Nuprl Lemma : function-eq-symmetry
∀[A:Type]. ∀[B:Base].
∀[f,g:Base]. (function-eq(A;a.B[a];f;g) ⇒ function-eq(A;a.B[a];g;f)) supposing base-type-family{i:l}(A;a.B[a])
Proof
Definitions occuring in Statement :
function-eq: function-eq(A;a.B[a];f;g),
base-type-family: base-type-family{i:l}(A;a.B[a]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
function-eq: function-eq(A;a.B[a];f;g),
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
equal-wf-base,
function-eq_wf,
base_wf,
base-type-family_wf,
base-type-family-implies,
subtype_rel_self,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
equalitySymmetry,
hypothesis,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
cumulativity,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
lambdaEquality,
dependent_functionElimination,
universeEquality,
applyEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:Base].
\mforall{}[f,g:Base]. (function-eq(A;a.B[a];f;g) {}\mRightarrow{} function-eq(A;a.B[a];g;f))
supposing base-type-family\{i:l\}(A;a.B[a])
Date html generated:
2016_10_21-AM-09_39_27
Last ObjectModification:
2016_07_12-AM-05_01_25
Theory : per!type
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