Nuprl Lemma : comp_nat_ind_tp
∀[P:ℕ ⟶ ℙ{k}]. ((∀i:ℕ. ((∀j:ℕ. P[j] supposing j < i) ⇒ P[i])) ⇒ {∀i:ℕ. P[i]})
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
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]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
false: False,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
nat_wf,
less_than_wf,
subtype_rel_self,
subtract_wf,
istype-int,
primrec-wf2,
all_wf,
isect_wf,
member-less_than,
less_than_transitivity1,
less_than_irreflexivity,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
istype-void,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
decidable__le,
not-le-2,
sq_stable__le,
zero-add,
add-zero,
le_wf,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
Error :inhabitedIsType,
hypothesisEquality,
Error :isectIsType,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
applyEquality,
instantiate,
universeEquality,
because_Cache,
natural_numberEquality,
Error :setIsType,
Error :lambdaEquality_alt,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
minusEquality,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}\{k\}]. ((\mforall{}i:\mBbbN{}. ((\mforall{}j:\mBbbN{}. P[j] supposing j < i) {}\mRightarrow{} P[i])) {}\mRightarrow{} \{\mforall{}i:\mBbbN{}. P[i]\})
Date html generated:
2019_06_20-AM-11_27_58
Last ObjectModification:
2018_10_06-AM-10_12_47
Theory : call!by!value_2
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