Assuming unrestricted shared filesystem usage for local execution.
Building DAG of jobs...
Using shell: /usr/bin/bash
Provided cores: 1 (use --cores to define parallelism)
Rules claiming more threads will be scaled down.
Job stats:
job      count
-----  -------
a            1
all          1
b1           1
b2           1
total        4

Select jobs to execute...
Execute 1 jobs...

[Wed Mar  6 10:04:25 2024]
localrule a:
    output: A.txt
    jobid: 3
    reason: Forced execution
    wildcards: name=A
    resources: tmpdir=/tmp

[Wed Mar  6 10:04:25 2024]
Finished job 3.
1 of 4 steps (25%) done
Select jobs to execute...
Execute 1 jobs...

[Wed Mar  6 10:04:25 2024]
localrule b1:
    input: A.txt
    output: A.b1.out
    jobid: 2
    reason: Forced execution
    resources: tmpdir=/tmp

[Wed Mar  6 10:04:25 2024]
Finished job 2.
2 of 4 steps (50%) done
Removing temporary output A.txt.
Select jobs to execute...
Execute 1 jobs...

[Wed Mar  6 10:04:25 2024]
localcheckpoint b2:
    input: A.b1.out
    output: A.b2.out
    jobid: 1
    reason: Forced execution
    resources: tmpdir=/tmp
DAG of jobs will be updated after completion.

[Wed Mar  6 10:04:25 2024]
Finished job 1.
3 of 4 steps (75%) done
Select jobs to execute...
Execute 1 jobs...

[Wed Mar  6 10:04:25 2024]
localrule a1:
    input: A.txt
    output: A.a1.out
    jobid: 6
    reason: Forced execution
    wildcards: name=A
    resources: tmpdir=/tmp

Waiting at most 5 seconds for missing files.
[...]
    raise IOError(
OSError: Missing files after 5 seconds. This might be due to filesystem latency. If that is the case, consider to increase the wait time with --latency-wait:
A.txt

rule a:
    output:
        temp("{name}.txt"),
    shell:
        "touch {output}"

rule a1:
    input:
        "{name}.txt",
    output:
        temp("{name}.a1.out"),
    shell:
        "touch {output}"

rule b1:
    input:
        "A.txt",
    output:
        temp("A.b1.out"),
    shell:
        "touch {output}"

checkpoint b2:
    input:
        "A.b1.out"
    output:
        "A.b2.out"
    shell:
        "echo A > {output}"

def in_func1(w):
    file = checkpoints.b2.get().output[0]
    return open(file, "r").read().strip()

rule all:
    input:
        lambda w: expand(rules.a1.output, name=in_func1(w)),
    default_target: True