Nuprl Lemma : WCPD_wf
∀F,H:(ℕ+ ⟶ ℤ) ⟶ 𝔹. ∀f:ℕ+ ⟶ ℤ. ∀G:n:ℕ+ ⟶ {g:ℕ+ ⟶ ℤ| f = g ∈ (ℕ+n ⟶ ℤ)} .
(WCPD(F;H;f;G) ∈ {n:ℕ+| F f = F (G n) ∧ H f = H (G n)} )
Proof
Definitions occuring in Statement :
WCPD: WCPD(F;H;f;G),
int_seg: {i..j-},
nat_plus: ℕ+,
bool: 𝔹,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
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 :
WCPD: WCPD(F;H;f;G),
weak-continuity-principle-nat+-int-bool-double-ext,
pi1: fst(t),
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
so_apply: x[s],
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
set_wf,
bool_wf,
exists_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat_plus,
nat_plus_wf,
subtype_rel_dep_function,
int_seg_wf,
equal_wf,
all_wf,
weak-continuity-principle-nat+-int-bool-double-ext
Rules used in proof :
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
productElimination,
productEquality,
independent_pairFormation,
independent_isectElimination,
intEquality,
rename,
setElimination,
natural_numberEquality,
setEquality,
functionEquality,
isectElimination,
introduction,
lambdaEquality,
because_Cache,
hypothesisEquality,
sqequalHypSubstitution,
sqequalRule,
functionExtensionality,
applyEquality,
hypothesis,
extract_by_obid,
instantiate,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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\} .
(WCPD(F;H;f;G) \mmember{} \{n:\mBbbN{}\msupplus{}| F f = F (G n) \mwedge{} H f = H (G n)\} )
Date html generated:
2017_09_29-PM-06_06_33
Last ObjectModification:
2017_09_12-PM-02_15_31
Theory : continuity
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