Nuprl Lemma : shadow_inequalities_wf
∀[n:{2...}]. ∀[ineqs:{L:ℤ List| ||L|| = n ∈ ℤ} List]. (shadow_inequalities(ineqs) ∈ {L:ℤ List| ||L|| = (n - 1) ∈ ℤ} L\000Cist)
Proof
Definitions occuring in Statement :
shadow_inequalities: shadow_inequalities(ineqs),
length: ||as||,
list: T List,
int_upper: {i...},
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,
shadow_inequalities: shadow_inequalities(ineqs),
so_lambda: λ2x.t[x],
int_upper: {i...},
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
or: P ∨ Q,
nil: [],
it: ⋅,
cons: [a / b],
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
listp: A List+,
guard: {T},
subtract: n - m,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
set_wf,
list_wf,
equal_wf,
length_wf,
list-cases,
nil_wf,
equal-wf-base-T,
subtract_wf,
product_subtype_list,
value-type-has-value,
int-value-type,
index-of-min_wf,
max_tl_coeffs_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
cons-listp,
shadow-inequalities_wf,
upper_subtype_nat,
istype-void,
cons_wf,
list_subtype_base,
int_subtype_base,
int_upper_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
subtype_rel_sets,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-member-int_seg2,
decidable__le,
not-le-2,
add-zero,
le-add-cancel2,
less_than_transitivity1,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesis,
Error :lambdaEquality_alt,
hypothesisEquality,
setElimination,
rename,
Error :universeIsType,
dependent_functionElimination,
unionElimination,
setEquality,
because_Cache,
natural_numberEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
callbyvalueReduce,
independent_isectElimination,
addEquality,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
Error :lambdaFormation_alt,
voidElimination,
independent_functionElimination,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
Error :equalityIsType1,
minusEquality,
Error :equalityIsType4,
Error :productIsType
Latex:
\mforall{}[n:\{2...\}]. \mforall{}[ineqs:\{L:\mBbbZ{} List| ||L|| = n\} List].
(shadow\_inequalities(ineqs) \mmember{} \{L:\mBbbZ{} List| ||L|| = (n - 1)\} List)
Date html generated:
2019_06_20-PM-00_50_34
Last ObjectModification:
2018_10_07-PM-08_45_57
Theory : omega
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