Nuprl Lemma : assert-equal-upto-finite-nat-seq
∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℕ]. (↑equal-upto-finite-nat-seq(n;f;g) ⇐⇒ f = g ∈ (ℕn ⟶ ℕ))
Proof
Definitions occuring in Statement :
equal-upto-finite-nat-seq: equal-upto-finite-nat-seq(n;f;g),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
equal-upto-finite-nat-seq: equal-upto-finite-nat-seq(n;f;g),
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,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
bfalse: ff,
band: p ∧b q,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
cand: A c∧ B,
squash: ↓T
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,
primrec0_lemma,
int_seg_wf,
int_seg_properties,
primrec_wf,
bool_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
btrue_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
eq_int_wf,
bfalse_wf,
subtract-1-ge-0,
istype-nat,
equal-upto-finite-nat-seq_wf,
true_wf,
primrec-unroll-1,
assert_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
istype-assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_eq_int,
subtype_rel_function,
nat_wf,
int_seg_subtype,
istype-false,
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,
subtype_rel_self,
equal_functionality_wrt_subtype_rel2,
equal_wf,
iff_weakening_equal,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
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 :functionIsType,
Error :dependent_set_memberEquality_alt,
unionElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
cumulativity,
Error :isect_memberFormation_alt,
Error :isectIsTypeImplies,
Error :equalityIstype,
Error :functionExtensionality_alt,
Error :productIsType,
productEquality,
intEquality,
sqequalBase,
promote_hyp,
addEquality,
minusEquality,
multiplyEquality,
functionEquality,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. (\muparrow{}equal-upto-finite-nat-seq(n;f;g) \mLeftarrow{}{}\mRightarrow{} f = g)
Date html generated:
2019_06_20-PM-03_03_18
Last ObjectModification:
2019_01_02-PM-00_36_10
Theory : continuity
Home
Index