Nuprl Lemma : product-Leibniz-type
∀A:Type. ∀B:A ⟶ Type. ((∀x,y:A. Dec(x = y ∈ A)) ⇒ (∀a:A. Leibniz-type{i:l}(B[a])) ⇒ Leibniz-type{i:l}(a:A × B[a]))
Proof
Definitions occuring in Statement :
Leibniz-type: Leibniz-type{i:l}(T),
decidable: Dec(P),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
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],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
pi1: fst(t),
top: Top,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
pi2: snd(t),
decidable: Dec(P),
guard: {T}
Lemmas referenced :
Leibniz-type_wf,
decidable_wf,
equal_wf,
istype-universe,
subtype_rel_self,
istype-void,
pi1_wf_top,
pi2_wf,
subtype_rel-equal,
subtype_rel_product,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
hypothesis,
promote_hyp,
thin,
productElimination,
functionIsType,
universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
because_Cache,
inhabitedIsType,
instantiate,
universeEquality,
productIsType,
unionIsType,
equalityIstype,
rename,
dependent_pairFormation_alt,
lambdaEquality_alt,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
setElimination,
unionElimination,
inrFormation_alt,
inlFormation_alt,
hyp_replacement,
dependent_pairEquality_alt
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type.
((\mforall{}x,y:A. Dec(x = y)) {}\mRightarrow{} (\mforall{}a:A. Leibniz-type\{i:l\}(B[a])) {}\mRightarrow{} Leibniz-type\{i:l\}(a:A \mtimes{} B[a]))
Date html generated:
2019_10_31-AM-07_26_05
Last ObjectModification:
2019_09_19-PM-06_44_54
Theory : constructive!algebra
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