Nuprl Lemma : increasing_inj
∀[k,m:ℕ]. ∀[f:ℕk ⟶ ℕm]. Inj(ℕk;ℕm;f) supposing increasing(f;k)
Proof
Definitions occuring in Statement :
inject: Inj(A;B;f),
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
false: False,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
equal_wf,
int_seg_wf,
increasing_wf,
nat_wf,
increasing_implies,
decidable__lt,
less_than_transitivity1,
lelt_wf,
le_weakening,
less_than_irreflexivity,
decidable__int_equal,
false_wf,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_isectElimination,
unionElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
independent_functionElimination,
voidElimination,
addEquality,
voidEquality,
intEquality
Latex:
\mforall{}[k,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}m]. Inj(\mBbbN{}k;\mBbbN{}m;f) supposing increasing(f;k)
Date html generated:
2017_04_14-AM-07_33_19
Last ObjectModification:
2017_02_27-PM-03_07_36
Theory : fun_1
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