Nuprl Lemma : split-at-first
∀[T:Type]. ∀[P:T ⟶ ℙ].
((∀x:T. Dec(P[x]))
⇒ (∀L:T List. ∃X,Y:T List. ((L = (X @ Y) ∈ (T List)) ∧ (∀x∈X.¬P[x]) ∧ P[hd(Y)] supposing ||Y|| ≥ 1 )))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
hd: hd(l),
length: ||as||,
append: as @ bs,
list: T List,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
uimplies: b supposing a,
ge: i ≥ j ,
le: A ≤ B,
not: ¬A,
false: False,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
exists_wf,
list_wf,
equal_wf,
append_wf,
length_wf,
length-append,
all_wf,
decidable_wf,
nil_wf,
list_ind_nil_lemma,
length_of_nil_lemma,
l_all_nil,
less_than'_wf,
ge_wf,
equal-wf-base-T,
l_all_wf2,
not_wf,
l_member_wf,
hd_wf,
cons_wf,
length_of_cons_lemma,
reduce_hd_cons_lemma,
list_ind_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
l_all_cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
independent_pairFormation,
productElimination,
independent_pairEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
setElimination,
setEquality,
isectEquality,
independent_isectElimination,
dependent_pairFormation,
unionElimination,
addEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:T. Dec(P[x]))
{}\mRightarrow{} (\mforall{}L:T List. \mexists{}X,Y:T List. ((L = (X @ Y)) \mwedge{} (\mforall{}x\mmember{}X.\mneg{}P[x]) \mwedge{} P[hd(Y)] supposing ||Y|| \mgeq{} 1 )))
Date html generated:
2018_05_21-PM-07_40_09
Last ObjectModification:
2017_07_26-PM-05_14_18
Theory : general
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