Nuprl Lemma : weak-continuity-truncated
∀[T:{T:Type| (T ⊆r ℕ) ∧ (↓T)} ]
∀F:(ℕ ⟶ T) ⟶ ℕ. ⇃(∀f:ℕ ⟶ T. ∃n:ℕ. ∀g:ℕ ⟶ T. ((f = g ∈ (ℕn ⟶ T)) ⇒ ((F f) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
set: {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 :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
weak-continuity-skolem: weak-continuity-skolem(T;F),
exists: ∃x:A. B[x],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
weak-continuity-skolem-truncated,
nat_wf,
set_wf,
subtype_rel_wf,
squash_wf,
weak-continuity-skolem_wf,
all_wf,
exists_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
implies-quotient-true,
decidable__le,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
functionEquality,
hypothesis,
setElimination,
rename,
instantiate,
universeEquality,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
functionExtensionality,
applyEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
productElimination,
independent_functionElimination,
dependent_pairFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[T:\{T:Type| (T \msubseteq{}r \mBbbN{}) \mwedge{} (\mdownarrow{}T)\} ]
\mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}. \00D9(\mforall{}f:\mBbbN{} {}\mrightarrow{} T. \mexists{}n:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} T. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2017_04_17-AM-09_54_06
Last ObjectModification:
2017_02_27-PM-05_48_50
Theory : continuity
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