`∀[A:Type]. ∀[n,m:ℕ]. ∀[f:A ⟶ (A + Top)]. ∀[x:A].`
`  uiff(↑can-apply(f^n + m;x);(↑can-apply(f^m;x)) ∧ (↑can-apply(f^n;do-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` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` top: `Top` and: `P ∧ Q` function: `x:A ⟶ B[x]` union: `left + right` add: `n + m` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` guard: `{T}` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` top: `Top` implies: `P `` Q` prop: `ℙ` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` squash: `↓T` true: `True` rev_uimplies: `rev_uimplies(P;Q)` can-apply: `can-apply(f;x)` p-compose: `f o g` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  can-apply-fun-exp-add assert_witness can-apply_wf p-fun-exp_wf subtype_rel_dep_function subtype_rel_union top_wf do-apply_wf assert_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf nat_wf assert_functionality_wrt_uiff p-compose_wf squash_wf true_wf p-fun-exp-add bool_wf equal-wf-T-base bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot equal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality independent_isectElimination hypothesis productElimination sqequalRule independent_pairEquality cumulativity functionExtensionality applyEquality lambdaEquality lambdaFormation isect_memberEquality voidElimination voidEquality independent_functionElimination dependent_set_memberEquality addEquality setElimination rename dependent_functionElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll unionEquality productEquality equalityTransitivity equalitySymmetry functionEquality universeEquality imageElimination imageMemberEquality baseClosed equalityElimination

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

Date html generated: 2017_10_01-AM-09_14_51
Last ObjectModification: 2017_07_26-PM-04_49_46

Theory : general

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