Nuprl Lemma : weak-continuity-principle-nat+-int-bool-double
∀F,H:(ℕ+ ⟶ ℤ) ⟶ 𝔹. ∀f:ℕ+ ⟶ ℤ. ∀G:n:ℕ+ ⟶ {g:ℕ+ ⟶ ℤ| f = g ∈ (ℕ+n ⟶ ℤ)} . ∃n:ℕ+. (F f = F (G n) ∧ H f = H (G n))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
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 :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
istype: istype(T),
cand: A c∧ B
Lemmas referenced :
weak-continuity-nat-int,
nat_plus_wf,
nat_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
eqtt_to_assert,
istype-false,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat_plus,
subtype_rel_self,
add-member-int_seg2,
nat_properties,
add-subtract-cancel,
implies-quotient-true2,
add-swap,
exists_wf,
all_wf,
equal_wf,
int_seg_subtype_nat,
equal-wf-base,
int_subtype_base,
trivial-quotient-true,
imax_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
add_nat_plus,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
subtype_rel_dep_function,
int_seg_subtype,
le_weakening,
imax_ub,
subtract-add-cancel,
decidable__equal_int,
equal-wf-base-T,
btrue_wf,
bfalse_wf,
squash-from-quotient,
mu_wf,
band_wf,
eq_bool_wf,
assert_of_eq_bool,
assert_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
mu-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :lambdaEquality_alt,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
hypothesis,
intEquality,
Error :dependent_set_memberEquality_alt,
isectElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
because_Cache,
Error :inhabitedIsType,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
addEquality,
minusEquality,
Error :setIsType,
Error :functionExtensionality_alt,
applyLambdaEquality,
Error :productIsType,
productEquality,
Error :equalityIsType4,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
baseApply,
closedConclusion,
Error :inrFormation_alt,
Error :inlFormation_alt,
hyp_replacement,
imageElimination
Latex:
\mforall{}F,H:(\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}) {}\mrightarrow{} \mBbbB{}. \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) \mwedge{} H f = H (G n))
Date html generated:
2019_06_20-PM-02_51_53
Last ObjectModification:
2018_10_05-PM-05_56_56
Theory : continuity
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