Nuprl Lemma : nc-0-comp-s
∀[I,K:fset(ℕ)]. ∀[i:ℕ]. ∀[f:K ⟶ I+i]. (i0) ⋅ s ⋅ f = f ∈ K ⟶ I+i supposing (f i) = 0 ∈ Point(dM(K))
Proof
Definitions occuring in Statement :
nc-0: (i0),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dM0: 0,
dM: dM(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nc-0: (i0),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
prop: ℙ,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
names: names(I),
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
top: Top,
sq_type: SQType(T),
squash: ↓T,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
nequal: a ≠ b ∈ T ,
nc-s: s
Lemmas referenced :
names_wf,
add-name_wf,
equal_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
dM0_wf,
names-hom_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
dM0-sq-empty,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
dM-lift_wf,
dma-hom_wf,
all_wf,
dM_inc_wf,
dM-lift-0,
nc-s_wf,
f-subset-add-name,
iff_weakening_equal,
dM-lift-0-sq,
not-added-name,
dM-lift-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
intEquality,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
baseClosed,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
voidElimination,
voidEquality,
imageElimination,
setEquality,
imageMemberEquality
Latex:
\mforall{}[I,K:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[f:K {}\mrightarrow{} I+i]. (i0) \mcdot{} s \mcdot{} f = f supposing (f i) = 0
Date html generated:
2017_10_05-AM-01_02_43
Last ObjectModification:
2017_07_28-AM-09_26_23
Theory : cubical!type!theory
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