Nuprl Lemma : q-linear-equal
∀[k:ℕ]. ∀[X:ℕ ⟶ ℚ]. ∀[y,z:ℚ List].
(q-linear(k;j.X[j];y) = q-linear(k;j.X[j];z) ∈ ℚ) supposing
((∀i:ℕk. (y[i] = z[i] ∈ ℚ)) and
(k ≤ ||z||) and
(k ≤ ||y||))
Proof
Definitions occuring in Statement :
q-linear: q-linear(k;i.X[i];y),
rationals: ℚ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
so_apply: x[s],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
le_wf,
length_wf,
rationals_wf,
list_wf,
nat_wf,
subtract-1-ge-0,
select_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-false,
equal_wf,
squash_wf,
true_wf,
istype-universe,
q-linear-base,
subtype_rel_self,
iff_weakening_equal,
qmul_wf,
lelt_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
qadd_wf,
satisfiable-full-omega-tt,
all_wf,
q-linear-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
functionIsTypeImplies,
functionIsType,
productElimination,
equalityIsType1,
because_Cache,
unionElimination,
applyEquality,
dependent_set_memberEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
lambdaFormation,
functionExtensionality,
dependent_set_memberEquality,
functionEquality,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
dependent_pairFormation,
lambdaEquality,
isect_memberFormation
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[X:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[y,z:\mBbbQ{} List].
(q-linear(k;j.X[j];y) = q-linear(k;j.X[j];z)) supposing
((\mforall{}i:\mBbbN{}k. (y[i] = z[i])) and
(k \mleq{} ||z||) and
(k \mleq{} ||y||))
Date html generated:
2019_10_16-PM-00_33_23
Last ObjectModification:
2018_10_10-AM-11_05_23
Theory : rationals
Home
Index