Nuprl Lemma : ancestral-logic-example1
(∀x,y:Dom.  ((P x y) 
⇒ (Q x y))) 
⇒ (∀x,y:Dom.  (TC(λa,b.P a b)(x,y) 
⇒ TC(λa,b.Q a b)(x,y)))
Proof
Definitions occuring in Statement : 
TC: TC(λx,y.F[x; y])(a,b)
, 
language1: language1{i:l}(D,A,B,C,P,Q,R,H.fml[D;A;B;C;P;Q;R;H])
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
apply: f a
Definitions unfolded in proof : 
language1: language1{i:l}(D,A,B,C,P,Q,R,H.fml[D;A;B;C;P;Q;R;H])
, 
uall: ∀[x:A]. B[x]
, 
!hyp_hide: x
, 
member: t ∈ T
, 
prop: ℙ
, 
implies: P 
⇒ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
Lemmas referenced : 
all_wf, 
TC-base, 
TC-min-uniform, 
TC_wf, 
TC-trans
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
sqequalHypSubstitution, 
functionEquality, 
cumulativity, 
hypothesisEquality, 
universeEquality, 
because_Cache, 
lambdaFormation, 
cut, 
lemma_by_obid, 
isectElimination, 
thin, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
hypothesis, 
addLevel, 
dependent_functionElimination, 
independent_functionElimination, 
levelHypothesis
Latex:
(\mforall{}x,y:Dom.    ((P  x  y)  {}\mRightarrow{}  (Q  x  y)))  {}\mRightarrow{}  (\mforall{}x,y:Dom.    (TC(\mlambda{}a,b.P  a  b)(x,y)  {}\mRightarrow{}  TC(\mlambda{}a,b.Q  a  b)(x,y)))
Date html generated:
2016_05_16-AM-09_08_27
Last ObjectModification:
2015_12_28-PM-07_03_25
Theory : first-order!and!ancestral!logic
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