Nuprl Lemma : strong-continuity2-implies-weak-skolem2
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ⇃(∃M:(ℕ ⟶ 𝔹) ⟶ ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕM f ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
weak-continuity-skolem: weak-continuity-skolem(T;F),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
pi1: fst(t),
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
equipollent_inversion,
equipollent-two,
istype-nat,
bool_wf,
compose_wf,
nat_wf,
int_seg_wf,
Kleene-M_wf,
int_seg_subtype_nat,
istype-false,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
subtype_rel_wf,
squash_wf,
implies-quotient-true,
basic-strong-continuity_wf,
weak-continuity-skolem_wf,
basic-implies-strong-continuity2,
strong-continuity2-weak-skolem,
weak-continuity-skolem_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
rename,
Error :functionIsType,
Error :universeIsType,
Error :lambdaEquality_alt,
applyEquality,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
productElimination,
sqequalRule,
Error :inhabitedIsType,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
independent_isectElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :productIsType,
imageMemberEquality,
baseClosed
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}M:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2019_06_20-PM-02_51_42
Last ObjectModification:
2019_02_09-PM-11_57_48
Theory : continuity
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