Nuprl Lemma : nh-comp-nc-m-eq
∀[I,K:fset(ℕ)]. ∀[i,j:ℕ]. ∀[f:K ⟶ I+i+j]. (i0) ⋅ s ⋅ s ⋅ f = m(i;j) ⋅ f ∈ K ⟶ I+i supposing (f i) = 0 ∈ Point(dM(K))
Proof
Definitions occuring in Statement :
nc-m: m(i;j),
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,
names: names(I),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
names-hom: I ⟶ J,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
dM0: 0,
top: Top,
compose: f o g,
nc-0: (i0),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
nc-s: s,
nc-m: m(i;j)
Lemmas referenced :
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
add-name_wf,
trivial-member-add-name2,
names_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,
dM0_wf,
names-hom_wf,
fset_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
nh-comp_wf,
nc-s_wf,
f-subset-add-name,
nc-0_wf,
nh-comp-nc-m,
iff_weakening_equal,
lattice-meet-0,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
dM0-sq-empty,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nh-comp-sq,
dM-lift_wf,
dma-hom_wf,
all_wf,
dM_inc_wf,
names-subtype,
dM-lift-0,
nh-comp-cancel,
nc-m_wf,
dM-lift-inc,
not-added-name
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
dependent_set_memberEquality,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
isectElimination,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
functionExtensionality,
instantiate,
productEquality,
cumulativity,
universeEquality,
setElimination,
rename,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
equalityElimination,
dependent_pairFormation,
promote_hyp,
int_eqEquality,
computeAll,
setEquality
Latex:
\mforall{}[I,K:fset(\mBbbN{})]. \mforall{}[i,j:\mBbbN{}]. \mforall{}[f:K {}\mrightarrow{} I+i+j]. (i0) \mcdot{} s \mcdot{} s \mcdot{} f = m(i;j) \mcdot{} f supposing (f i) = 0
Date html generated:
2017_10_05-AM-01_03_21
Last ObjectModification:
2017_07_28-AM-09_26_38
Theory : cubical!type!theory
Home
Index