Nuprl Lemma : weak-continuity-principle-nat+-int-nat
∀F:(ℕ+ ⟶ ℤ) ⟶ ℕ. ∀f:ℕ+ ⟶ ℤ. ∀G:n:ℕ+ ⟶ {g:ℕ+ ⟶ ℤ| f = g ∈ (ℕ+n ⟶ ℤ)} . ∃n:ℕ+. ((F f) = (F (G n)) ∈ ℕ)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
squash: ↓T,
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
nat: ℕ,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
mu-property,
int_term_value_add_lemma,
itermAdd_wf,
add_nat_plus,
assert_wf,
assert_of_eq_int,
eq_int_wf,
mu_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
subtract-add-cancel,
lelt_wf,
add-subtract-cancel,
add-member-int_seg2,
set_wf,
nat_properties,
add-swap,
int_seg_subtype_nat,
all_wf,
exists_wf,
squash-from-quotient,
subtype_rel_self,
int_seg_subtype_nat_plus,
subtype_rel_dep_function,
int_seg_wf,
equal_wf,
nat_plus_wf,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
false_wf,
decidable__lt,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
subtract_wf,
nat_wf,
weak-continuity-nat-int
Rules used in proof :
hyp_replacement,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
imageElimination,
setEquality,
minusEquality,
productElimination,
addEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
rename,
setElimination,
isectElimination,
dependent_set_memberEquality,
hypothesis,
intEquality,
because_Cache,
functionEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}F:(\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. \mforall{}G:n:\mBbbN{}\msupplus{} {}\mrightarrow{} \{g:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| f = g\} . \mexists{}n:\mBbbN{}\msupplus{}. ((F f) = (F (G n)))
Date html generated:
2017_09_29-PM-06_06_24
Last ObjectModification:
2017_09_09-PM-07_33_12
Theory : continuity
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