### Nuprl Lemma : mtype-sqequal

`∀[L:MutualRectypeSpec]. ∀[i:mKinds].  (mtype(L;i) ~ mrec(L;i))`

Proof

Definitions occuring in Statement :  mtype: `mtype(L;i)` mkinds: `mKinds` mrec: `mrec(L;i)` mrec_spec: `MutualRectypeSpec` uall: `∀[x:A]. B[x]` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` mtype: `mtype(L;i)` mrec_spec: `MutualRectypeSpec` so_lambda: `λ2x.t[x]` so_apply: `x[s]` top: `Top` mkinds: `mKinds` all: `∀x:A. B[x]` 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]` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` cons: `[a / b]` decidable: `Dec(P)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` pi1: `fst(t)` has-value: `(a)↓`
Lemmas referenced :  apply_alist-eager-map-atom list_wf pi1_wf istype-atom istype-void mkinds_wf mrec_spec_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_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 ge_wf istype-less_than list-cases eager_map_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf eager_map_cons_lemma value-type-has-value top_wf list-value-type eager-map_wf product-value-type istype-nat istype-top
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination productEquality cumulativity atomEquality unionEquality universeEquality hypothesis hypothesisEquality Error :lambdaEquality_alt,  sqequalRule Error :productIsType,  Error :universeIsType,  Error :isect_memberEquality_alt,  voidElimination axiomSqEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :lambdaFormation_alt,  setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  int_eqEquality dependent_functionElimination independent_pairFormation Error :functionIsTypeImplies,  unionElimination promote_hyp hypothesis_subsumption productElimination Error :equalityIstype,  because_Cache Error :dependent_set_memberEquality_alt,  equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase callbyvalueReduce sqleReflexivity independent_pairEquality Error :functionIsType,  functionExtensionality

Latex:
\mforall{}[L:MutualRectypeSpec].  \mforall{}[i:mKinds].    (mtype(L;i)  \msim{}  mrec(L;i))

Date html generated: 2019_06_20-PM-02_14_53
Last ObjectModification: 2019_02_28-PM-02_06_33

Theory : tuples

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