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Question 35598: Induction Proof Obligations

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0070-35598 · q-20260914-induction-proof-obligation-0003

Original question and givens

For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 3 finite example(s). Which induction role is missing?

  • G001 — For a proposition family P(n) beginning at 0, a draft supplies base case, induction step, conclusion and checks 3 finite example(s). Which induction role is missing?

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-induction-proof-obligation, [base_index, finite_examples_checked, mode, provided_roles, step_relation, target_role]]

After

derived = [induction, null, null]

Conditions: none

Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.

Step identifiers and dependencies

Rule: R70_SOURCE_bank_specific_interface · Uses: G001

S002 · implies

Before

source roles = [base_case]

After

derived = [prove P(0)]

Conditions: none

Keep induction roles distinct:P(0)is the proved start;the IH for arbitrary fixed n is a CONDITIONAL assumption,not the universal conclusion;the step derives the next case under that assumption;only then apply induction.

Step identifiers and dependencies

Rule: Induction_DEFINE_each_distinct_proof_obligation · Uses: G001, S001

S003 · implies

Before

source roles = [induction_hypothesis]

After

derived = [assume P(n) for arbitrary fixed n>=0]

Conditions: none

Keep induction roles distinct:P(0)is the proved start;the IH for arbitrary fixed n is a CONDITIONAL assumption,not the universal conclusion;the step derives the next case under that assumption;only then apply induction.

Step identifiers and dependencies

Rule: Induction_DEFINE_each_distinct_proof_obligation · Uses: G001, S002

S004 · implies

Before

source roles = [induction_step]

After

derived = [under that assumption derive P(n+1)]

Conditions: none

Keep induction roles distinct:P(0)is the proved start;the IH for arbitrary fixed n is a CONDITIONAL assumption,not the universal conclusion;the step derives the next case under that assumption;only then apply induction.

Step identifiers and dependencies

Rule: Induction_DEFINE_each_distinct_proof_obligation · Uses: G001, S003

S005 · implies

Before

source roles = [conclusion]

After

derived = [infer every n>=0 by induction after base and step]

Conditions: none

Keep induction roles distinct:P(0)is the proved start;the IH for arbitrary fixed n is a CONDITIONAL assumption,not the universal conclusion;the step derives the next case under that assumption;only then apply induction.

Step identifiers and dependencies

Rule: Induction_DEFINE_each_distinct_proof_obligation · Uses: G001, S004

S006 · implies

Before

source roles = [missing_role, [base_case, induction_step, conclusion]]

After

derived = [induction_hypothesis]

Conditions: none

For a missing-role task,subtract the THREE provided roles from all FOUR.For identification,use the SINGLE given clause’s role.Do not copy target_role instead of deriving it.

Step identifiers and dependencies

Rule: Induction_DERIVE_role_from_provided_draft · Uses: G001, S005

S007 · implies

Before

source roles = [3, induction_hypothesis]

After

derived = [role identified; arbitrary theorem NOT proved]

Conditions: none

These n FINITE examples neither prove all natural cases nor discharge a missing base/step.This task identifies an obligation;it does not prove an unspecified theorem P.

Step identifiers and dependencies

Rule: Induction_FINITE_examples_do_NOT_discharge_universal_obligation · Uses: G001, S006

Reverse checks

  • S007 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S006 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.

Result

induction hypothesis

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: aac29d20970642429da3b5a8fed3502a728abe3917938379c619cd906052f0ba
  • candidate record SHA-256: 831712c60b3058d3408af206605a0f5e08006ce797e1421d3c23bb654158c0d3
  • semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
  • question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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