Step
*
of Lemma
bl-exists-first
∀[A:Type]. ∀P:A ⟶ 𝔹. ∀L:A List. (↑(∃x∈L.P[x])_b
⇐⇒ ∃i:ℕ||L||. ((↑P[L[i]]) ∧ (∀j:ℕi. (¬↑P[L[j]]))))
BY
{ ((UnivCD THENA Auto) THEN (RWO "assert-bl-exists" 0 THENA Auto) THEN ListInd (-1) THEN Reduce 0) }
1
1. [A] : Type
2. P : A ⟶ 𝔹
⊢ (∃x∈[]. ↑P[x])
⇐⇒ ∃i:ℕ0. ((↑P[⊥]) ∧ (∀j:ℕi. (¬↑P[⊥])))
2
1. [A] : Type
2. P : A ⟶ 𝔹
3. u : A
4. v : A List
5. (∃x∈v. ↑P[x])
⇐⇒ ∃i:ℕ||v||. ((↑P[v[i]]) ∧ (∀j:ℕi. (¬↑P[v[j]])))
⊢ (∃x∈[u / v]. ↑P[x])
⇐⇒ ∃i:ℕ||v|| + 1. ((↑P[[u / v][i]]) ∧ (∀j:ℕi. (¬↑P[[u / v][j]])))
Latex:
Latex:
\mforall{}[A:Type]
\mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. (\muparrow{}(\mexists{}x\mmember{}L.P[x])\_b \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. ((\muparrow{}P[L[i]]) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}P[L[j]]))))
By
Latex:
((UnivCD THENA Auto) THEN (RWO "assert-bl-exists" 0 THENA Auto) THEN ListInd (-1) THEN Reduce 0)
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