ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A pushout along an isomorphism recovers the other group

Example

Let $f:K\to G$ be an isomorphism and $h:K\to H$ any homomorphism. The pushout is $H$ , with legs $h\circ f^{-1}:G\to H$ and $\mathrm{id}_H$ . For example, pushing $C_4\xleftarrow{\cong}C_4\to C_2$ along reduction modulo $2$ gives $C_2$ .

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Given homomorphisms $f:K\to G$ and $h:K\to H$ as in def-group-homomorphism, a pushout is a group $P$ with homomorphisms $i_G:G\to P$ and $i_H:H\to P$ such that $i_G\circ f=i_H\circ h$ , and such that every compatible pair $u:G\to Q$ , $v:H\to Q$ factors through a unique $w:P\to Q$ with $w\circ i_G=u$ and $w\circ i_H=v$ . The maps $f,h$ need not be injective. (Pushouts of group homomorphisms).

[L2]

Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$ . An isomorphism $f:G\to H$ is a bijective group homomorphism (def-group-homomorphism, def-injection-surjection-bijection). When $G=H$ , it is an automorphism of $G$ . Write $\operatorname{Aut}(G):=\{f:G\to G:f\text{ is an automorphism}\}.$ (Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$ ).

[L3]

The inverse of a bijective group homomorphism is a group homomorphism. If $f:G\to H$ is a bijective group homomorphism, then its set-theoretic inverse $f^{-1}:H\to G$ is a group homomorphism. (The inverse of a bijective group homomorphism is a group homomorphism).

[L4]

For every $n\in\mathbb N$ , view $n$ as its canonical nonnegative integer and put $n\mathbb Z:=\{nk:k\in\mathbb Z\}$ . Then the left cosets of $n\mathbb Z$ in $(\mathbb Z,+)$ are exactly the congruence classes modulo $n$ , and coset addition is the published addition of congruence classes. Thus $(\mathbb Z,+)/n\mathbb Z=(\mathbb Z/n,+)$ as the same group on the same underlying set. This includes $n=0$ and $n=1$ . (For every $n\in\mathbb N$ , the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$ ).

Verification

technique · direct

1.1

The two displayed legs agree on $K$ : $(h\circ f^{-1})\circ f=h$ .

givenL1L2L3L4

2.1

For a compatible pair $u:G\to Q$ , $v:H\to Q$ , one has $u=v\circ h\circ f^{-1}$ , so $v$ is the unique mediator from $H$ .

step 1.1

3.1

In the cyclic example this says the $C_4$ leg is reduction modulo $2$ , the $C_2$ leg is the identity, and every compatible cocone factors uniquely through $C_2$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.