Nuprl Lemma : base-equal-partial
∀[A:Type]
∀[a,b:Base].
a = b ∈ partial(A) supposing (((a)↓ ⇐⇒ (b)↓) ∧ a = b ∈ A supposing (a)↓) ∧ (¬is-exception(a)) ∧ (¬is-exception(b))
supposing value-type(A)
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
has-value: (a)↓,
is-exception: is-exception(t),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
partial: partial(T),
so_lambda: λ2x y.t[x; y],
base-partial: base-partial(T),
so_apply: x[s1;s2],
all: ∀x:A. B[x],
cand: A c∧ B,
per-partial: per-partial(T;x;y),
uiff: uiff(P;Q),
has-value: (a)↓
Lemmas referenced :
equal-wf-base,
and_wf,
iff_wf,
has-value_wf_base,
isect_wf,
not_wf,
is-exception_wf,
base_wf,
value-type_wf,
quotient-member-eq,
base-partial_wf,
per-partial_wf,
per-partial-equiv_rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
universeEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
introduction,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
isectEquality,
axiomSqleEquality
Latex:
\mforall{}[A:Type]
\mforall{}[a,b:Base].
a = b
supposing (((a)\mdownarrow{} \mLeftarrow{}{}\mRightarrow{} (b)\mdownarrow{}) \mwedge{} a = b supposing (a)\mdownarrow{}) \mwedge{} (\mneg{}is-exception(a)) \mwedge{} (\mneg{}is-exception(b))
supposing value-type(A)
Date html generated:
2016_05_14-AM-06_09_42
Last ObjectModification:
2015_12_26-AM-11_52_16
Theory : partial_1
Home
Index