Nuprl Lemma : strong-continuity2-half-squash
∀[T:Type]. ∀F:(ℕ ⟶ T) ⟶ ℕ. ⇃(strong-continuity2(T;F)) supposing (T ⊆r ℕ) ∧ (↓T)
Proof
Definitions occuring in Statement :
strong-continuity2: strong-continuity2(T;F),
quotient: x,y:A//B[x; y],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
and: P ∧ Q,
true: True,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
all: ∀x:A. B[x],
prop: ℙ,
sq-basic-strong-continuity: sq-basic-strong-continuity(T;F),
sq_exists: ∃x:A [B[x]],
basic-strong-continuity: basic-strong-continuity(T;F),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
guard: {T}
Lemmas referenced :
Kleene-M_wf,
subtype_rel_wf,
nat_wf,
squash_wf,
subtype_rel_self,
quotient_wf,
basic-strong-continuity_wf,
true_wf,
equiv_rel_true,
implies-quotient-true2,
strong-continuity2_wf,
trivial-quotient-true,
basic-implies-strong-continuity2-ext,
istype-nat,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
imageElimination,
imageMemberEquality,
hypothesisEquality,
baseClosed,
rename,
Error :lambdaFormation_alt,
extract_by_obid,
isectElimination,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :universeIsType,
applyEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
independent_isectElimination,
independent_functionElimination,
because_Cache,
Error :functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}. \00D9(strong-continuity2(T;F)) supposing (T \msubseteq{}r \mBbbN{}) \mwedge{} (\mdownarrow{}T)
Date html generated:
2019_06_20-PM-02_51_16
Last ObjectModification:
2019_02_09-PM-11_56_09
Theory : continuity
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