theorem |
| delete-insert | ⊦ (s -- a) ++ a = s ++ a | A User Theorem |
| delete-shift | ⊦ a ∈ s → (s -- a = s0 ↔ s = s0 ++ a ∧ ¬ a ∈ s0) | A User Theorem |
| delete-subset | ⊦ (s -- a) ++ a0 ⊆ s ++ a0 | A User Theorem |
| delsw | ⊦ a ≠ a0 → (s ++ a0) -- a = (s -- a) ++ a0 | A User Theorem |
| delsw-01 | ⊦ s -- a0 -- a = s -- a -- a0 | A User Theorem |
| delsw-02 | ⊦ s -- a0 -- a = s -- a -- a0 ↔ true | A User Theorem |
| delsw-suc | ⊦ a ≠ a0 → ((s ++ a0) -- a = (s -- a) ++ a0 ↔ true) | A User Theorem |
| diff-add-del | ⊦ s1 \ (s2 ++ a) = (s1 \ s2) -- a | A User Theorem |
| diff-add-del-01 | ⊦ ¬ a ∈ s1 → s1 \ (s2 ++ a) = s1 \ s2 | A User Theorem |
| diff-del | ⊦ s = s1 \ s2 ∧ ¬ a ∈ s → (s = (s1 -- a) \ (s2 -- a) ↔ true) | A User Theorem |
| diff-del-01 | ⊦ ¬ a ∈ s → s \ s0 ⊆ (s \ s0) -- a | A User Theorem |
| diff-del-02 | ⊦ ¬ a ∈ s → s \ s0 ⊆ ((s \ s0) -- a) ++ a0 | A User Theorem |
| diff-del-minus | ⊦ (s -- a) \ (s1 -- a) = (s -- a) \ s1 | A User Theorem |
| diff-del-minus-2 | ⊦ ¬ a ∈ s1 → (s -- a) \ s1 = (s \ s1) -- a | A User Theorem |
| diff-del-minus-3 | ⊦ a ∈ s1 → (s -- a) \ s1 = s \ s1 | A User Theorem |
| diff-del-plus-1 | ⊦ a ∈ s1 → (s ++ a) \ s1 = s \ s1 | A User Theorem |
| diff-del-plus-2 | ⊦ ¬ a ∈ s1 → (s ++ a) \ s1 = (s \ s1) ++ a | A User Theorem |
| diff-diff-right | ⊦ a ∈ s2 → (a ∈ s0 \ s1 \ s2 ↔ a ∈ s0) | A User Theorem |
| diff-e | ⊦ s \ ∅ = s | A User Theorem |
| diff-e-01 | ⊦ ∅ \ s = ∅ | A User Theorem |
| diff-empty | ⊦ s1 \ s2 = ∅ ∧ a ∈ s1 → a ∈ s2 | A User Theorem |
| diff-idem | ⊦ s \ s = ∅ | A User Theorem |
| diff-int | ⊦ (s \ s0) ∩ s0 = ∅ | A User Theorem |
| diff-int-01 | ⊦ (s \ (s0 ∪ s1)) ∩ s0 = ∅ | A User Theorem |
| diff-nin | ⊦ s = s1 \ s2 ∧ a ∈ s2 → ¬ a ∈ s | A User Theorem |
| diff-one | ⊦ (∅ ++ a) \ s ⊆ s0 ++ a | A User Theorem |
| diff-rc | ⊦ (s \ s1) \ s2 = (s \ s2) \ s1 | A User Theorem |
| diff-sub | ⊦ s \ s0 ⊆ s | A User Theorem |
| diff-sub-size | s = s0 \ s1, s1 ⊆ s0 ⊦ # s = # s0 - # s1 | A User Theorem |
| diff-subset | ⊦ s \ (s0 ∪ s1) ⊆ s \ s1 | A User Theorem |
| diff-subset-01 | ⊦ ((s -- a) ++ a0) \ s0 ⊆ ((s \ s0) -- a) ++ a0 | A User Theorem |
| diff-supset-empty | ⊦ s \ (s ∪ s0) = ∅ | A User Theorem |
| diff-union | ⊦ a ∈ s0 → (a ∈ (s \ s0) ∪ s1 ↔ a ∈ s1) | A User Theorem |
| diff-union-01 | ⊦ ¬ a ∈ s → (a ∈ (s \ s0) ∪ s1 ↔ a ∈ s1) | A User Theorem |
| diff-union-02 | ⊦ ¬ a ∈ s → (a ∈ ((s ∪ s2) \ s0) ∪ s1 ↔ a ∈ (s2 \ s0) ∪ s1) | A User Theorem |
| diff-union-03 | ⊦ (s ∪ s0) \ s1 = (s \ s1) ∪ (s0 \ s1) | A User Theorem |
| disjoint-01 | ⊦ disjoint(s, s0) ∧ a ∈ s0 → (a ∈ s ↔ false) | A User Theorem |
| disjoint-02 | ⊦ disjoint(s, s0) ∧ a ∈ s → (a ∈ s0 ↔ false) | A User Theorem |
| disjoint-all | ⊦ disjoint(s, s0) ↔ (∀ a. ¬ a ∈ s ∨ ¬ a ∈ s0) | A User Theorem |
| disjoint-diff | ⊦ disjoint(s, s0) → s \ s0 = s | A User Theorem |
| disjoint-diff-subset | ⊦ disjoint(s0, s1) → (s0 ⊆ s2 \ s1 ↔ s0 ⊆ s2) | A User Theorem |
| disjoint-diff1 | ⊦ disjoint(s0 \ s1, s1) | A User Theorem |
| disjoint-diff2 | ⊦ disjoint(s1, s0 \ s1) | A User Theorem |
| disjoint-empty | ⊦ disjoint(s, ∅) | A User Theorem |
| disjoint-empty2 | ⊦ disjoint(∅, s) | A User Theorem |
| disjoint-notempty-01 | ⊦ disjoint(s, s0) → (disjoint(s, (s0 -- a) ++ a0) ↔ ¬ a0 ∈ s) | A User Theorem |
| disjoint-rec | ⊦ disjoint(s, s0 ++ a) ↔ disjoint(s, s0) ∧ ¬ a ∈ s | A User Theorem |
| disjoint-rec2 | ⊦ disjoint(s0 ++ a, s) ↔ disjoint(s0, s) ∧ ¬ a ∈ s | A User Theorem |
| disjoint-rem | ⊦ disjoint(s0, s1) → (s0 -- a) ∩ s1 = ∅ | A User Theorem |
| disjoint-sym | ⊦ disjoint(s, s0) → (disjoint(s0, s) ↔ true) | A User Theorem |
| disjointu-empty | ⊦ disjoint(s, s0 ∪ s1) ↔ disjoint(s, s0) ∧ disjoint(s, s1) | A User Theorem |
| disjointu-empty2 | ⊦ disjoint(s0 ∪ s1, s) ↔ disjoint(s0, s) ∧ disjoint(s1, s) | A User Theorem |
| equal-add-in | ⊦ s ++ a = s0 ++ a1 ∧ s0 ++ a1 = (s1 -- a2) ++ a0 ∧ a1 ≠ a0 → a1 ∈ s1 -- a2 | A User Theorem |
| filter-add | ⊦ filter(s ++ a, p) = (p(a) ⊃ filter(s, p) ++ a;filter(s, p)) | A User Theorem |
| filter-empty | ⊦ filter(∅, p) = ∅ | A User Theorem |
| filter-union | ⊦ filter(s1 ∪ s2, p) = filter(s1, p) ∪ filter(s2, p) | A User Theorem |
| insdel-sub | ⊦ ¬ s -- a ⊆ s0 ∧ a ∈ s → ¬ s ⊆ s0 ++ a | A User Theorem |
| int-a | ⊦ (s ∩ s0) ∩ s1 = s ∩ s0 ∩ s1 | A User Theorem |
| int-c | ⊦ s ∩ s0 = s0 ∩ s | A User Theorem |
| int-del-rew | ⊦ s ∩ (s0 -- a) = (s ∩ s0) -- a | A User Theorem |
| int-e | ⊦ s ∩ ∅ = ∅ | A User Theorem |
| int-eq | ⊦ s = s ∩ s0 ↔ s ⊆ s0 | A User Theorem |
| int-idem | ⊦ s ∩ s = s | A User Theorem |
| int-inse | ⊦ s ∩ (s0 ++ a) = ∅ → s ∩ s0 = ∅ ∧ ¬ a ∈ s | A User Theorem |
| int-inserte | ⊦ a ∈ s → s ∩ (s0 ++ a) ≠ ∅ | A User Theorem |
| int-subs01 | ⊦ s ∩ s0 ⊆ s | A User Theorem |
| intdel-empty | ⊦ (s -- a) ∩ s0 ≠ ∅ → s ∩ s0 ≠ ∅ | A User Theorem |
| intersect-add-rem | ⊦ (s0 ++ a) ∩ (s1 -- a) = (s0 ∩ s1) -- a | A User Theorem |
| intersection-insert | ⊦ ¬ a ∈ s → s ∩ (s0 ++ a) = s ∩ s0 | A User Theorem |
| intersection-shift | ⊦ a ∈ s → s ∩ (s0 ++ a) = (s ∩ s0) ++ a | A User Theorem |
| intersection-size | ⊦ ¬ # s0 + # s < #(s0 ∩ s) | A User Theorem |
| intexin | s ∩ s0 ≠ ∅ ⊦ ∃ a. a ∈ s ∧ a ∈ s0 | A User Theorem |
| k-delete-rec | ⊦ ¬ a0 ∈ s ∧ a ≠ a0 → (s ++ a0) -- a = (s -- a) ++ a0 | A User Theorem |
| k-diff-rec1 | ⊦ ¬ a ∈ s ∧ a ∈ s0 → s \ s0 = s \ (s0 -- a) | A User Theorem |
| k-diff-rec2 | ⊦ a ∈ s ∧ a ∈ s0 → s \ s0 = (s -- a) \ (s0 -- a) | A User Theorem |
| k-intersect-base | ⊦ ∅ ∩ s = ∅ | A User Theorem |
| k-intersect-rec1 | ⊦ a ∈ s ∧ a ∈ s0 → s ∩ s0 = ((s -- a) ∩ (s0 -- a)) ++ a | A User Theorem |
| k-intersect-rec2 | ⊦ ¬ a ∈ s ∨ ¬ a ∈ s0 → s ∩ s0 = (s -- a) ∩ (s0 -- a) | A User Theorem |
| k-union-base | ⊦ ∅ ∪ s = s | A User Theorem |
| k-union-rec1 | ⊦ ¬ a ∈ s ∧ a ∈ s0 → (s ++ a) ∪ s0 = s ∪ s0 | A User Theorem |
| k-union-rec2 | ⊦ ¬ a ∈ s ∧ ¬ a ∈ s0 → (s ++ a) ∪ s0 = (s ∪ s0) ++ a | A User Theorem |
| one-in | ⊦ a ∈ {a0} ↔ a = a0 | A User Theorem |
| one-min | ⊦ s = ∅ ++ a → s -- a = ∅ | A User Theorem |
| rem-disjoint | ⊦ disjoint(s \ s0, s1) ∧ (s2 \ s) ∩ s1 = ∅ → (s2 \ s0) ∩ s1 = ∅ | A User Theorem |
| rem-union-ins | ⊦ (s0 ∪ (s1 -- a)) ++ a = (s0 ∪ s1) ++ a | A User Theorem |
| selsub-diff | s1 ⊆ s2 ⊦ s1 = s2, ∃ s. s ⊆ s2 ∧ s1 = s2 \ s | A User Theorem |
| size-del-leq | ⊦ # s ≤ #(s -- a) ↔ ¬ a ∈ s | A User Theorem |
| size-eq | ⊦ s0 = s1 → (# s0 = # s1 ↔ true) | A User Theorem |
| size-eq-ext | ∀ a. a ∈ s0 ↔ a ∈ s1 ⊦ # s0 = # s1 ↔ true | A User Theorem |
| size-union | ⊦ s ∩ s0 = ∅ → #(s ∪ s0) = # s + # s0 | A User Theorem |
| sub-ade | ⊦ s ⊆ s0 → (s -- a) ++ a0 ⊆ (s0 -- a) ++ a0 | A User Theorem |
| sub-ade-01 | ⊦ s ⊆ s0 → (s -- a) ++ a0 ++ a1 ⊆ (s0 -- a) ++ a0 ++ a1 | A User Theorem |
| sub-ade-02 | ⊦ s ⊆ s0 ∧ ¬ a ∈ s → s ⊆ (s0 -- a) ++ a0 | A User Theorem |
| sub-ade-03 | ⊦ s ⊆ s0 ∧ ¬ a ∈ s → s ⊆ (s0 -- a) ++ a0 ++ a1 | A User Theorem |
| sub-del | ⊦ s ⊆ s0 → s -- a ⊆ s0 -- a | A User Theorem |
| sub-del-01 | ⊦ s ⊆ s0 ∧ ¬ a ∈ s → s ⊆ s0 -- a | A User Theorem |
| sub-del-02 | ⊦ s -- a ⊆ s | A User Theorem |
| sub-del-03 | ⊦ s ⊆ s0 → s -- a ⊆ s0 | A User Theorem |
| sub-del-04 | ⊦ s ⊆ s0 -- a ↔ s ⊆ s0 ∧ ¬ a ∈ s | A User Theorem |
| sub-del-05 | ⊦ a ≠ a0 ∧ ¬ a ∈ s → s -- a0 ⊆ s -- a0 -- a | A User Theorem |
| sub-rem-inter-rem | ⊦ a ∈ s0 → (s \ (s0 -- a)) ∩ (s1 -- a) = (s \ s0) ∩ s1 | A User Theorem |
| sub-union | ⊦ s ⊆ s0 → s ⊆ s0 ∪ s1 | A User Theorem |
| sub-union-01 | ⊦ s ⊆ s0 → s ⊆ (s0 ∪ s1) ++ a | A User Theorem |
| sub-union-02 | ⊦ s ⊆ (s ∪ s0) ++ a ++ a0 | A User Theorem |
| sub-union-03 | ⊦ s ++ a ⊆ (s ∪ s0) ++ a | A User Theorem |
| sub-union-04 | ⊦ s ++ a ⊆ (s ∪ s0) ++ a0 ++ a | A User Theorem |
| sub-union-05 | ⊦ s ⊆ (s ∪ s0) ++ a ++ a0 ++ a1 | A User Theorem |
| subset-del | ⊦ s ⊆ s -- a ↔ ¬ a ∈ s | A User Theorem |
| subset-disjoint-union | ⊦ s ⊆ s0 ↔ (∃ s1. s ∪ s1 = s0 ∧ disjoint(s, s1)) | A User Theorem |
| subset-union-left-rem | ⊦ s0 ∪ s1 ⊆ s2 ∪ s1 ↔ s0 ⊆ s2 ∪ s1 | A User Theorem |
| subseteq-disjoint-empty | ⊦ s0 ⊆ s1 ∧ disjoint(s1, s2) → disjoint(s0, s2) | A User Theorem |
| subseteq-rem-del | ⊦ a ∈ s0 ∧ s0 ⊆ s1 → s1 \ (s0 -- a) = (s1 \ s0) ++ a | A User Theorem |
| subsub-del | ⊦ s -- a ⊆ s0 ∧ ¬ s ⊆ s0 ∧ a ∈ s → ¬ a ∈ s0 | A User Theorem |
| una | ⊦ (s ∪ s0) ∪ s1 = s ∪ s0 ∪ s1 | A User Theorem |
| unadd | ⊦ (s ++ a) ∪ s0 = (s ∪ s0) ++ a | A User Theorem |
| unadd-01 | ⊦ s ∪ (s0 ++ a) = (s ∪ s0) ++ a | A User Theorem |
| unall | ⊦ s = s0 ∪ s ↔ (∀ a. a ∈ s0 → a ∈ s) | A User Theorem |
| unc | ⊦ s ∪ s0 = s0 ∪ s | A User Theorem |
| undel | ⊦ a ∈ s → s ∪ (s0 -- a) = s ∪ s0 | A User Theorem |
| undel-01 | ⊦ (s ∪ s0) -- a = s ∪ (s0 -- a) ↔ ¬ a ∈ s | A User Theorem |
| undel-02 | ⊦ (s ∪ s0) -- a = (s -- a) ∪ (s0 -- a) | A User Theorem |
| undel-03 | ⊦ ¬ a ∈ s → s ∪ (s0 -- a) = (s ∪ s0) -- a | A User Theorem |
| undel-04 | ⊦ (s -- a) ∪ (s0 -- a) = (s ∪ s0) -- a | A User Theorem |
| undel-05 | ⊦ ¬ a ∈ s ∧ ¬ a ∈ s0 → (s -- a) ∪ (s0 -- a) = s ∪ s0 | A User Theorem |
| undel-06 | ⊦ (s -- a -- a0) ∪ s0 = (s -- a0 -- a) ∪ s0 ↔ true | A User Theorem |
| une | ⊦ s ∪ ∅ = s | A User Theorem |
| une-01 | ⊦ s1 ∪ s2 = ∅ ↔ s1 = ∅ ∧ s2 = ∅ | A User Theorem |
| uneq | ⊦ s ∪ s0 = s0 ↔ s ⊆ s0 | A User Theorem |
| unidem | ⊦ s ∪ s = s | A User Theorem |
| unin | ⊦ ¬ a ∈ s → (a ∈ s ∪ s0 ↔ a ∈ s0) | A User Theorem |
| unin-01 | ⊦ a ∈ s ∪ s0 ↔ ¬(¬ a ∈ s ∧ ¬ a ∈ s0) | A User Theorem |
| unindel | ⊦ ¬ a ∈ s ∪ s0 → ¬ a ∈ (s -- a0) ∪ s0 | A User Theorem |
| uninsert | ⊦ a ∈ s → (s ∪ s0) ++ a = s ∪ s0 | A User Theorem |
| uninsubdel | ⊦ ¬ a ∈ s ∪ s0 ∧ s1 ⊆ s → ¬ a ∈ (s1 -- a0) ∪ s0 | A User Theorem |
| union-del | ⊦ a ∈ (s -- a) ∪ s0 ↔ a ∈ s0 | A User Theorem |
| union-diff | ⊦ ¬ a ∈ s0 ∧ ¬ a ∈ s1 → (a ∈ s \ (s0 ∪ s1) ↔ a ∈ s) | A User Theorem |
| union-diff-union-same | ⊦ (s0 ∪ s1) \ (s2 ∪ s1) = s0 \ (s2 ∪ s1) | A User Theorem |
| union-difference | ⊦ s ⊆ s0 → s ∪ (s0 \ s) = s0 | A User Theorem |
| union-inter-inter-union | ⊦ (s0 ∪ s1) ∩ s2 = (s0 ∩ s2) ∪ (s1 ∩ s2) | A User Theorem |
| union-one | ⊦ a ≠ a0 → (a ∈ {a0} ∪ s ↔ a ∈ s) | A User Theorem |
| union-rem-add | ⊦ a ∈ s0 → ((s0 -- a) ∪ s1) ++ a = s0 ∪ s1 | A User Theorem |
| union-rem-unchdg | ⊦ (s0 -- a) ∪ s1 = s0 ∪ s1 ↔ a ∈ s1 ∨ ¬ a ∈ s0 | A User Theorem |
| union-rewone | ⊦ s ++ a = s ∪ {a} | A User Theorem |
| union-size | ⊦ ¬ #(s ∪ s0) < # s | A User Theorem |
| union-size-01 | ⊦ ¬ a ∈ s → ¬ #((s ∪ s0) ++ a) ≤ # s | A User Theorem |
| union-size-02 | ⊦ ¬ a ∈ s ∧ a ∈ s0 → #(s ∪ s0) = #(s ∪ (s0 -- a)) + 1 | A User Theorem |
| union-size-03 | ⊦ #(s0 ∪ s1) = # s0 + # s1 - #(s0 ∩ s1) | A User Theorem |
| union-size-insert | ⊦ ¬ a ∈ s ∧ ¬ a ∈ s0 → #((s ∪ s0) ++ a) = #(s ∪ s0) + 1 | A User Theorem |
| union-size-le | ⊦ #(s0 ∪ s1) ≤ # s0 + # s1 | A User Theorem |
| union-size-ls-in | ⊦ a ∈ s0 ∧ a ∈ s1 → #(s0 ∪ s1) < # s0 + # s1 | A User Theorem |
| union-size-not-less | ⊦ ¬ # s0 + # s1 < #(s0 ∪ s1) | A User Theorem |
| union-size-not-less-01 | ⊦ # s0 ≤ m ∧ # s1 ≤ n → ¬ m + n < #(s0 ∪ s1) | A User Theorem |
| union-size-not-less-02 | ⊦ # s0 ≤ m ∧ # s1 ≤ n → ¬ m + n + 1 < #(s0 ∪ s1) | A User Theorem |
| union-sizeeq | ⊦ #(s ∪ s0) = # s ↔ s0 ⊆ s | A User Theorem |
| union-sizele | ⊦ #(s ∪ s0) ≤ # s ↔ s0 ⊆ s | A User Theorem |
| union-sizele-01 | ⊦ #((s ∪ s0) ++ a) ≤ # s ↔ s0 ⊆ s ∧ a ∈ s | A User Theorem |
| union-sub | ⊦ s0 ⊆ s → s ∪ s0 = s | A User Theorem |
| union-sub-01 | ⊦ s0 ⊆ s → s0 ∪ s = s | A User Theorem |
| union-sub-rem-disjoint | ⊦ disjoint(s0, s2) ∧ disjoint(s1, s2) → (s0 ∪ s1 ⊆ s3 \ s2 ↔ s0 ∪ s1 ⊆ s3) | A User Theorem |
| union-subset | ⊦ s ⊆ s0 → s ∪ s1 ⊆ s0 ∪ s1 | A User Theorem |
| union-subset-01 | ⊦ ¬ s ∪ s1 ⊆ s0 ∪ s1 → ¬ s ⊆ s0 | A User Theorem |
| union-subset-02 | ⊦ s ⊆ s0 ∧ s1 ⊆ s2 → s ∪ s1 ⊆ (s0 ∪ s2) ++ a | A User Theorem |
| union-subset-03 | ⊦ s ⊆ s0 ∧ s1 ⊆ s2 ∧ s3 ⊆ s4 → s ∪ s1 ∪ s3 ⊆ s0 ∪ s2 ∪ s4 | A User Theorem |
| union-subset-04 | ⊦ s ⊆ s0 ∧ s1 ⊆ s2 ∧ s3 ⊆ s4 → s ∪ s1 ∪ s3 ⊆ (s0 ∪ s2 ∪ s4) ++ a | A User Theorem |
| union-subset-05 | ⊦ s ⊆ s0 → s ∪ s1 ⊆ (s0 ∪ s1) ++ a | A User Theorem |
| unsub | ⊦ s ⊆ s ∪ s0 | A User Theorem |
| unsub-union-01 | ⊦ s1 ∪ s2 ⊆ s ↔ s1 ⊆ s ∧ s2 ⊆ s | A User Theorem |
| unsubdel | ⊦ s ⊆ s0 ∧ s1 ⊆ s2 ∧ ¬ a ∈ s ∧ ¬ a ∈ s1 → s ∪ s1 ⊆ (s0 -- a) ∪ (s2 -- a) | A User Theorem |
| unsubdel-01 | ⊦ s ⊆ s0 ∧ ¬ a ∈ s → (s0 -- a) ∪ s = s0 -- a | A User Theorem |
| unsubins | ⊦ s ++ a ⊆ s ∪ s0 ↔ a ∈ s ∪ s0 | A User Theorem |
| unsubins-01 | ⊦ s ⊆ (s ∪ s0) ++ a | A User Theorem |
| unsubsub | ⊦ s ⊆ s0 ∧ s1 ⊆ s2 → s ∪ s1 ⊆ s0 ∪ s2 | A User Theorem |