Nuprl Lemma : all-quotient-true3
∀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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
canonicalizable: canonicalizable(T),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
guard: {T},
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
iff_weakening_equal,
implies-quotient-true,
equiv_rel_true,
true_wf,
quotient_wf,
all_wf,
subtype_rel_self,
isect2_subtype_rel,
subtype_rel_dep_function,
isect2_subtype_rel2,
base_wf,
isect2_wf,
all-quotient-true2,
canonicalizable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
instantiate,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
functionExtensionality,
because_Cache,
isect_memberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}T:Type. (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:
2016_05_14-PM-09_42_10
Last ObjectModification:
2016_05_10-AM-11_31_10
Theory : continuity
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