Nuprl Lemma : mk_deq-subtype
∀[T,S:Type]. ∀[p:∀x,y:S. Dec(x = y ∈ S)]. (mk_deq(p) ∈ EqDecider(T)) supposing strong-subtype(T;S)
Proof
Definitions occuring in Statement :
mk_deq: mk_deq(p),
deq: EqDecider(T),
strong-subtype: strong-subtype(A;B),
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
deq: EqDecider(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
strong-subtype-implies,
mk_deq_wf,
subtype_rel_sets,
bool_wf,
all_wf,
iff_wf,
equal_wf,
assert_wf,
subtype_rel_set,
subtype_rel_dep_function,
subtype_rel_self,
decidable_wf,
strong-subtype_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
functionEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
because_Cache,
productElimination,
lambdaFormation,
independent_pairFormation,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination
Latex:
\mforall{}[T,S:Type]. \mforall{}[p:\mforall{}x,y:S. Dec(x = y)]. (mk\_deq(p) \mmember{} EqDecider(T)) supposing strong-subtype(T;S)
Date html generated:
2016_05_14-AM-06_06_34
Last ObjectModification:
2015_12_26-AM-11_46_51
Theory : equality!deciders
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