Nuprl Lemma : shadow-inequalities_wf
∀[n:ℕ]. ∀[ineqs:{L:ℤ List| ||L|| = n ∈ ℤ} List]. ∀[i:ℕn].
(shadow-inequalities(i;ineqs) ∈ {L:ℤ List| ||L|| = (n - 1) ∈ ℤ} List)
Proof
Definitions occuring in Statement :
shadow-inequalities: shadow-inequalities(i;ineqs),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
shadow-inequalities: shadow-inequalities(i;ineqs),
has-value: (a)↓
Lemmas referenced :
int_seg_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
set_subtype_base,
le_wf,
istype-int,
istype-nat,
l_member_wf,
subtype_rel_list,
equal-wf-base-T,
subtype_base_sq,
select_wf,
sq_stable__le,
set_wf,
equal_wf,
less_than_transitivity1,
length_wf,
le_weakening,
list-set-type,
map_wf,
squash_wf,
true_wf,
istype-universe,
length-list-delete,
int_seg_subtype_nat,
istype-false,
subtract_wf,
subtype_rel_self,
iff_weakening_equal,
list-delete_wf,
filter_wf5,
eq_int_wf,
evalall-reduce,
list-valueall-type,
set-valueall-type,
int-valueall-type,
list-value-type,
value-type-has-value,
eager-append_wf,
set-value-type,
eager-product-map_wf,
istype-less_than,
equal_functionality_wrt_subtype_rel2,
length-shadow-vec,
istype-le,
shadow-vec_wf,
lt_int_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
setEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
lambdaFormation,
lambdaEquality,
productElimination,
instantiate,
cumulativity,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
imageElimination,
lambdaFormation_alt,
equalityIstype,
setIsType,
sqequalBase,
universeEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
callbyvalueReduce,
equalityIsType4,
productIsType,
dependent_pairFormation_alt
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[ineqs:\{L:\mBbbZ{} List| ||L|| = n\} List]. \mforall{}[i:\mBbbN{}n].
(shadow-inequalities(i;ineqs) \mmember{} \{L:\mBbbZ{} List| ||L|| = (n - 1)\} List)
Date html generated:
2020_05_19-PM-09_39_17
Last ObjectModification:
2020_01_04-PM-07_59_08
Theory : omega
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