Nuprl Lemma : eq-finite-seqs-iff-eq-upto
∀a,b:ℕ ⟶ ℕ. ∀x:ℕ. (↑eq-finite-seqs(a;b;x) ⇐⇒ a = b ∈ (ℕx ⟶ ℕ))
Proof
Definitions occuring in Statement :
eq-finite-seqs: eq-finite-seqs(a;b;x),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
eq-finite-seqs: eq-finite-seqs(a;b;x),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
band: p ∧b q,
subtract: n - m
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,
istype-less_than,
assert_witness,
istype-assert,
eq-finite-seqs_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
int_seg_wf,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
subtract-1-ge-0,
istype-nat,
int_seg_properties,
primrec0_lemma,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
le_wf,
int_subtype_base,
bool_cases,
band_wf,
btrue_wf,
eq_int_wf,
bfalse_wf,
equal-wf-base,
iff_transitivity,
assert_of_band,
assert_of_eq_int,
decidable__equal_nat,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
int_seg_subtype,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
productElimination,
independent_pairEquality,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
unionElimination,
Error :equalityIstype,
Error :functionIsType,
because_Cache,
applyEquality,
Error :functionExtensionality_alt,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
Error :productIsType,
intEquality,
sqequalBase,
productEquality,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
addEquality,
minusEquality,
multiplyEquality,
functionEquality
Latex:
\mforall{}a,b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}x:\mBbbN{}. (\muparrow{}eq-finite-seqs(a;b;x) \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2019_06_20-PM-03_07_31
Last ObjectModification:
2018_12_19-PM-05_36_41
Theory : continuity
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