`∀[A:Type]. ∀[n,m:ℕ]. ∀[f:A ⟶ (A + Top)]. ∀[x:A].`
`  {(↑can-apply(f^m;x)) ∧ (↑can-apply(f^n;do-apply(f^m;x))) ∧ (do-apply(f^n + m;x) = do-apply(f^n;do-apply(f^m;x)) ∈ A)} `
`  supposing ↑can-apply(f^n + 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` guard: `{T}` and: `P ∧ Q` function: `x:A ⟶ B[x]` union: `left + right` add: `n + m` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` subtype_rel: `A ⊆r B` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` top: `Top` true: `True` guard: `{T}` uiff: `uiff(P;Q)` and: `P ∧ Q` implies: `P `` Q` 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` cand: `A c∧ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  assert_functionality_wrt_uiff can-apply_wf p-fun-exp_wf p-compose_wf top_wf squash_wf true_wf p-fun-exp-add subtype_rel_dep_function subtype_rel_union assert_witness 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 can-apply-compose equal_wf iff_weakening_equal do-apply-compose
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache applyEquality sqequalRule cumulativity hypothesisEquality functionExtensionality independent_isectElimination lambdaEquality imageElimination equalityTransitivity equalitySymmetry functionEquality unionEquality lambdaFormation isect_memberEquality voidElimination voidEquality natural_numberEquality imageMemberEquality baseClosed productElimination independent_pairEquality independent_functionElimination axiomEquality dependent_set_memberEquality addEquality setElimination rename dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll universeEquality

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

Date html generated: 2017_10_01-AM-09_14_45
Last ObjectModification: 2017_07_26-PM-04_49_42

Theory : general

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