# Question 9974: Synthetic Implication/Conjunction Term Type against a Claim [Return to catalogue page 50](https://ocme.evemisslab.com/en/questions/page/50/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r07-lean-proof-term-foundations/#q-20261013-advance-20261019-r07-lean-proof-term-0024) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0018-09974 · q-20261013-advance-20261019-r07-lean-proof-term-0024 ## Original question and givens Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair23","type":["and","P23","Q23"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair23"}}, claimed_type="P23". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?). - G001 — Use the synthetic small propositional calculus or synthetic receipt specified here, not actual Lean execution. The original source pulse did not run Lean builds for these questions. A statement, tactic log and kernel-checked proof are different obligations. Context=[{"name":"pair23","type":["and","P23","Q23"]}], term={"kind":"fst","pair":{"kind":"var","name":"pair23"}}, claimed_type="P23". Atomic types are uppercase names; [imp,A,B] and [and,A,B] are implication and conjunction. Terms use var, lam, app, pair, fst, snd; this toy rule forbids rebinding a context name. Give (inferred type, claimed type, exact type match?). ## 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. - [mko-lean-proof-term](https://ocme.evemisslab.com/en/mko/mko-lean-proof-term/) ## Step-by-step derivation S001 · impliesBefore synthetic proposition/term calculus or supplied synthetic receipt After no actual Lean build,source-digest check or kernel acceptance Conditions: none Apply only the specified small calculus or synthetic receipt protocol.Statement,tactic completion,term presence and kernel acceptance are separate.No Lean is run and the teaching guide is not promoted to a formal proof. Step identifiers and dependenciesRule: synthetic_proofterm_not_actual_kernel · Uses: G001 S002 · impliesBefore raw context=[{"name":"pair23","type":["and","P23","Q23"]}] After validated English-locale context=[["pair23",["and","P23","Q23"]]] Conditions: none Validate every context name and type,reject duplicates,and display in the specified English order without translating opaque names. Step identifiers and dependenciesRule: synthetic_whole_context_validation · Uses: G001, S001 S003 · impliesBefore path=root/pair;term={"kind":"var","name":"pair23"};context=[["pair23",["and","P23","Q23"]]] After variable lookup type=[and, P23, Q23] Conditions: none Look up the exact name in the current lexical context.Do not treat a similar name or a missing binding as an assumption. Step identifiers and dependenciesRule: synthetic_variable_typing · Uses: G001, S002 S004 · impliesBefore path=root;term={"kind":"fst","pair":{"kind":"var","name":"pair23"}};context=[["pair23",["and","P23","Q23"]]];pair type=[and, P23, Q23] After fst projection type=P23 Conditions: none Project the corresponding component of an and type:fst selects the first,snd the second.An imp type cannot stand in for a pair. Step identifiers and dependenciesRule: synthetic_conjunction_projection · Uses: G001, S003 S005 · impliesBefore inferred type=P23;declared claim=P23 After exact structural type match=Yes Conditions: none Compare the inferred type structurally with the claim.Do not fill the inference from the claim.A match belongs only to this synthetic calculus. Step identifiers and dependenciesRule: synthetic_claim_checked_not_assumed · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (P23, P23, Yes) 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: c32cdcccbf7900f81bf711f762f27bf2e3ea614690bdab2797e3cb2b7008febf - candidate record SHA-256: a10dbd0dd1906a6fe6a20b0e03d0150f5c22f05928d0a5a52951bb15484f04be - semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c - question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r07-lean-proof-term-0024.json)