Nuprl Lemma : function-eq-symmetry-type-function
∀A:Type. ∀B:type-function{i:l}(A). ∀f,g:Base. (function-eq(A;a.B[a];f;g) ⇒ function-eq(A;a.B[a];g;f))
Proof
Definitions occuring in Statement :
type-function: type-function{i:l}(A),
function-eq: function-eq(A;a.B[a];f;g),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
base: Base,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
base-type-family: base-type-family{i:l}(A;a.B[a]),
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
squash: ↓T,
label: ...$L... t,
true: True
Lemmas referenced :
type-function-eta,
function-eq_wf_type_function,
function-eq_wf,
function-eq-symmetry,
equal-wf-base,
apply_wf_type-function,
equal_wf,
and_wf,
type-function_wf,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
because_Cache,
cut,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeEquality,
pointwiseFunctionality,
isect_memberFormation,
introduction,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
applyEquality,
lambdaEquality,
setElimination,
rename,
productElimination,
setEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
independent_functionElimination,
functionEquality,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}A:Type. \mforall{}B:type-function\{i:l\}(A). \mforall{}f,g:Base.
(function-eq(A;a.B[a];f;g) {}\mRightarrow{} function-eq(A;a.B[a];g;f))
Date html generated:
2016_05_13-PM-03_53_51
Last ObjectModification:
2016_01_14-PM-07_15_43
Theory : per!type
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