Nuprl Lemma : test-squash-simp
∀[B:ℤ ⟶ ℤ ⟶ ℙ]. ∀[A:ℤ ⟶ ℙ].
(↓∃x:ℤ
((↓∃v:ℤ. (↓A[x] ∨ (↓A[v]))) ∧ (↓(∃u:ℤ. (↓((↓B[x;u]) ∧ (↓B[u;u])) ∨ ((↓B[u;x]) ∧ B[x;x]))) ∨ (↓∃n:ℕ. (↓4 ≤ n))))
⇐⇒ ↓∃x:ℤ. ((∃v:ℤ. (A[x] ∨ A[v])) ∧ ((∃u:ℤ. ((B[x;u] ∧ B[u;u]) ∨ (B[u;x] ∧ B[x;x]))) ∨ (∃n:ℕ. (4 ≤ n)))))
Proof
Definitions occuring in Statement :
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
le: A ≤ B,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
exists: ∃x:A. B[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
nat: ℕ,
rev_implies: P ⇐ Q,
squash: ↓T,
hint: hint(t),
true: True,
le: A ≤ B,
guard: {T},
cand: A c∧ B
Lemmas referenced :
iff_wf,
and_wf,
le_wf,
nat_wf,
or_wf,
exists_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
lambdaEquality,
productEquality,
because_Cache,
applyEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
addLevel,
productElimination,
impliesFunctionality,
imageElimination,
unionElimination,
dependent_pairFormation,
inlFormation,
imageMemberEquality,
baseClosed,
inrFormation,
universeEquality,
promote_hyp,
independent_pairEquality,
dependent_functionElimination,
functionEquality,
cumulativity,
isect_memberEquality
Latex:
\mforall{}[B:\mBbbZ{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[A:\mBbbZ{} {}\mrightarrow{} \mBbbP{}].
(\mdownarrow{}\mexists{}x:\mBbbZ{}
((\mdownarrow{}\mexists{}v:\mBbbZ{}. (\mdownarrow{}A[x] \mvee{} (\mdownarrow{}A[v])))
\mwedge{} (\mdownarrow{}(\mexists{}u:\mBbbZ{}. (\mdownarrow{}((\mdownarrow{}B[x;u]) \mwedge{} (\mdownarrow{}B[u;u])) \mvee{} ((\mdownarrow{}B[u;x]) \mwedge{} B[x;x]))) \mvee{} (\mdownarrow{}\mexists{}n:\mBbbN{}. (\mdownarrow{}4 \mleq{} n))))
\mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}x:\mBbbZ{}
((\mexists{}v:\mBbbZ{}. (A[x] \mvee{} A[v]))
\mwedge{} ((\mexists{}u:\mBbbZ{}. ((B[x;u] \mwedge{} B[u;u]) \mvee{} (B[u;x] \mwedge{} B[x;x]))) \mvee{} (\mexists{}n:\mBbbN{}. (4 \mleq{} n)))))
Date html generated:
2016_05_15-PM-07_49_34
Last ObjectModification:
2016_01_16-AM-09_35_05
Theory : general
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