Nuprl Lemma : l_all-squash-exists-list
∀[A,B:Type]. ∀[as:A List]. ∀[P:A ⟶ B ⟶ ℙ].
↓∃bs:(A × B) List. ((map(λx.(fst(x));bs) = as ∈ (A List)) ∧ (∀x∈bs.↓P[fst(x);snd(x)])) supposing (∀x∈as.↓∃y:B. P[x;y])
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
exists: ∃x:A. B[x],
and: P ∧ Q,
pi1: fst(t),
top: Top,
cand: A c∧ B,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
pi2: snd(t)
Lemmas referenced :
list_induction,
l_all_wf,
l_member_wf,
squash_wf,
exists_wf,
list_wf,
equal_wf,
map_wf,
length_wf,
length-map,
nil_wf,
map_nil_lemma,
l_all_nil,
equal-wf-T-base,
pi1_wf,
length_of_nil_lemma,
pi2_wf,
l_all_wf_nil,
l_all_cons,
cons_wf,
map_cons_lemma,
true_wf,
iff_weakening_equal,
length_of_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
lambdaFormation,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
productEquality,
productElimination,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_functionElimination,
independent_pairFormation,
independent_pairEquality,
baseClosed,
imageMemberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[as:A List]. \mforall{}[P:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
\mdownarrow{}\mexists{}bs:(A \mtimes{} B) List. ((map(\mlambda{}x.(fst(x));bs) = as) \mwedge{} (\mforall{}x\mmember{}bs.\mdownarrow{}P[fst(x);snd(x)]))
supposing (\mforall{}x\mmember{}as.\mdownarrow{}\mexists{}y:B. P[x;y])
Date html generated:
2017_10_01-AM-08_55_26
Last ObjectModification:
2017_07_26-PM-04_37_27
Theory : bags
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