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Question 35610: 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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R0070-35610 · q-20260914-induction-proof-obligation-0015

Original question and givens

For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 15 finite example(s). Which induction role is missing?

  • G001 — For a proposition family P(n) beginning at 0, a draft supplies base case, induction hypothesis, induction step and checks 15 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_hypothesis, induction_step]]

After

derived = [conclusion]

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 = [15, conclusion]

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

conclusion

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: 5e97d271f5372cbb07c5c9ec6c37db0d4173600ec231d38277f644e9582c6cde
  • candidate record SHA-256: 51e4282901df3634a6886800a09da53247079f18a625daf8bec4d5a83815212d
  • semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
  • question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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