Nuprl Lemma : comparison-connex
∀[T:Type]. ∀cmp:comparison(T). Connex(T;x,y.0 ≤ (cmp x y))
Proof
Definitions occuring in Statement :
comparison: comparison(T),
connex: Connex(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
connex: Connex(T;x,y.R[x; y]),
comparison: comparison(T),
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
iff_weakening_equal,
true_wf,
squash_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_or_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
intformle_wf,
intformor_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
decidable__or,
sq_stable_from_decidable,
le_wf,
or_wf,
comparison_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
cut,
lemma_by_obid,
dependent_functionElimination,
hypothesis,
universeEquality,
isectElimination,
natural_numberEquality,
applyEquality,
productElimination,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
because_Cache,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}cmp:comparison(T). Connex(T;x,y.0 \mleq{} (cmp x y))
Date html generated:
2016_05_14-PM-02_38_27
Last ObjectModification:
2016_01_15-AM-07_41_08
Theory : list_1
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