Nuprl Lemma : int_upper_ind_uniform
∀i:ℤ. ∀[E:{i...} ⟶ ℙ{u}]. ((∀[k:{i...}]. ((∀[j:{i..k-}]. E[j]) ⇒ E[k])) ⇒ {∀[k:{i...}]. E[k]})
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
uwellfounded: uWellFnd(A;x,y.R[x; y])
Lemmas referenced :
uall_wf,
int_upper_wf,
int_seg_wf,
subtype_rel_sets,
lelt_wf,
le_wf,
int_upper_uwell_founded,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
functionEquality,
because_Cache,
setElimination,
rename,
intEquality,
independent_isectElimination,
setEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}[E:\{i...\} {}\mrightarrow{} \mBbbP{}\{u\}]. ((\mforall{}[k:\{i...\}]. ((\mforall{}[j:\{i..k\msupminus{}\}]. E[j]) {}\mRightarrow{} E[k])) {}\mRightarrow{} \{\mforall{}[k:\{i...\}]. E[k]\})
Date html generated:
2016_05_14-AM-07_26_14
Last ObjectModification:
2015_12_26-PM-01_27_56
Theory : int_2
Home
Index