Nuprl Lemma : complete_nat_ind_with_y
∀[P:ℕ ⟶ ℙ{k}]. ((∀i:ℕ. ((∀j:ℕi. P[j]) ⇒ P[i])) ⇒ (∀i:ℕ. P[i]))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
complete_nat_measure_ind,
nat_wf,
int_seg_wf,
all_wf,
less_than_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_sets,
le_wf,
lelt_wf
Rules used in proof :
cut,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
isect_memberFormation,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
cumulativity,
setEquality,
applyEquality,
because_Cache,
universeEquality,
functionEquality,
independent_isectElimination,
independent_pairFormation,
intEquality,
productElimination
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}\{k\}]. ((\mforall{}i:\mBbbN{}. ((\mforall{}j:\mBbbN{}i. P[j]) {}\mRightarrow{} P[i])) {}\mRightarrow{} (\mforall{}i:\mBbbN{}. P[i]))
Date html generated:
2016_05_13-PM-04_03_08
Last ObjectModification:
2015_12_26-AM-10_56_14
Theory : int_1
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