Nuprl Lemma : decidable-equality-implies-Leibniz-type
∀T:Type. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ Leibniz-type{i:l}(T))
Proof
Definitions occuring in Statement :
Leibniz-type: Leibniz-type{i:l}(T),
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Leibniz-type: Leibniz-type{i:l}(T),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
not_wf,
equal_wf,
istype-void,
subtype_rel_self,
decidable_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_pairFormation_alt,
lambdaEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
universeIsType,
sqequalRule,
dependent_functionElimination,
unionElimination,
inrFormation_alt,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
voidElimination,
equalityIstype,
functionIsType,
inlFormation_alt,
because_Cache,
independent_pairFormation,
productIsType,
applyEquality,
instantiate,
unionIsType,
universeEquality
Latex:
\mforall{}T:Type. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} Leibniz-type\{i:l\}(T))
Date html generated:
2019_10_31-AM-07_25_49
Last ObjectModification:
2019_09_19-PM-04_39_28
Theory : constructive!algebra
Home
Index