Nuprl Lemma : finite-function-type-equality
∀[k:ℕ]. ∀[F:ℕk ⟶ Type]. ((i:ℕk ⟶ F[i]) = (i:ℕk ⟶ map(λi.F[i];upto(k))[i]) ∈ Type)
Proof
Definitions occuring in Statement :
upto: upto(n),
select: L[n],
map: map(f;as),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
squash: ↓T,
prop: ℙ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
int_seg_wf,
equal_wf,
squash_wf,
true_wf,
map_select,
upto_wf,
length_upto,
lelt_wf,
length_wf,
iff_weakening_equal,
nat_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
select_upto
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
because_Cache,
cumulativity,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:\mBbbN{}k {}\mrightarrow{} Type]. ((i:\mBbbN{}k {}\mrightarrow{} F[i]) = (i:\mBbbN{}k {}\mrightarrow{} map(\mlambda{}i.F[i];upto(k))[i]))
Date html generated:
2018_05_21-PM-08_03_16
Last ObjectModification:
2017_07_26-PM-05_39_29
Theory : general
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