### Nuprl Lemma : base-equal-partial

`∀[A:Type]`
`  ∀[a,b:Base].`
`    a = b ∈ partial(A) supposing (((a)↓ `⇐⇒` (b)↓) ∧ a = b ∈ A supposing (a)↓) ∧ (¬is-exception(a)) ∧ (¬is-exception(b)) `
`  supposing value-type(A)`

Proof

Definitions occuring in Statement :  partial: `partial(T)` value-type: `value-type(T)` has-value: `(a)↓` is-exception: `is-exception(t)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` not: `¬A` and: `P ∧ Q` base: `Base` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` partial: `partial(T)` so_lambda: `λ2x y.t[x; y]` base-partial: `base-partial(T)` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` cand: `A c∧ B` per-partial: `per-partial(T;x;y)` uiff: `uiff(P;Q)` has-value: `(a)↓`
Lemmas referenced :  equal-wf-base and_wf iff_wf has-value_wf_base isect_wf not_wf is-exception_wf base_wf value-type_wf quotient-member-eq base-partial_wf per-partial_wf per-partial-equiv_rel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis sqequalRule lambdaEquality because_Cache universeEquality setElimination rename independent_isectElimination productElimination dependent_functionElimination introduction independent_functionElimination equalityTransitivity equalitySymmetry axiomEquality independent_pairFormation dependent_set_memberEquality productEquality isectEquality axiomSqleEquality

Latex:
\mforall{}[A:Type]
\mforall{}[a,b:Base].
a  =  b
supposing  (((a)\mdownarrow{}  \mLeftarrow{}{}\mRightarrow{}  (b)\mdownarrow{})  \mwedge{}  a  =  b  supposing  (a)\mdownarrow{})  \mwedge{}  (\mneg{}is-exception(a))  \mwedge{}  (\mneg{}is-exception(b))
supposing  value-type(A)

Date html generated: 2016_05_14-AM-06_09_42
Last ObjectModification: 2015_12_26-AM-11_52_16

Theory : partial_1

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