diff --git a/specs/core/0_beacon-chain.md b/specs/core/0_beacon-chain.md
index 9a54822d3..3bc95f717 100644
--- a/specs/core/0_beacon-chain.md
+++ b/specs/core/0_beacon-chain.md
@@ -815,8 +815,12 @@ def get_permuted_index(index: int, list_size: int, seed: Bytes32) -> int:
 ### `get_split_offset`
 
 ```python
-def get_split_offset(list_length: int, split_count: int, index: int) -> int:
-    return (list_length * index) // split_count
+def get_split_offset(list_size: int, chunks: int, index: int) -> int:
+  """
+  Returns a value such that for a list L, chunk count k and index i,
+  split(L, k)[i] == L[get_split_offset(len(L), k, i): get_split_offset(len(L), k, i+1)]
+  """
+  return (list_size * index) // chunks
 ```
 
 ### `get_epoch_committee_count`
diff --git a/specs/core/1_shard-data-chains.md b/specs/core/1_shard-data-chains.md
index c76f9ba08..92cee4d19 100644
--- a/specs/core/1_shard-data-chains.md
+++ b/specs/core/1_shard-data-chains.md
@@ -19,7 +19,6 @@ At the current stage, Phase 1, while fundamentally feature-complete, is still su
             - [Signature domains](#signature-domains)
 - [Shard chains and crosslink data](#shard-chains-and-crosslink-data)
     - [Helper functions](#helper-functions)
-            - [`get_split_offset`](#get_split_offset)
             - [`get_shuffled_committee`](#get_shuffled_committee)
             - [`get_persistent_committee`](#get_persistent_committee)
             - [`get_shard_proposer_index`](#get_shard_proposer_index)
@@ -122,17 +121,6 @@ Phase 1 depends upon all of the constants defined in [Phase 0](0_beacon-chain.md
 
 ## Helper functions
 
-#### `get_split_offset`
-
-````python
-def get_split_offset(list_size: int, chunks: int, index: int) -> int:
-  """
-  Returns a value such that for a list L, chunk count k and index i,
-  split(L, k)[i] == L[get_split_offset(len(L), k, i): get_split_offset(len(L), k, i+1)]
-  """
-  return (list_size * index) // chunks
-````
-
 #### `get_shuffled_committee`
 
 ```python
diff --git a/tests/phase0/block_processing/test_process_block_header.py b/tests/phase0/block_processing/test_process_block_header.py
index 4ec7e336f..4981b656c 100644
--- a/tests/phase0/block_processing/test_process_block_header.py
+++ b/tests/phase0/block_processing/test_process_block_header.py
@@ -4,6 +4,8 @@ import pytest
 
 from build.phase0.spec import (
     get_beacon_proposer_index,
+    cache_state,
+    advance_slot,
     process_block_header,
 )
 from tests.phase0.helpers import (
@@ -14,13 +16,56 @@ from tests.phase0.helpers import (
 pytestmark = pytest.mark.header
 
 
+def prepare_state_for_header_processing(state):
+    cache_state(state)
+    advance_slot(state)
+
+
+def run_block_header_processing(state, block, valid=True):
+    """
+    Run ``process_block_header`` returning the pre and post state.
+    If ``valid == False``, run expecting ``AssertionError``
+    """
+    prepare_state_for_header_processing(state)
+    post_state = deepcopy(state)
+
+    if not valid:
+        with pytest.raises(AssertionError):
+            process_block_header(post_state, block)
+        return state, None
+
+    process_block_header(post_state, block)
+    return state, post_state
+
+
+def test_success(state):
+    block = build_empty_block_for_next_slot(state)
+    pre_state, post_state = run_block_header_processing(state, block)
+    return state, block, post_state
+
+
+def test_invalid_slot(state):
+    block = build_empty_block_for_next_slot(state)
+    block.slot = state.slot + 2  # invalid slot
+
+    pre_state, post_state = run_block_header_processing(state, block, valid=False)
+    return pre_state, block, None
+
+
+def test_invalid_previous_block_root(state):
+    block = build_empty_block_for_next_slot(state)
+    block.previous_block_root = b'\12'*32  # invalid prev root
+
+    pre_state, post_state = run_block_header_processing(state, block, valid=False)
+    return pre_state, block, None
+
+
 def test_proposer_slashed(state):
-    pre_state = deepcopy(state)
+    # set proposer to slashed
+    proposer_index = get_beacon_proposer_index(state, state.slot + 1)
+    state.validator_registry[proposer_index].slashed = True
 
-    block = build_empty_block_for_next_slot(pre_state)
-    proposer_index = get_beacon_proposer_index(pre_state, block.slot)
-    pre_state.validator_registry[proposer_index].slashed = True
-    with pytest.raises(AssertionError):
-        process_block_header(pre_state, block)
+    block = build_empty_block_for_next_slot(state)
 
-    return state, [block], None
+    pre_state, post_state = run_block_header_processing(state, block, valid=False)
+    return pre_state, block, None