Nuprl Lemma : nc-e'-p
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[z:Point(dM(I))]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ J} ].
((i/z) ⋅ f = f,i=j ⋅ (j/dM-lift(J;I;f) z) ∈ J ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-p: (i/z),
add-name: I+i,
nh-comp: g ⋅ f,
dM-lift: dM-lift(I;J;f),
names-hom: I ⟶ J,
dM: dM(I),
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
top: Top,
names-hom: I ⟶ J,
compose: f o g,
nc-p: (i/z),
nc-e': g,i=j,
names: names(I),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
squash: ↓T,
true: True,
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,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A
Lemmas referenced :
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
names-hom_wf,
names_wf,
add-name_wf,
nh-comp-sq,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
dM-lift_wf,
dM-lift-inc,
nc-p_wf,
trivial-member-add-name1,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
dma-hom_wf,
all_wf,
dM_inc_wf,
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,
dM-lift-is-id,
f-subset-add-name,
f-subset_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
instantiate,
productEquality,
cumulativity,
universeEquality,
voidElimination,
voidEquality,
functionExtensionality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
imageElimination,
dependent_functionElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
setEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[z:Point(dM(I))]. \mforall{}[i:\mBbbN{}]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ].
((i/z) \mcdot{} f = f,i=j \mcdot{} (j/dM-lift(J;I;f) z))
Date html generated:
2017_10_05-AM-01_06_48
Last ObjectModification:
2017_07_28-AM-09_28_03
Theory : cubical!type!theory
Home
Index