Nuprl Lemma : comparison-antisym
∀[T:Type]. ∀cmp:comparison(T). AntiSym(T;x,y.0 ≤ (cmp x y)) supposing ∀x,y:T. (((cmp x y) = 0 ∈ ℤ) ⇒ (x = y ∈ T))
Proof
Definitions occuring in Statement :
comparison: comparison(T),
anti_sym: AntiSym(T;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
anti_sym: AntiSym(T;x,y.R[x; y]),
implies: P ⇒ Q,
comparison: comparison(T),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
comparison_wf,
equal_wf,
all_wf,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMinus_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
iff_weakening_equal,
true_wf,
squash_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
intEquality,
natural_numberEquality,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
because_Cache,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
functionEquality
Latex:
\mforall{}[T:Type]
\mforall{}cmp:comparison(T). AntiSym(T;x,y.0 \mleq{} (cmp x y)) supposing \mforall{}x,y:T. (((cmp x y) = 0) {}\mRightarrow{} (x = y))
Date html generated:
2016_05_14-PM-02_38_36
Last ObjectModification:
2016_01_15-AM-07_39_23
Theory : list_1
Home
Index