### Nuprl Lemma : word-rel-diamond

`∀[X:Type]`
`  ∀x,y,z:(X + X) List.`
`    (word-rel(X;x;y)`
`    `` word-rel(X;x;z)`
`    `` ((y = z ∈ ((X + X) List)) ∨ (∃w:(X + X) List. (word-rel(X;y;w) ∧ word-rel(X;z;w)))))`

Proof

Definitions occuring in Statement :  word-rel: `word-rel(X;w1;w2)` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` union: `left + right` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` nat: `ℕ` ge: `i ≥ j ` less_than: `a < b` squash: `↓T` word-rel: `word-rel(X;w1;w2)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` append: `as @ bs` so_lambda: so_lambda3 so_apply: `x[s1;s2;s3]` uiff: `uiff(P;Q)` cons: `[a / b]` true: `True` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  int_seg_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf int_seg_wf decidable__equal_int subtract_wf int_seg_subtype false_wf decidable__le intformnot_wf itermSubtract_wf intformeq_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_formula_prop_eq_lemma le_wf length_wf non_neg_length nat_properties decidable__lt lelt_wf less_than_wf word-rel_wf all_wf list_wf or_wf equal_wf exists_wf set_wf primrec-wf2 nat_wf itermAdd_wf int_term_value_add_lemma length_wf_nat append_wf list-cases list_ind_nil_lemma list_ind_cons_lemma cons_one_one cons_wf product_subtype_list reduce_tl_cons_lemma and_wf tl_wf squash_wf true_wf reduce_hd_cons_lemma hd_wf ge_wf length_cons_ge_one subtype_rel_list top_wf inverse-letters_wf inverse-inverse-letters nil_wf length-append nil-append length_of_cons_lemma length_cons length_append iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination natural_numberEquality because_Cache hypothesisEquality hypothesis setElimination rename productElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll unionElimination addLevel applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality levelHypothesis hypothesis_subsumption dependent_set_memberEquality unionEquality cumulativity imageElimination independent_functionElimination functionEquality productEquality addEquality universeEquality hyp_replacement promote_hyp inlFormation imageMemberEquality baseClosed inrFormation instantiate equalityUniverse

Latex:
\mforall{}[X:Type]
\mforall{}x,y,z:(X  +  X)  List.
(word-rel(X;x;y)
{}\mRightarrow{}  word-rel(X;x;z)
{}\mRightarrow{}  ((y  =  z)  \mvee{}  (\mexists{}w:(X  +  X)  List.  (word-rel(X;y;w)  \mwedge{}  word-rel(X;z;w)))))

Date html generated: 2020_05_20-AM-08_22_02
Last ObjectModification: 2017_07_28-AM-09_18_36

Theory : free!groups

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