Nuprl Lemma : qeq-sym
Sym(ℤ ⋃ (ℤ × ℤ-o);r,s.qeq(r;s) = tt)
Proof
Definitions occuring in Statement :
qeq: qeq(r;s),
sym: Sym(T;x,y.E[x; y]),
int_nzero: ℤ-o,
b-union: A ⋃ B,
btrue: tt,
bool: 𝔹,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
sym: Sym(T;x,y.E[x; y]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
qeq: qeq(r;s),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
btrue: tt,
squash: ↓T,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
valueall-type-has-valueall,
b-union_wf,
int_nzero_wf,
bunion-valueall-type,
int-valueall-type,
product-valueall-type,
set-valueall-type,
nequal_wf,
evalall-reduce,
equal_wf,
squash_wf,
true_wf,
eq_int_eq_true,
eqtt_to_assert,
assert_of_eq_int,
iff_weakening_equal,
equal-wf-T-base,
bool_wf,
qeq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
thin,
intEquality,
productEquality,
hypothesis,
independent_isectElimination,
because_Cache,
lambdaEquality,
independent_functionElimination,
hypothesisEquality,
natural_numberEquality,
callbyvalueReduce,
imageElimination,
productElimination,
unionElimination,
equalityElimination,
isintReduceTrue,
applyEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
addLevel,
imageMemberEquality,
baseClosed,
multiplyEquality,
setElimination,
rename
Latex:
Sym(\mBbbZ{} \mcup{} (\mBbbZ{} \mtimes{} \mBbbZ{}\msupminus{}\msupzero{});r,s.qeq(r;s) = tt)
Date html generated:
2018_05_21-PM-11_43_44
Last ObjectModification:
2017_07_26-PM-06_42_56
Theory : rationals
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