Nuprl Lemma : assert-finite-fun-deq
∀[T:Type]. ∀[k:ℕ]. ∀[eq:EqDecider(T)]. ∀[f,g:ℕk ⟶ T]. uiff(↑(finite-fun-deq(k;eq) f g);f = g ∈ (ℕk ⟶ T))
Proof
Definitions occuring in Statement :
finite-fun-deq: finite-fun-deq(k;eq),
deq: EqDecider(T),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
deq: EqDecider(T),
nat: ℕ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
iff_weakening_uiff,
assert_wf,
finite-fun-deq_wf,
equal_wf,
int_seg_wf,
assert-deq,
istype-assert,
assert_witness,
deq_wf,
istype-nat,
istype-universe
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation_alt,
hypothesis,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
because_Cache,
sqequalHypSubstitution,
productElimination,
thin,
independent_isectElimination,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
functionEquality,
natural_numberEquality,
independent_functionElimination,
promote_hyp,
functionIsType,
universeIsType,
instantiate,
universeEquality,
independent_pairEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[T:Type]. \mforall{}[k:\mBbbN{}]. \mforall{}[eq:EqDecider(T)]. \mforall{}[f,g:\mBbbN{}k {}\mrightarrow{} T]. uiff(\muparrow{}(finite-fun-deq(k;eq) f g);f = g)
Date html generated:
2020_05_19-PM-09_36_36
Last ObjectModification:
2019_10_18-PM-00_01_10
Theory : equality!deciders
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