Nuprl Lemma : nc-e-comp
∀I:fset(ℕ). ∀i,j,k:ℕ. ((¬j ∈ I) ⇒ (e(k;j) ⋅ e(j;i) = e(k;i) ∈ I+i ⟶ I+k))
Proof
Definitions occuring in Statement :
nc-e: e(i;j),
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
nat-deq: NatDeq,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
names-hom: I ⟶ J,
nc-e: e(i;j),
nh-comp: g ⋅ f,
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),
member: t ∈ T,
uall: ∀[x:A]. B[x],
names: names(I),
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
sq_stable: SqStable(P),
nat-deq: NatDeq,
int-deq: IntDeq
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
lattice-point_wf,
dM_wf,
add-name_wf,
dM-lift-inc,
nc-e_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,
dM_inc_wf,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
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,
not-added-name,
names-subtype,
f-subset-add-name,
names_wf,
not_wf,
nat-deq_wf,
fset_wf,
int_subtype_base,
sq_stable__fset-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
intEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}i,j,k:\mBbbN{}. ((\mneg{}j \mmember{} I) {}\mRightarrow{} (e(k;j) \mcdot{} e(j;i) = e(k;i)))
Date html generated:
2017_10_05-AM-01_03_50
Last ObjectModification:
2017_07_28-AM-09_26_50
Theory : cubical!type!theory
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