Nuprl Lemma : function-mono
∀A:Type. ∀B:A ⟶ Type. (((A ⊆r Base) ∧ (∀a:A. mono(B[a])) ∧ (↓∃a:A. value-type(B[a]))) ⇒ mono(a:A ⟶ B[a]))
Proof
Definitions occuring in Statement :
mono: mono(T),
value-type: value-type(T),
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
mono: mono(T),
squash: ↓T,
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
is-above_wf,
base_wf,
subtype_rel_wf,
all_wf,
mono_wf,
squash_wf,
exists_wf,
value-type_wf,
and_wf,
equal_wf,
has-value_wf_base,
is-exception_wf,
equal-wf-base-T,
sqle_wf_base,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
imageElimination,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
productEquality,
sqequalRule,
lambdaEquality,
universeEquality,
dependent_functionElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
dependent_pairFormation,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
applyLambdaEquality,
setElimination,
divergentSqle,
sqleRule,
sqleReflexivity,
because_Cache,
hyp_replacement,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type.
(((A \msubseteq{}r Base) \mwedge{} (\mforall{}a:A. mono(B[a])) \mwedge{} (\mdownarrow{}\mexists{}a:A. value-type(B[a]))) {}\mRightarrow{} mono(a:A {}\mrightarrow{} B[a]))
Date html generated:
2017_04_14-AM-07_37_21
Last ObjectModification:
2017_02_27-PM-03_09_39
Theory : subtype_1
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