Nuprl Lemma : pcw-pp_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. (PartialPath ∈ Type)
Proof
Definitions occuring in Statement :
pcw-pp: PartialPath,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pcw-pp: PartialPath,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
nat_wf,
int_seg_wf,
pcw-step_wf,
all_wf,
subtract_wf,
pcw-steprel_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
lelt_wf,
add-member-int_seg2,
decidable__le,
not-le-2,
zero-add,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
setEquality,
productEquality,
lemma_by_obid,
hypothesis,
functionEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
cumulativity,
lambdaEquality,
applyEquality,
because_Cache,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P].
(PartialPath \mmember{} Type)
Date html generated:
2016_05_14-AM-06_12_56
Last ObjectModification:
2015_12_26-PM-00_05_52
Theory : co-recursion
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