Nuprl Lemma : implies-member-rat-cube-faces
∀k:ℕ. ∀c:ℚCube(k).
∀f:{f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)} . (f ∈ rat-cube-faces(k;c)) supposing ↑Inhabited(c)
Proof
Definitions occuring in Statement :
rat-cube-faces: rat-cube-faces(k;c),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
pi2: snd(t),
upper-rc-face: upper-rc-face(c;j),
nequal: a ≠ b ∈ T ,
pi1: fst(t),
lower-rc-face: lower-rc-face(c;j),
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
bnot: ¬bb,
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
eq_int: (i =z j),
subtract: n - m,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rat-cube-dimension: dim(c),
rat-cube-face: c ≤ d,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
rational-cube: ℚCube(k),
nat: ℕ,
rat-cube-faces: rat-cube-faces(k;c),
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
member_upto2,
rational-interval_wf,
istype-universe,
equal_wf,
rat-point-interval_wf,
member_singleton,
cons_member,
le_wf,
int_term_value_add_lemma,
itermAdd_wf,
sum_le,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
add_functionality_wrt_eq,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
less_than_wf,
ifthenelse_wf,
sum_wf,
istype-false,
int_seg_subtype_nat,
Error :isolate_summand2,
false_wf,
subtract-is-int-iff,
decidable__not,
decidable__cand,
not_wf,
int_seg_cases,
int_seg_subtype_special,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
itermSubtract_wf,
intformeq_wf,
decidable__equal_int,
equal-wf-base,
decidable__exists_int_seg,
rat-interval-face-dimension,
assert-inhabited-rat-cube,
istype-true,
btrue_wf,
iff_imp_equal_bool,
bool_subtype_base,
bool_wf,
subtype_base_sq,
inhabited-rat-cube-face,
istype-nat,
subtract_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
l_member_wf,
member-mapfilter,
istype-assert,
nil_wf,
upper-rc-face_wf,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
lower-rc-face_wf,
cons_wf,
rational-cube_wf,
list_wf,
rat-interval-dimension_wf,
eq_int_wf,
upto_wf,
int_seg_wf,
mapfilter_wf,
member-concat,
decidable__rat-cube-face,
rat-cube-face_wf,
sq_stable_from_decidable,
inhabited-rat-cube_wf,
assert_witness
Rules used in proof :
functionExtensionality,
unionIsType,
inrFormation_alt,
inlFormation_alt,
functionIsType,
hyp_replacement,
equalityElimination,
universeEquality,
closedConclusion,
baseApply,
pointwiseFunctionality,
productEquality,
hypothesis_subsumption,
applyLambdaEquality,
cumulativity,
instantiate,
sqequalBase,
addEquality,
minusEquality,
intEquality,
equalityIstype,
promote_hyp,
setIsType,
productIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_set_memberEquality_alt,
universeIsType,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
natural_numberEquality,
because_Cache,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
productElimination,
dependent_functionElimination,
setElimination,
rename,
independent_functionElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k).
\mforall{}f:\{f:\mBbbQ{}Cube(k)| f \mleq{} c \mwedge{} (dim(f) = (dim(c) - 1))\} . (f \mmember{} rat-cube-faces(k;c))
supposing \muparrow{}Inhabited(c)
Date html generated:
2019_10_29-AM-07_57_27
Last ObjectModification:
2019_10_18-PM-06_30_13
Theory : rationals
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