Nuprl Lemma : assert-fset-pairwise
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[s:fset(T)].
uiff(↑fset-pairwise(x,y.R[x;y];s);∀x,y:T. (↑R[x;y]) supposing (x ∈ s and y ∈ s))
Proof
Definitions occuring in Statement :
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
fset-all: fset-all(s;x.P[x]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
fset-all-iff,
iff_weakening_uiff,
fset-all_wf,
uall_wf,
isect_wf,
fset-member_wf,
assert_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
all_wf,
uiff_wf,
fset-pairwise_wf,
fset_wf,
bool_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
independent_pairFormation,
isect_memberFormation,
lambdaFormation,
hypothesis,
independent_isectElimination,
lambdaEquality,
applyEquality,
independent_functionElimination,
because_Cache,
productElimination,
isect_memberEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
addLevel,
functionEquality,
universeEquality,
independent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
uiff(\muparrow{}fset-pairwise(x,y.R[x;y];s);\mforall{}x,y:T. (\muparrow{}R[x;y]) supposing (x \mmember{} s and y \mmember{} s))
Date html generated:
2019_06_20-PM-01_59_27
Last ObjectModification:
2018_08_24-PM-11_34_36
Theory : finite!sets
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