Nuprl Lemma : union-Leibniz-type
∀A,B:Type. (Leibniz-type{i:l}(A) ⇒ Leibniz-type{i:l}(B) ⇒ Leibniz-type{i:l}(A + B))
Proof
Definitions occuring in Statement :
Leibniz-type: Leibniz-type{i:l}(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
Leibniz-type: Leibniz-type{i:l}(T),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
or: P ∨ Q,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
isl: isl(x),
prop: ℙ,
uall: ∀[x:A]. B[x],
guard: {T}
Lemmas referenced :
true_wf,
istype-true,
istype-void,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
subtype_rel_self,
Leibniz-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
lambdaEquality_alt,
cut,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
inhabitedIsType,
unionElimination,
sqequalRule,
applyEquality,
introduction,
extract_by_obid,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
unionIsType,
universeIsType,
independent_pairFormation,
inlFormation_alt,
natural_numberEquality,
inrFormation_alt,
because_Cache,
applyLambdaEquality,
promote_hyp,
voidElimination,
inlEquality_alt,
functionIsType,
dependent_set_memberEquality_alt,
productIsType,
setElimination,
rename,
inrEquality_alt,
instantiate,
isectElimination,
cumulativity,
universeEquality
Latex:
\mforall{}A,B:Type. (Leibniz-type\{i:l\}(A) {}\mRightarrow{} Leibniz-type\{i:l\}(B) {}\mRightarrow{} Leibniz-type\{i:l\}(A + B))
Date html generated:
2019_10_31-AM-07_25_59
Last ObjectModification:
2019_09_19-PM-06_50_43
Theory : constructive!algebra
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