Nuprl Lemma : subtype-deq
∀[A,B:Type]. (EqDecider(B) ⊆r EqDecider(A)) supposing ((∀x,y:A. ((x = y ∈ B) ⇒ (x = y ∈ A))) and (A ⊆r B))
Proof
Definitions occuring in Statement :
deq: EqDecider(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
deq: EqDecider(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
equal_wf,
assert_wf,
all_wf,
iff_wf,
deq_wf,
subtype_rel_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesisEquality,
applyEquality,
extract_by_obid,
isectElimination,
sqequalRule,
functionEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
because_Cache,
independent_pairFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination
Latex:
\mforall{}[A,B:Type].
(EqDecider(B) \msubseteq{}r EqDecider(A)) supposing ((\mforall{}x,y:A. ((x = y) {}\mRightarrow{} (x = y))) and (A \msubseteq{}r B))
Date html generated:
2019_06_20-PM-00_32_04
Last ObjectModification:
2018_09_17-PM-05_40_27
Theory : equality!deciders
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