Nuprl Lemma : project-seq_wf
∀[T:ℕ ⟶ Type]. ∀[n:ℕ]. ∀[s:ℕn ⟶ (i:ℕ × T[i])]. project-seq(s) ∈ i:ℕn ⟶ T[i] supposing ∀i:ℕn. ((fst((s i))) = i ∈ ℤ)
Proof
Definitions occuring in Statement :
project-seq: project-seq(s),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
false: False,
less_than': less_than'(a;b),
guard: {T},
sq_type: SQType(T),
sq_stable: SqStable(P),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
pi2: snd(t),
pi1: fst(t),
implies: P ⇒ Q,
project-seq: project-seq(s),
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-false,
int_seg_subtype_nat,
subtype_rel_self,
istype-le,
sq_stable__le,
subtype_base_sq,
istype-universe,
istype-nat,
less_than_wf,
and_wf,
int_subtype_base,
le_wf,
set_subtype_base,
nat_wf,
pi1_wf,
istype-int,
int_seg_wf
Rules used in proof :
independent_pairFormation,
dependent_set_memberEquality_alt,
baseClosed,
imageMemberEquality,
cumulativity,
universeEquality,
instantiate,
productIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
sqequalBase,
imageElimination,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
rename,
setElimination,
natural_numberEquality,
isectElimination,
extract_by_obid,
universeIsType,
functionIsType,
axiomEquality,
because_Cache,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productElimination,
lambdaFormation_alt,
hypothesis,
inhabitedIsType,
applyEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
functionExtensionality,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:\mBbbN{} {}\mrightarrow{} Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n {}\mrightarrow{} (i:\mBbbN{} \mtimes{} T[i])].
project-seq(s) \mmember{} i:\mBbbN{}n {}\mrightarrow{} T[i] supposing \mforall{}i:\mBbbN{}n. ((fst((s i))) = i)
Date html generated:
2019_10_15-AM-10_20_09
Last ObjectModification:
2019_10_07-PM-04_08_18
Theory : bar-induction
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