Nuprl Lemma : genFAN_wf
∀[T:ℕ ⟶ Type]
∀[max:∀i:ℕ. Bounded(T[i])]. ∀[X:n:ℕ ⟶ (i:ℕn ⟶ T[i]) ⟶ ℙ].
∀[d:∀n:ℕ. ∀s:i:ℕn ⟶ T[i]. Dec(X[n;s])]. (genFAN(max;d) ∈ {k:ℕ| ∀f:i:ℕ ⟶ T[i]. ∃n:ℕk. X[n;f]} )
supposing ∀f:i:ℕ ⟶ T[i]. (↓∃n:ℕ. X[n;f])
supposing ∀i:ℕ. T[i]
Proof
Definitions occuring in Statement :
genFAN: genFAN(max;d),
bounded-type: Bounded(T),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
lelt: i ≤ j < k,
int_seg: {i..j-},
sq_exists: ∃x:A [B[x]],
istype: istype(T),
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
simple_more_general_fan_theorem-ext,
int_seg_decide: int_seg_decide(d;i;j),
ifthenelse: if b then t else f fi ,
bar_recursion: bar_recursion,
genFAN: genFAN(max;d),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
decidable_wf,
istype-false,
int_seg_subtype_nat,
subtype_rel_dep_function,
nat_wf,
squash_wf,
int_seg_wf,
bounded-type_wf,
istype-nat,
simple_more_general_fan_theorem-ext
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
axiomEquality,
instantiate,
productElimination,
productIsType,
setIsType,
independent_pairFormation,
independent_isectElimination,
productEquality,
universeEquality,
setElimination,
natural_numberEquality,
because_Cache,
universeIsType,
extract_by_obid,
functionIsType,
isectIsType,
independent_functionElimination,
dependent_functionElimination,
equalityIstype,
rename,
sqequalHypSubstitution,
thin,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
lambdaEquality_alt,
applyEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:\mBbbN{} {}\mrightarrow{} Type]
\mforall{}[max:\mforall{}i:\mBbbN{}. Bounded(T[i])]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{}n {}\mrightarrow{} T[i]) {}\mrightarrow{} \mBbbP{}].
\mforall{}[d:\mforall{}n:\mBbbN{}. \mforall{}s:i:\mBbbN{}n {}\mrightarrow{} T[i]. Dec(X[n;s])]. (genFAN(max;d) \mmember{} \{k:\mBbbN{}| \mforall{}f:i:\mBbbN{} {}\mrightarrow{} T[i]. \mexists{}n:\mBbbN{}k. X[n;f]\} \000C)
supposing \mforall{}f:i:\mBbbN{} {}\mrightarrow{} T[i]. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f])
supposing \mforall{}i:\mBbbN{}. T[i]
Date html generated:
2019_10_15-AM-10_20_21
Last ObjectModification:
2019_10_07-PM-04_41_41
Theory : bar-induction
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