### Nuprl Lemma : p-fun-exp-add-sq

`∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[x:A]. ∀[m,n:ℕ].  f^n + m x ~ f^n do-apply(f^m;x) supposing ↑can-apply(f^m;x)`

Proof

Definitions occuring in Statement :  p-fun-exp: `f^n` do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` nat: `ℕ` assert: `↑b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` top: `Top` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` add: `n + m` universe: `Type` sqequal: `s ~ 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` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` p-fun-exp: `f^n` do-apply: `do-apply(f;x)` p-id: `p-id()` outl: `outl(x)` sq_type: `SQType(T)` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` p-compose: `f o g` can-apply: `can-apply(f;x)` squash: `↓T` true: `True`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf 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 less_than_wf assert_wf can-apply_wf p-fun-exp_wf false_wf le_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf top_wf subtype_rel_dep_function subtype_rel_union primrec0_lemma add-zero decidable__equal_int subtype_base_sq int_subtype_base zero-add inl-do-apply lt_int_wf bool_wf equal-wf-T-base equal-wf-base intformeq_wf int_formula_prop_eq_lemma le_int_wf bnot_wf itermAdd_wf int_term_value_add_lemma primrec-unroll uiff_transitivity eqtt_to_assert assert_of_lt_int eqff_to_assert assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int equal_wf can-apply-fun-exp not_wf assert_of_bnot do-apply_wf p-fun-exp-add1-sq bool_subtype_base isl_wf squash_wf true_wf p-compose_wf subtract-add-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation sqequalAxiom cumulativity because_Cache functionExtensionality applyEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality unionElimination functionEquality unionEquality universeEquality instantiate addEquality baseClosed baseApply closedConclusion equalityElimination productElimination imageElimination imageMemberEquality hyp_replacement applyLambdaEquality

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  (A  +  Top)].  \mforall{}[x:A].  \mforall{}[m,n:\mBbbN{}].
f\^{}n  +  m  x  \msim{}  f\^{}n  do-apply(f\^{}m;x)  supposing  \muparrow{}can-apply(f\^{}m;x)

Date html generated: 2018_05_21-PM-06_29_39
Last ObjectModification: 2018_05_19-PM-04_40_42

Theory : general

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