### Nuprl Lemma : hd-before

`∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ L) `` hd(L) before x ∈ L supposing ¬(x = hd(L) ∈ T)) supposing 0 < ||L||`

Proof

Definitions occuring in Statement :  l_before: `x before y ∈ l` l_member: `(x ∈ l)` hd: `hd(l)` length: `||as||` list: `T List` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` not: `¬A` false: `False` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` less_than': `less_than'(a;b)` cons: `[a / b]` iff: `P `⇐⇒` Q` cand: `A c∧ B` rev_implies: `P `` Q`
Lemmas referenced :  member-less_than length_wf equal_wf hd_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf list-cases length_of_nil_lemma product_subtype_list reduce_hd_cons_lemma length_of_cons_lemma not_wf l_member_wf less_than_wf list_wf cons_member l_before_wf cons_before
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality cumulativity hypothesisEquality hypothesis independent_isectElimination rename sqequalRule lambdaEquality dependent_functionElimination voidElimination because_Cache unionElimination imageElimination productElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll promote_hyp hypothesis_subsumption universeEquality independent_functionElimination inlFormation

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List.  \mforall{}x:T.  ((x  \mmember{}  L)  {}\mRightarrow{}  hd(L)  before  x  \mmember{}  L  supposing  \mneg{}(x  =  hd(L)))  supposing  0  <  ||L||

Date html generated: 2017_04_17-AM-08_47_53
Last ObjectModification: 2017_02_27-PM-05_06_24

Theory : list_1

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