Nuprl Lemma : b_all-squash-exists-bag
∀[A,B:Type]. ∀[as:bag(A)]. ∀[P:A ⟶ B ⟶ ℙ].
↓∃bs:bag(A × B). ((bag-map(λx.(fst(x));bs) = as ∈ bag(A)) ∧ b_all(A × B;bs;x.let a,b = x in ↓P[a;b]))
supposing b_all(A;as;x.↓∃y:B. P[x;y])
Proof
Definitions occuring in Statement :
b_all: b_all(T;b;x.P[x]),
bag-map: bag-map(f;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
pi1: fst(t),
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
spread: spread def,
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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
and: P ∧ Q,
pi1: fst(t),
all: ∀x:A. B[x],
cand: A c∧ B,
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
nil: [],
it: ⋅,
b_all: b_all(T;b;x.P[x]),
empty-bag: {},
uiff: uiff(P;Q),
false: False,
cons-bag: x.b,
iff: P ⇐⇒ Q,
top: Top,
true: True,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
bag_to_squash_list,
b_all_wf,
squash_wf,
exists_wf,
list_induction,
list-subtype-bag,
bag_wf,
equal_wf,
bag-map_wf,
list_wf,
nil_wf,
cons_wf,
bag-member-empty-iff,
bag-member_wf,
equal-wf-T-base,
pi1_wf,
b_all-cons,
sq_stable__squash,
cons-bag_wf,
bag-map-cons,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
rename,
functionEquality,
because_Cache,
independent_isectElimination,
productEquality,
independent_functionElimination,
lambdaFormation,
voidEquality,
voidElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
isect_memberEquality,
equalityTransitivity,
universeEquality,
dependent_pairFormation,
independent_pairFormation,
independent_pairEquality,
spreadEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[as:bag(A)]. \mforall{}[P:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
\mdownarrow{}\mexists{}bs:bag(A \mtimes{} B). ((bag-map(\mlambda{}x.(fst(x));bs) = as) \mwedge{} b\_all(A \mtimes{} B;bs;x.let a,b = x in \mdownarrow{}P[a;b]))
supposing b\_all(A;as;x.\mdownarrow{}\mexists{}y:B. P[x;y])
Date html generated:
2017_10_01-AM-08_55_18
Last ObjectModification:
2017_07_26-PM-04_37_18
Theory : bags
Home
Index