Step
*
1
1
3
of Lemma
decidable-bar-rec_wf
1. B : n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ
2. Q : n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ
3. bar : ∀s:ℕ ⟶ ℕ. (↓∃n:ℕ. B[n;s])
4. dec : ∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ∨ (¬B[n;s]))
5. base : ∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s]
⇒ Q[n;s])
6. ind : ∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. Q[n + 1;s.m@n])
⇒ Q[n;s])
7. n1 : ℕ
8. s1 : ℕn1 ⟶ ℕ
9. w : B n1 s1
⊢ decidable-bar-rec(dec;base;ind;n1;s1) ∈ Q[n1;s1]
BY
{ (RecUnfold `decidable-bar-rec` 0⋅
THEN GenConclAtAddr [2;1]
THEN DVar `v'
THEN Reduce 0
THEN Try (Complete ((D (-2) THEN Auto)))
THEN Try (CpltAuto)) }
Latex:
Latex:
1. B : n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}
2. Q : n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}
3. bar : \mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (\mdownarrow{}\mexists{}n:\mBbbN{}. B[n;s])
4. dec : \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] \mvee{} (\mneg{}B[n;s]))
5. base : \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s])
6. ind : \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. ((\mforall{}m:\mBbbN{}. Q[n + 1;s.m@n]) {}\mRightarrow{} Q[n;s])
7. n1 : \mBbbN{}
8. s1 : \mBbbN{}n1 {}\mrightarrow{} \mBbbN{}
9. w : B n1 s1
\mvdash{} decidable-bar-rec(dec;base;ind;n1;s1) \mmember{} Q[n1;s1]
By
Latex:
(RecUnfold `decidable-bar-rec` 0\mcdot{}
THEN GenConclAtAddr [2;1]
THEN DVar `v'
THEN Reduce 0
THEN Try (Complete ((D (-2) THEN Auto)))
THEN Try (CpltAuto))
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