Nuprl Lemma : evodd-enum-surjection
Surj(ℕ;b:𝔹 × (pw-evenodd() b);λn.evodd-enum(n))
Proof
Definitions occuring in Statement :
evodd-enum: evodd-enum(n),
pw-evenodd: pw-evenodd(),
surject: Surj(A;B;f),
nat: ℕ,
bool: 𝔹,
apply: f a,
lambda: λx.A[x],
product: x:A × B[x]
Definitions unfolded in proof :
surject: Surj(A;B;f),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
evodd-enum: evodd-enum(n),
primrec: primrec(n;b;c),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
true: True,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
bool_wf,
pw-evenodd_wf,
evodd-induction2-ext,
exists_wf,
nat_wf,
equal_wf,
evodd-enum_wf,
false_wf,
le_wf,
btrue_wf,
evodd-zero_wf,
equal-wf-T-base,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
bnot_wf,
evodd-succ_wf,
subtype_rel-equal,
squash_wf,
true_wf,
bnot_bnot_elim,
iff_weakening_equal,
primrec-unroll,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
le_antisymmetry_iff,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
productElimination,
thin,
sqequalRule,
cut,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
productEquality,
introduction,
extract_by_obid,
applyEquality,
isectElimination,
lambdaEquality,
dependent_pairEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
baseClosed,
addEquality,
setElimination,
rename,
unionElimination,
voidElimination,
independent_isectElimination,
imageMemberEquality,
imageElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityElimination,
promote_hyp,
cumulativity,
hyp_replacement,
applyLambdaEquality
Latex:
Surj(\mBbbN{};b:\mBbbB{} \mtimes{} (pw-evenodd() b);\mlambda{}n.evodd-enum(n))
Date html generated:
2017_04_14-AM-07_43_27
Last ObjectModification:
2017_02_27-PM-03_14_06
Theory : co-recursion
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