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Step * 2 of Lemma monotone-bar-induction-strict3


1. B : n:ℕ ⟶ {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}  ⟶ ℙ@i'
2. Q : n:ℕ ⟶ {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}  ⟶ ℙ@i'
3. ∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .
     (B[n;s] ⇒ (∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) ⇒ B[n + 1;s.m@n])))@i
4. ∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .  (B[n;s] ⇒ ⇃(Q[n;s]))@i
5. ∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .
     ((∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) ⇒ ⇃(Q[n + 1;s.m@n]))) ⇒ ⇃(Q[n;s]))@i
6. ∀alpha:StrictInc. ∃m:ℕ. B[m;alpha]@i
⊢ ⇃(Q[0;λx.⊥])
BY
{ (InstLemma `monotone-bar-induction-strict2` [⌜B⌝;⌜Q⌝]⋅ THEN Auto) }

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Latex:

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Latex:

1.  B  :  n:\mBbbN{}  {}\mrightarrow{}  \{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}    {}\mrightarrow{}  \mBbbP{}@i'
2.  Q  :  n:\mBbbN{}  {}\mrightarrow{}  \{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}    {}\mrightarrow{}  \mBbbP{}@i'
3.  \mforall{}n:\mBbbN{}.  \mforall{}s:\{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}  .
          (B[n;s]  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  (strictly-increasing-seq(n  +  1;s.m@n)  {}\mRightarrow{}  B[n  +  1;s.m@n])))@i
4.  \mforall{}n:\mBbbN{}.  \mforall{}s:\{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}  .    (B[n;s]  {}\mRightarrow{}  \00D9(Q[n;s]))@i
5.  \mforall{}n:\mBbbN{}.  \mforall{}s:\{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}  .
          ((\mforall{}m:\mBbbN{}.  (strictly-increasing-seq(n  +  1;s.m@n)  {}\mRightarrow{}  \00D9(Q[n  +  1;s.m@n])))  {}\mRightarrow{}  \00D9(Q[n;s]))@i
6.  \mforall{}alpha:StrictInc.  \mexists{}m:\mBbbN{}.  B[m;alpha]@i
\mvdash{}  \00D9(Q[0;\mlambda{}x.\mbot{}])


By

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Latex:
(InstLemma  `monotone-bar-induction-strict2`  [\mkleeneopen{}B\mkleeneclose{};\mkleeneopen{}Q\mkleeneclose{}]\mcdot{}  THEN  Auto)

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