Nuprl Lemma : immediate-rc-face-implies
∀k:ℕ. ∀f,c:ℚCube(k).
(0 < dim(c)
⇒ immediate-rc-face(k;f;c)
⇒ (∃i:ℕk
((dim(c i) = 1 ∈ ℤ)
∧ (∀j:ℕk. ((¬(j = i ∈ ℤ)) ⇒ ((f j) = (c j) ∈ ℚInterval)))
∧ (((f i) = [fst((c i))] ∈ ℚInterval) ∨ ((f i) = [snd((c i))] ∈ ℚInterval)))))
Proof
Definitions occuring in Statement :
immediate-rc-face: immediate-rc-face(k;f;c),
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
rat-interval-dimension: dim(I),
rat-point-interval: [a],
rational-interval: ℚInterval,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rat-cube-dimension: dim(c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
immediate-rc-face: immediate-rc-face(k;f;c),
bfalse: ff,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
rational-interval: ℚInterval,
rat-interval-dimension: dim(I),
rat-interval-face: I ≤ J,
pi1: fst(t),
pi2: snd(t),
rat-point-interval: [a],
inhabited-rat-interval: Inhabited(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
not: ¬A,
rev_implies: P ⇐ Q,
true: True,
nat: ℕ,
decidable: Dec(P),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
inhabited-rat-cube_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert-inhabited-rat-cube,
eqff_to_assert,
assert_of_bnot,
immediate-rc-face_wf,
istype-less_than,
rat-cube-dimension_wf,
rational-cube_wf,
istype-nat,
pi2_wf,
rationals_wf,
pi1_wf_top,
istype-void,
equal_wf,
rational-interval_wf,
q_less_wf,
assert-q_less-eq,
iff_weakening_equal,
qle_wf,
bool_cases_sqequal,
assert-bnot,
qless_wf,
qless_complement_qorder,
qle_antisymmetry,
int_subtype_base,
or_wf,
equal-wf-base,
subtype_rel_self,
assert-q_le-eq,
istype-assert,
q_le_wf,
int_seg_wf,
decidable__exists_int_seg,
rat-interval-dimension_wf,
set_subtype_base,
lelt_wf,
rat-point-interval_wf,
decidable__cand,
decidable__equal_int,
decidable__or,
decidable__equal_rational-interval,
subtype_rel_product,
top_wf,
subtract_wf,
nat_properties,
full-omega-unsat,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_properties,
subtract-is-int-iff,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
false_wf,
rat-interval-dimension-single,
squash_wf,
true_wf,
istype-universe,
iff_imp_equal_bool,
btrue_wf,
istype-true,
decidable__lt,
sum_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
istype-le,
isolate_summand2,
int_seg_subtype_nat,
istype-false,
ifthenelse_wf,
eq_int_wf,
less_than_wf,
add_functionality_wrt_eq,
assert_of_eq_int,
neg_assert_of_eq_int,
itermAdd_wf,
int_term_value_add_lemma,
sum_le,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
sqequalRule,
imageElimination,
voidElimination,
universeIsType,
natural_numberEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
applyLambdaEquality,
independent_pairEquality,
isect_memberEquality_alt,
promote_hyp,
hyp_replacement,
equalityElimination,
inrFormation_alt,
independent_pairFormation,
inlFormation_alt,
equalityIstype,
dependent_pairFormation_alt,
productIsType,
baseClosed,
sqequalBase,
intEquality,
functionEquality,
productEquality,
unionIsType,
unionEquality,
functionExtensionality,
minusEquality,
addEquality,
approximateComputation,
int_eqEquality,
functionIsType,
pointwiseFunctionality,
baseApply,
closedConclusion,
imageMemberEquality,
universeEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}k:\mBbbN{}. \mforall{}f,c:\mBbbQ{}Cube(k).
(0 < dim(c)
{}\mRightarrow{} immediate-rc-face(k;f;c)
{}\mRightarrow{} (\mexists{}i:\mBbbN{}k
((dim(c i) = 1)
\mwedge{} (\mforall{}j:\mBbbN{}k. ((\mneg{}(j = i)) {}\mRightarrow{} ((f j) = (c j))))
\mwedge{} (((f i) = [fst((c i))]) \mvee{} ((f i) = [snd((c i))])))))
Date html generated:
2020_05_20-AM-09_20_06
Last ObjectModification:
2019_11_01-PM-02_51_10
Theory : rationals
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