Step
*
2
2
of Lemma
rng-pp-nontrivial-1
1. r : IntegDom{i}@i'
2. eq : ∀x,y:|r|. Dec(x = y ∈ |r|)@i
3. l : {p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} @i
4. [%2] : ¬((l 1) = 0 ∈ |r|)
5. ∀i,j:ℕ3.
((¬((l i) = 0 ∈ |r|))
⇒ (¬(i = j ∈ ℤ))
⇒ (λx.if (x =z j) then l i
if (x =z i) then -r (l j)
else 0
fi ∈ {p:ℕ3 ⟶ |r|| (¬(p = 0 ∈ (ℕ3 ⟶ |r|))) ∧ ((p . l) = 0 ∈ |r|)} ))
⊢ ∃a:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} . (∃b:{p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} [(((a . l) = 0 ∈ |r|) ∧ ((b . l)\000C = 0 ∈ |r|) ∧ (¬((a x b) = 0 ∈ (ℕ3 ⟶ |r|))))])
BY
{ ((InstHyp [⌜1⌝;⌜0⌝] (-1)⋅ THENA Auto)
THEN (InstHyp [⌜1⌝;⌜2⌝] (-2)⋅ THENA Auto)
THEN (InstConcl [λx.if (x =z 0) then l 1
if (x =z 1) then -r (l 0)
else 0
fi
; λx.if (x =z 2) then l 1
if (x =z 1) then -r (l 2)
else 0
fi ]⋅
THENA Auto
)
THEN Auto
THEN Try ((GenConclAtAddr [2;3] THEN Auto))) }
1
1. r : IntegDom{i}@i'
2. eq : ∀x,y:|r|. Dec(x = y ∈ |r|)@i
3. l : {p:ℕ3 ⟶ |r|| ¬(p = 0 ∈ (ℕ3 ⟶ |r|))} @i
4. ¬((l 1) = 0 ∈ |r|)
5. ∀i,j:ℕ3.
((¬((l i) = 0 ∈ |r|))
⇒ (¬(i = j ∈ ℤ))
⇒ (λx.if (x =z j) then l i
if (x =z i) then -r (l j)
else 0
fi ∈ {p:ℕ3 ⟶ |r|| (¬(p = 0 ∈ (ℕ3 ⟶ |r|))) ∧ ((p . l) = 0 ∈ |r|)} ))
6. λx.if (x =z 0) then l 1
if (x =z 1) then -r (l 0)
else 0
fi ∈ {p:ℕ3 ⟶ |r|| (¬(p = 0 ∈ (ℕ3 ⟶ |r|))) ∧ ((p . l) = 0 ∈ |r|)}
7. λx.if (x =z 2) then l 1
if (x =z 1) then -r (l 2)
else 0
fi ∈ {p:ℕ3 ⟶ |r|| (¬(p = 0 ∈ (ℕ3 ⟶ |r|))) ∧ ((p . l) = 0 ∈ |r|)}
8. (λx.if (x =z 0) then l 1 if (x =z 1) then -r (l 0) else 0 fi . l) = 0 ∈ |r|
9. (λx.if (x =z 2) then l 1 if (x =z 1) then -r (l 2) else 0 fi . l) = 0 ∈ |r|
⊢ ¬((λx.if (x =z 0) then l 1
if (x =z 1) then -r (l 0)
else 0
fi x λx.if (x =z 2) then l 1
if (x =z 1) then -r (l 2)
else 0
fi )
= 0
∈ (ℕ3 ⟶ |r|))
Latex:
Latex:
1. r : IntegDom\{i\}@i'
2. eq : \mforall{}x,y:|r|. Dec(x = y)@i
3. l : \{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} @i
4. [\%2] : \mneg{}((l 1) = 0)
5. \mforall{}i,j:\mBbbN{}3.
((\mneg{}((l i) = 0))
{}\mRightarrow{} (\mneg{}(i = j))
{}\mRightarrow{} (\mlambda{}x.if (x =\msubz{} j) then l i
if (x =\msubz{} i) then -r (l j)
else 0
fi \mmember{} \{p:\mBbbN{}3 {}\mrightarrow{} |r|| (\mneg{}(p = 0)) \mwedge{} ((p . l) = 0)\} ))
\mvdash{} \mexists{}a:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} . (\mexists{}b:\{p:\mBbbN{}3 {}\mrightarrow{} |r|| \mneg{}(p = 0)\} [(((a . l) = 0) \mwedge{} ((b . l) = 0) \mwedge{} (\mneg{}((a \000Cx b) = 0)))])
By
Latex:
((InstHyp [\mkleeneopen{}1\mkleeneclose{};\mkleeneopen{}0\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN (InstHyp [\mkleeneopen{}1\mkleeneclose{};\mkleeneopen{}2\mkleeneclose{}] (-2)\mcdot{} THENA Auto)
THEN (InstConcl [\mlambda{}x.if (x =\msubz{} 0) then l 1
if (x =\msubz{} 1) then -r (l 0)
else 0
fi
; \mlambda{}x.if (x =\msubz{} 2) then l 1
if (x =\msubz{} 1) then -r (l 2)
else 0
fi ]\mcdot{}
THENA Auto
)
THEN Auto
THEN Try ((GenConclAtAddr [2;3] THEN Auto)))
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