Nuprl Lemma : trivial-same-context-set
∀X,Y:?CubicalContext.
((Y = bind-provision(X;ctxt.OK(ctxt)) ∈ ?CubicalContext)
⇒ context-ok(X)
⇒ {(X = Y ∈ ?CubicalContext) ∧ (context-set(X) = context-set(Y) ∈ CubicalSet''')})
Proof
Definitions occuring in Statement :
context-set: context-set(ctxt),
context-ok: context-ok(ctxt),
cubical-context: ?CubicalContext,
cubical_set: CubicalSet,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
equal: s = t ∈ T,
bind-provision: bind-provision(x;t.f[t]),
provision: provision(ok; v)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-context: ?CubicalContext,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
trivial-same-cubical-context,
cubical-context_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical_set_wf,
context-set_wf,
context-ok_wf,
subtype_rel_self,
iff_weakening_equal,
bind-provision_wf,
cubical_context_wf,
provision_wf,
istype-true,
allowed_wf,
allow_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
inhabitedIsType,
universeIsType,
independent_pairFormation,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
because_Cache,
natural_numberEquality,
productElimination,
equalityIstype,
cumulativity,
isect_memberEquality_alt,
setElimination,
rename,
setIsType,
productIsType
Latex:
\mforall{}X,Y:?CubicalContext.
((Y = bind-provision(X;ctxt.OK(ctxt)))
{}\mRightarrow{} context-ok(X)
{}\mRightarrow{} \{(X = Y) \mwedge{} (context-set(X) = context-set(Y))\})
Date html generated:
2020_05_20-PM-08_07_19
Last ObjectModification:
2020_05_18-PM-03_02_49
Theory : cubical!type!theory
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