Nuprl Lemma : all-quotient-true
∀T:Type. (⇃(canonicalizable(T)) ⇒ (∀P:T ⟶ ℙ. (∀t:T. ⇃(P[t]) ⇐⇒ ⇃(∀t:T. P[t]))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
canonicalizable: canonicalizable(T),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
true: True,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
true: True,
quotient: x,y:A//B[x; y],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
rev_implies: P ⇐ Q
Lemmas referenced :
prop-truncation-quot,
implies-quotient-true,
squash_wf,
dep-fun-equiv_wf,
equal-wf-base,
quotient-member-eq,
all-quotient-dependent,
all_wf,
quotient_wf,
true_wf,
equiv_rel_true,
canonicalizable_wf
Rules used in proof :
productEquality,
natural_numberEquality,
pertypeElimination,
pointwiseFunctionalityForEquality,
rename,
productElimination,
independent_functionElimination,
dependent_functionElimination,
promote_hyp,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
hypothesis,
independent_isectElimination,
functionEquality,
universeEquality
Latex:
\mforall{}T:Type. (\00D9(canonicalizable(T)) {}\mRightarrow{} (\mforall{}P:T {}\mrightarrow{} \mBbbP{}. (\mforall{}t:T. \00D9(P[t]) \mLeftarrow{}{}\mRightarrow{} \00D9(\mforall{}t:T. P[t]))))
Date html generated:
2017_09_29-PM-06_07_48
Last ObjectModification:
2017_09_07-PM-05_50_23
Theory : continuity
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