Nuprl Lemma : s-comp-nc-1'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. (s ⋅ (j1) = s ∈ I+i ⟶ I)
Proof
Definitions occuring in Statement :
nc-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nc-s: s,
nc-1: (i1),
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),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
all: ∀x:A. B[x],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
names: names(I),
uiff: uiff(P;Q),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
dM_wf,
add-name_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,
dM-lift-inc,
nc-1_wf,
dM_inc_wf,
names-subtype,
f-subset-add-name,
iff_weakening_equal,
names_wf,
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
fset-member-add-name,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
because_Cache,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
intEquality,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
inrFormation,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I\} ]. (s \mcdot{} (j1) = s)
Date html generated:
2017_10_05-AM-01_02_52
Last ObjectModification:
2017_07_28-AM-09_26_28
Theory : cubical!type!theory
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