### Nuprl Lemma : p-fun-exp-add1-sq

`∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[x:A]. ∀[n:ℕ].  f^n + 1 x ~ f^n do-apply(f;x) supposing ↑can-apply(f;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` natural_number: `\$n` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  do-apply: `do-apply(f;x)` can-apply: `can-apply(f;x)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` isl: `isl(x)` p-fun-exp: `f^n` top: `Top` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` not: `¬A` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` prop: `ℙ` p-id: `p-id()` p-compose: `f o g` outl: `outl(x)` ifthenelse: `if b then t else f fi ` btrue: `tt` assert: `↑b` bfalse: `ff` subtype_rel: `A ⊆r B` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  istype-assert btrue_wf bfalse_wf istype-nat istype-top istype-universe simple-primrec-add istype-void istype-le primrec1_lemma 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 primrec0_lemma subtract-1-ge-0 istype-true not_wf bnot_wf assert_wf int_subtype_base equal-wf-base bool_wf eq_int_wf primrec-unroll uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot equal_wf int_formula_prop_eq_lemma intformeq_wf satisfiable-full-omega-tt
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt introduction cut hypothesis axiomSqEquality extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality inhabitedIsType lambdaFormation_alt unionElimination equalityIstype equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination isect_memberEquality_alt isectIsTypeImplies universeIsType functionIsType unionIsType because_Cache instantiate universeEquality voidElimination dependent_set_memberEquality_alt natural_numberEquality independent_pairFormation promote_hyp setElimination rename intWeakElimination independent_isectElimination approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality functionIsTypeImplies intEquality baseClosed closedConclusion baseApply voidEquality isect_memberEquality lambdaFormation equalityElimination productElimination impliesFunctionality computeAll lambdaEquality dependent_pairFormation

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  (A  +  Top)].  \mforall{}[x:A].  \mforall{}[n:\mBbbN{}].
f\^{}n  +  1  x  \msim{}  f\^{}n  do-apply(f;x)  supposing  \muparrow{}can-apply(f;x)

Date html generated: 2019_10_15-AM-11_07_44
Last ObjectModification: 2019_06_26-PM-04_19_11

Theory : general

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