Nuprl Lemma : face-of-face-pairity
∀k:ℕ. ∀a,b,c:ℚCube(k).
(1 < dim(c)
⇒ immediate-rc-face(k;a;b)
⇒ immediate-rc-face(k;b;c)
⇒ (∃!b':ℚCube(k). (immediate-rc-face(k;a;b') ∧ immediate-rc-face(k;b';c) ∧ (¬(b' = b ∈ ℚCube(k))))))
Proof
Definitions occuring in Statement :
immediate-rc-face: immediate-rc-face(k;f;c),
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
nat: ℕ,
less_than: a < b,
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
immediate-rc-face: immediate-rc-face(k;f;c),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
subtype_rel: A ⊆r B,
rat-cube-dimension: dim(c),
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
less_than': less_than'(a;b),
rational-cube: ℚCube(k),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
rat-interval-dimension: dim(I),
rat-point-interval: [a],
rational-interval: ℚInterval,
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
nequal: a ≠ b ∈ T ,
subtract: n - m,
exists!: ∃!x:T. P[x],
cand: A c∧ B,
rat-cube-face: c ≤ d,
label: ...$L... t,
top: Top,
rat-interval-face: I ≤ J
Lemmas referenced :
immediate-rc-face-implies,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermSubtract_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
rat-cube-dimension_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
decidable__equal_int,
inhabited-rat-cube_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
immediate-rc-face_wf,
istype-less_than,
rational-cube_wf,
istype-nat,
ifthenelse_wf,
eq_int_wf,
rational-interval_wf,
assert-inhabited-rat-cube,
assert_of_eq_int,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
int_subtype_base,
set_subtype_base,
lelt_wf,
q_less_wf,
assert-q_less-eq,
iff_weakening_equal,
qless_irreflexivity,
qless_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rat-interval-dimension_wf,
int_seg_wf,
subtype_rel_self,
isolate_summand2,
int_seg_subtype_nat,
istype-false,
sum_wf,
subtract_wf,
add_functionality_wrt_eq,
add-associates,
zero-add,
add-commutes,
add-swap,
istype-void,
rat-interval-face_wf,
rat-interval-face-self,
rat-point-interval_wf,
pi1_wf_top,
rationals_wf,
subtype_rel_product,
top_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
uiff_transitivity2,
iff_transitivity,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
isectElimination,
hypothesis,
setElimination,
rename,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
imageElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
dependent_set_memberEquality_alt,
applyEquality,
inhabitedIsType,
instantiate,
cumulativity,
minusEquality,
equalityElimination,
equalityIstype,
promote_hyp,
intEquality,
sqequalBase,
universeEquality,
imageMemberEquality,
baseClosed,
addEquality,
functionIsType,
productIsType,
isect_memberEquality_alt,
inlFormation_alt,
unionIsType,
inrFormation_alt,
applyLambdaEquality,
functionExtensionality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,c:\mBbbQ{}Cube(k).
(1 < dim(c)
{}\mRightarrow{} immediate-rc-face(k;a;b)
{}\mRightarrow{} immediate-rc-face(k;b;c)
{}\mRightarrow{} (\mexists{}!b':\mBbbQ{}Cube(k). (immediate-rc-face(k;a;b') \mwedge{} immediate-rc-face(k;b';c) \mwedge{} (\mneg{}(b' = b)))))
Date html generated:
2020_05_20-AM-09_23_01
Last ObjectModification:
2020_01_04-PM-10_19_09
Theory : rationals
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