Nuprl Lemma : add-has-value-iff
∀[x,y:partial(ℕ)]. uiff((x + y)↓;(x)↓ ∧ (y)↓)
Proof
Definitions occuring in Statement :
partial: partial(T),
nat: ℕ,
has-value: (a)↓,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
add: n + m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
has-value: (a)↓,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
guard: {T},
cand: A c∧ B
Lemmas referenced :
has-value_wf_base,
partial_subtype_base,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
has-value_wf-partial,
set-value-type,
int-value-type,
partial_wf,
add-has-value-partial-nat,
value-type-has-value,
termination
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomSqleEquality,
hypothesis,
Error :universeIsType,
extract_by_obid,
isectElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
dependent_functionElimination,
independent_functionElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
Error :productIsType,
because_Cache,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
addEquality,
setElimination,
rename
Latex:
\mforall{}[x,y:partial(\mBbbN{})]. uiff((x + y)\mdownarrow{};(x)\mdownarrow{} \mwedge{} (y)\mdownarrow{})
Date html generated:
2019_06_20-PM-00_34_46
Last ObjectModification:
2019_02_21-PM-06_02_10
Theory : partial_1
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