### Nuprl Lemma : int-prod-1

`∀[n:ℕ]. (Π(1 | x < n) = 1 ∈ ℤ)`

Proof

Definitions occuring in Statement :  int-prod: `Π(f[x] | x < k)` nat: `ℕ` uall: `∀[x:A]. B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` int-prod: `Π(f[x] | x < k)` lt_int: `i <z j` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` subtype_rel: `A ⊆r B` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` decidable: `Dec(P)`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than primrec-unroll subtract-1-ge-0 lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf assert-bnot iff_weakening_uiff assert_wf less_than_wf decidable__equal_int intformnot_wf intformeq_wf itermMultiply_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma istype-nat
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination Error :lambdaFormation_alt,  natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality promote_hyp instantiate cumulativity Error :equalityIsType1

Latex:
\mforall{}[n:\mBbbN{}].  (\mPi{}(1  |  x  <  n)  =  1)

Date html generated: 2019_06_20-PM-01_18_41
Last ObjectModification: 2018_10_19-PM-00_59_57

Theory : int_2

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