Nuprl Lemma : not-has-value-decidable-quot
∀[E:(∀x:Base. ((x)↓ ∨ (¬(x)↓))) ⟶ (∀x:Base. ((x)↓ ∨ (¬(x)↓))) ⟶ ℙ]
¬(f,g:∀x:Base. ((x)↓ ∨ (¬(x)↓))//E[f;g]) supposing EquivRel(∀x:Base. ((x)↓ ∨ (¬(x)↓));f,g.E[f;g])
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
has-value: (a)↓,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
function: x:A ⟶ B[x],
base: Base
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
quotient_wf,
all_wf,
base_wf,
or_wf,
has-value_wf_base,
not_wf,
equiv_rel_wf,
false_wf,
equal_wf,
equal-wf-base,
not_has-value_decidable_on_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
rename,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
productEquality
Latex:
\mforall{}[E:(\mforall{}x:Base. ((x)\mdownarrow{} \mvee{} (\mneg{}(x)\mdownarrow{}))) {}\mrightarrow{} (\mforall{}x:Base. ((x)\mdownarrow{} \mvee{} (\mneg{}(x)\mdownarrow{}))) {}\mrightarrow{} \mBbbP{}]
\mneg{}(f,g:\mforall{}x:Base. ((x)\mdownarrow{} \mvee{} (\mneg{}(x)\mdownarrow{}))//E[f;g]) supposing EquivRel(\mforall{}x:Base. ((x)\mdownarrow{} \mvee{} (\mneg{}(x)\mdownarrow{}));f,g.E[f;g])
Date html generated:
2016_05_14-AM-06_08_32
Last ObjectModification:
2015_12_26-AM-11_48_21
Theory : quot_1
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