Nuprl Lemma : k-ext-iff
∀[k:ℕ]. ∀[A,B:ℕk ⟶ Type]. uiff(A ≡ B;∀i:ℕk. A i ≡ B i)
Proof
Definitions occuring in Statement :
k-ext: A ≡ B,
int_seg: {i..j-},
nat: ℕ,
ext-eq: A ≡ B,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
k-ext: A ≡ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
k-subtype: A ⊆ B,
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
k-subtype_wf,
all_wf,
ext-eq_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
lambdaEquality,
independent_pairEquality,
axiomEquality,
because_Cache,
productEquality,
functionExtensionality,
applyEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[A,B:\mBbbN{}k {}\mrightarrow{} Type]. uiff(A \mequiv{} B;\mforall{}i:\mBbbN{}k. A i \mequiv{} B i)
Date html generated:
2018_05_21-PM-00_09_03
Last ObjectModification:
2017_10_18-PM-02_32_19
Theory : co-recursion
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