Nuprl Lemma : pcw-pp-tail_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. ∀[pp:PartialPath].
pcw-pp-tail(pp) ∈ PartialPath supposing ¬↑pcw-pp-null(pp)
Proof
Definitions occuring in Statement :
pcw-pp-tail: pcw-pp-tail(pp),
pcw-pp-null: pcw-pp-null(pp),
pcw-pp: PartialPath,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
not: ¬A,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
pcw-pp: PartialPath,
pcw-pp-null: pcw-pp-null(pp),
not: ¬A,
implies: P ⇒ Q,
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
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],
pcw-pp-tail: pcw-pp-tail(pp),
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
false: False,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b
Lemmas referenced :
assert_of_le_int,
le_wf,
not_wf,
assert_wf,
pcw-pp-null_wf,
pcw-pp_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-zero,
add-zero,
add-commutes,
zero-add,
minus-one-mul,
minus-one-mul-top,
nat_wf,
minus-minus,
add-associates,
add-swap,
add_functionality_wrt_le,
le-add-cancel,
add-member-int_seg2,
le-add-cancel2,
lelt_wf,
int_seg_wf,
pcw-step_wf,
all_wf,
pcw-steprel_wf,
decidable__lt,
not-lt-2,
less-iff-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
hypothesis,
lambdaFormation,
independent_functionElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
promote_hyp,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality,
dependent_pairEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
addEquality,
voidEquality,
intEquality,
minusEquality
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].
\mforall{}[pp:PartialPath].
pcw-pp-tail(pp) \mmember{} PartialPath supposing \mneg{}\muparrow{}pcw-pp-null(pp)
Date html generated:
2016_05_14-AM-06_13_07
Last ObjectModification:
2015_12_26-PM-00_05_55
Theory : co-recursion
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