Nuprl Lemma : strict-comparison-trans
∀[T:Type]. ∀cmp:comparison(T). Trans(T;x,y.0 < cmp x y)
Proof
Definitions occuring in Statement :
comparison: comparison(T),
trans: Trans(T;x,y.E[x; y]),
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
trans: Trans(T;x,y.E[x; y]),
implies: P ⇒ Q,
comparison: comparison(T),
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q
Lemmas referenced :
int_term_value_minus_lemma,
itermMinus_wf,
iff_weakening_equal,
le_wf,
equal-wf-T-base,
equal_wf,
all_wf,
comparison-anti,
true_wf,
squash_wf,
int_subtype_base,
subtype_base_sq,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__lt,
decidable__equal_int,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
member-less_than,
comparison_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
hypothesis,
lemma_by_obid,
isectElimination,
natural_numberEquality,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
cumulativity,
sqequalRule,
lambdaEquality,
independent_isectElimination,
because_Cache,
universeEquality,
independent_functionElimination,
functionExtensionality,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
instantiate,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
productEquality,
minusEquality,
functionEquality,
baseClosed,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}cmp:comparison(T). Trans(T;x,y.0 < cmp x y)
Date html generated:
2016_05_14-PM-02_38_18
Last ObjectModification:
2016_04_08-AM-01_26_10
Theory : list_1
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