### Nuprl Lemma : accum-induction-factorial

`∀n:ℕ. (∃x:{ℤ| ((n)! = x ∈ ℤ)})`

Proof

Definitions occuring in Statement :  fact: `(n)!` nat: `ℕ` all: `∀x:A. B[x]` sq_exists: `∃x:{A| B[x]}` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` so_apply: `x[s]` implies: `P `` Q` all: `∀x:A. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` sq_exists: `∃x:{A| B[x]}` sq_stable: `SqStable(P)` squash: `↓T` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top`
Lemmas referenced :  int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties sq_stable__equal le_wf false_wf primrec-induction-factorial nat_wf nat_plus_wf fact_wf equal_wf sq_exists_wf accum-induction-ext
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin sqequalRule lambdaEquality intEquality hypothesisEquality hypothesis applyEquality setElimination rename independent_functionElimination dependent_functionElimination dependent_set_memberEquality natural_numberEquality independent_pairFormation lambdaFormation addEquality equalityTransitivity equalitySymmetry introduction imageMemberEquality baseClosed imageElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache

Latex:
\mforall{}n:\mBbbN{}.  (\mexists{}x:\{\mBbbZ{}|  ((n)!  =  x)\})

Date html generated: 2016_05_15-PM-04_09_55
Last ObjectModification: 2016_01_16-AM-11_05_35

Theory : general

Home Index