Nuprl Lemma : uniform-continuity-pi2-dec
∀T:Type. ∀F:(ℕ ⟶ 𝔹) ⟶ T. ∀n:ℕ. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ Dec(ucB(T;F;n)))
Proof
Definitions occuring in Statement :
uniform-continuity-pi2: ucB(T;F;n),
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uniform-continuity-pi2: ucB(T;F;n),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
guard: {T},
false: False
Lemmas referenced :
all_wf,
decidable_wf,
equal_wf,
nat_wf,
bool_wf,
decidable__all_int_seg,
int_seg_wf,
decidable__bool,
decidable__not,
ext2Cantor_wf,
btrue_wf,
bfalse_wf,
simple-finite-cantor-decider_wf,
not_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
universeEquality,
instantiate,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
independent_functionElimination,
applyEquality,
introduction,
unionElimination,
productElimination,
inrFormation,
inlFormation,
dependent_pairFormation,
voidElimination
Latex:
\mforall{}T:Type. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. \mforall{}n:\mBbbN{}. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} Dec(ucB(T;F;n)))
Date html generated:
2016_05_14-PM-09_38_35
Last ObjectModification:
2015_12_26-PM-09_49_19
Theory : continuity
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