Nuprl Lemma : equipollent-nat-rationals-ext
ℕ ~ ℚ
Proof
Definitions occuring in Statement :
rationals: ℚ,
equipollent: A ~ B,
nat: ℕ
Definitions unfolded in proof :
member: t ∈ T,
pi1: fst(t),
remainder: n rem m,
eq_int: (i =z j),
btrue: tt,
it: ⋅,
bfalse: ff,
divide: n ÷ m,
ifthenelse: if b then t else f fi ,
compose: f o g,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
subtract: n - m,
code-pair: code-pair(a;b),
triangular-num: t(n),
bottom: ⊥,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
equipollent-nat-rationals,
equipollent_functionality_wrt_equipollent2,
equipollent-rationals-ext,
equipollent-nat-subset-ext,
decidable__assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_eq_int,
assoced_nelim,
bool_cases,
member_map,
select_member,
any: any x,
equipollent_functionality_wrt_equipollent,
equipollent-nat-squared,
equipollent_weakening_ext-eq,
product_functionality_wrt_equipollent_left,
equipollent-int-nat,
product_functionality_wrt_equipollent_right,
equipollent-int_upper-nat,
equipollent-product-com,
equipollent_transitivity,
code-pair-bijection,
equipollent_inversion,
biject-int-nat,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
equipollent-nat-rationals,
strictness-spread,
istype-void,
lifting-strict-spread,
strict4-spread,
lifting-strict-decide,
strict4-decide,
lifting-strict-int_eq,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
lifting-strict-less,
equipollent_functionality_wrt_equipollent2,
equipollent-rationals-ext,
equipollent-nat-subset-ext,
decidable__assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_eq_int,
assoced_nelim,
bool_cases,
member_map,
select_member,
equipollent_functionality_wrt_equipollent,
equipollent-nat-squared,
equipollent_weakening_ext-eq,
product_functionality_wrt_equipollent_left,
equipollent-int-nat,
product_functionality_wrt_equipollent_right,
equipollent-int_upper-nat,
equipollent-product-com,
equipollent_transitivity,
code-pair-bijection,
equipollent_inversion,
biject-int-nat
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
isect_memberEquality_alt,
voidElimination,
baseClosed,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueIntEq,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
universeIsType,
int_eqExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt,
inhabitedIsType,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
sqleReflexivity,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
spreadExceptionCases,
axiomSqleEquality,
callbyvalueAdd,
addExceptionCases
Latex:
\mBbbN{} \msim{} \mBbbQ{}
Date html generated:
2019_10_16-AM-11_47_53
Last ObjectModification:
2019_06_26-PM-04_10_11
Theory : rationals
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