Nuprl Lemma : pcw-path-rel_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. ∀[f,g:Path].
(pcw-path-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];f;g) ∈ ℙ)
Proof
Definitions occuring in Statement :
pcw-path-rel: pcw-path-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];f;g),
pcw-path: Path,
uall: ∀[x:A]. B[x],
prop: ℙ,
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-path-rel: pcw-path-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];f;g),
prop: ℙ,
and: P ∧ Q,
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],
nat: ℕ,
pcw-path: Path,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Lemmas referenced :
pcw-consistent-paths_wf,
exists_wf,
nat_wf,
all_wf,
int_seg_wf,
not_wf,
pcw-final-step_wf,
int_seg_subtype_nat,
false_wf,
pcw-path_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
productEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
hypothesis,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
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]. \mforall{}[f,g:Path].
(pcw-path-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];f;g) \mmember{} \mBbbP{})
Date html generated:
2016_05_14-AM-06_12_51
Last ObjectModification:
2015_12_26-PM-00_05_56
Theory : co-recursion
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