diff --git a/configs/mainnet/phase0.yaml b/configs/mainnet/phase0.yaml
index 616baaf34..48a479d80 100644
--- a/configs/mainnet/phase0.yaml
+++ b/configs/mainnet/phase0.yaml
@@ -27,6 +27,8 @@ HYSTERESIS_QUOTIENT: 4
 HYSTERESIS_DOWNWARD_MULTIPLIER: 1
 # 5 (plus 1.25)
 HYSTERESIS_UPWARD_MULTIPLIER: 5
+# 3
+PROPORTIONAL_SLASHING_MULTIPLIER: 3
 
 
 # Fork Choice
diff --git a/configs/minimal/phase0.yaml b/configs/minimal/phase0.yaml
index 78cd60177..02afa369c 100644
--- a/configs/minimal/phase0.yaml
+++ b/configs/minimal/phase0.yaml
@@ -26,7 +26,8 @@ HYSTERESIS_QUOTIENT: 4
 HYSTERESIS_DOWNWARD_MULTIPLIER: 1
 # 5 (plus 1.25)
 HYSTERESIS_UPWARD_MULTIPLIER: 5
-
+# 3
+PROPORTIONAL_SLASHING_MULTIPLIER: 3
 
 
 # Fork Choice
diff --git a/setup.py b/setup.py
index 427f08bc6..fa91aa13c 100644
--- a/setup.py
+++ b/setup.py
@@ -125,6 +125,7 @@ from eth2spec.config.config_util import apply_constants_config
 from typing import (
     Any, Dict, Set, Sequence, NewType, Tuple, TypeVar, Callable, Optional
 )
+from typing import List as PyList
 
 from dataclasses import (
     dataclass,
@@ -152,8 +153,10 @@ GeneralizedIndex = NewType('GeneralizedIndex', int)
 SSZObject = TypeVar('SSZObject', bound=View)
 '''
 SUNDRY_CONSTANTS_FUNCTIONS = '''
-def ceillog2(x: uint64) -> int:
-    return (x - 1).bit_length()
+def ceillog2(x: int) -> uint64:
+    if x < 1:
+        raise ValueError(f"ceillog2 accepts only positive values, x={x}")
+    return uint64((x - 1).bit_length())
 '''
 PHASE0_SUNDRY_FUNCTIONS = '''
 def get_eth1_data(block: Eth1Block) -> Eth1Data:
diff --git a/specs/phase0/beacon-chain.md b/specs/phase0/beacon-chain.md
index ca7b0b858..db0b9a434 100644
--- a/specs/phase0/beacon-chain.md
+++ b/specs/phase0/beacon-chain.md
@@ -195,6 +195,7 @@ The following values are (non-configurable) constants used throughout the specif
 | `HYSTERESIS_QUOTIENT` | `uint64(4)` |
 | `HYSTERESIS_DOWNWARD_MULTIPLIER` | `uint64(1)` |
 | `HYSTERESIS_UPWARD_MULTIPLIER` | `uint64(5)` |
+| `PROPORTIONAL_SLASHING_MULTIPLIER` | `uint64(3)` |
 
 - For the safety of committees, `TARGET_COMMITTEE_SIZE` exceeds [the recommended minimum committee size of 111](http://web.archive.org/web/20190504131341/https://vitalik.ca/files/Ithaca201807_Sharding.pdf); with sufficient active validators (at least `SLOTS_PER_EPOCH * TARGET_COMMITTEE_SIZE`), the shuffling algorithm ensures committee sizes of at least `TARGET_COMMITTEE_SIZE`. (Unbiasable randomness with a Verifiable Delay Function (VDF) will improve committee robustness and lower the safe minimum committee size.)
 
@@ -773,7 +774,7 @@ def compute_committee(indices: Sequence[ValidatorIndex],
     Return the committee corresponding to ``indices``, ``seed``, ``index``, and committee ``count``.
     """
     start = (len(indices) * index) // count
-    end = (len(indices) * (index + 1)) // count
+    end = (len(indices) * uint64(index + 1)) // count
     return [indices[compute_shuffled_index(uint64(i), uint64(len(indices)), seed)] for i in range(start, end)]
 ```
 
@@ -1454,7 +1455,7 @@ def get_inclusion_delay_deltas(state: BeaconState) -> Tuple[Sequence[Gwei], Sequ
             if index in get_attesting_indices(state, a.data, a.aggregation_bits)
         ], key=lambda a: a.inclusion_delay)
         rewards[attestation.proposer_index] += get_proposer_reward(state, index)
-        max_attester_reward = get_base_reward(state, index) - get_proposer_reward(state, index)
+        max_attester_reward = Gwei(get_base_reward(state, index) - get_proposer_reward(state, index))
         rewards[index] += Gwei(max_attester_reward // attestation.inclusion_delay)
 
     # No penalties associated with inclusion delay
@@ -1554,10 +1555,11 @@ def process_registry_updates(state: BeaconState) -> None:
 def process_slashings(state: BeaconState) -> None:
     epoch = get_current_epoch(state)
     total_balance = get_total_active_balance(state)
+    adjusted_total_slashing_balance = min(sum(state.slashings) * PROPORTIONAL_SLASHING_MULTIPLIER, total_balance)
     for index, validator in enumerate(state.validators):
         if validator.slashed and epoch + EPOCHS_PER_SLASHINGS_VECTOR // 2 == validator.withdrawable_epoch:
             increment = EFFECTIVE_BALANCE_INCREMENT  # Factored out from penalty numerator to avoid uint64 overflow
-            penalty_numerator = validator.effective_balance // increment * min(sum(state.slashings) * 3, total_balance)
+            penalty_numerator = validator.effective_balance // increment * adjusted_total_slashing_balance
             penalty = penalty_numerator // total_balance * increment
             decrease_balance(state, ValidatorIndex(index), penalty)
 ```
@@ -1574,7 +1576,7 @@ def process_final_updates(state: BeaconState) -> None:
     # Update effective balances with hysteresis
     for index, validator in enumerate(state.validators):
         balance = state.balances[index]
-        HYSTERESIS_INCREMENT = EFFECTIVE_BALANCE_INCREMENT // HYSTERESIS_QUOTIENT
+        HYSTERESIS_INCREMENT = uint64(EFFECTIVE_BALANCE_INCREMENT // HYSTERESIS_QUOTIENT)
         DOWNWARD_THRESHOLD = HYSTERESIS_INCREMENT * HYSTERESIS_DOWNWARD_MULTIPLIER
         UPWARD_THRESHOLD = HYSTERESIS_INCREMENT * HYSTERESIS_UPWARD_MULTIPLIER
         if (
diff --git a/specs/phase0/p2p-interface.md b/specs/phase0/p2p-interface.md
index a6068cfa1..1ecbc4ec9 100644
--- a/specs/phase0/p2p-interface.md
+++ b/specs/phase0/p2p-interface.md
@@ -395,9 +395,9 @@ The following validations MUST pass before forwarding the `attestation` on the s
 - _[REJECT]_ The attestation's epoch matches its target -- i.e. `attestation.data.target.epoch ==
   compute_epoch_at_slot(attestation.data.slot)`
 - _[REJECT]_ The attestation is unaggregated --
-  that is, it has exactly one participating validator (`len([bit in bit attestation.aggregation_bits if bit]) == 1`, i.e. exactly 1 bit is set).
+  that is, it has exactly one participating validator (`len([bit for bit in attestation.aggregation_bits if bit]) == 1`, i.e. exactly 1 bit is set).
 - _[REJECT]_ The number of aggregation bits matches the committee size -- i.e.
-  `len(attestation.aggregation_bits) == len(get_beacon_committee(state, data.slot, data.index))`
+  `len(attestation.aggregation_bits) == len(get_beacon_committee(state, data.slot, data.index))`.
 - _[IGNORE]_ There has been no other valid attestation seen on an attestation subnet
   that has an identical `attestation.data.target.epoch` and participating validator index.
 - _[REJECT]_ The signature of `attestation` is valid.
@@ -413,7 +413,6 @@ The following validations MUST pass before forwarding the `attestation` on the s
 
 
 
-
 #### Attestations and Aggregation
 
 Attestation broadcasting is grouped into subnets defined by a topic.
diff --git a/specs/phase0/validator.md b/specs/phase0/validator.md
index 86cb8ec1c..95411f1b6 100644
--- a/specs/phase0/validator.md
+++ b/specs/phase0/validator.md
@@ -168,7 +168,7 @@ def get_committee_assignment(state: BeaconState,
         * ``assignment[2]`` is the slot at which the committee is assigned
     Return None if no assignment.
     """
-    next_epoch = get_current_epoch(state) + 1
+    next_epoch = Epoch(get_current_epoch(state) + 1)
     assert epoch <= next_epoch
 
     start_slot = compute_start_slot_at_epoch(epoch)
@@ -460,7 +460,7 @@ def compute_subnet_for_attestation(committees_per_slot: uint64, slot: Slot, comm
     Compute the correct subnet for an attestation for Phase 0.
     Note, this mimics expected Phase 1 behavior where attestations will be mapped to their shard subnet.
     """
-    slots_since_epoch_start = slot % SLOTS_PER_EPOCH
+    slots_since_epoch_start = uint64(slot % SLOTS_PER_EPOCH)
     committees_since_epoch_start = committees_per_slot * slots_since_epoch_start
 
     return uint64((committees_since_epoch_start + committee_index) % ATTESTATION_SUBNET_COUNT)
diff --git a/specs/phase1/beacon-chain.md b/specs/phase1/beacon-chain.md
index 81a8ddcc8..fd6b63d9c 100644
--- a/specs/phase1/beacon-chain.md
+++ b/specs/phase1/beacon-chain.md
@@ -105,20 +105,20 @@ Configuration is not namespaced. Instead it is strictly an extension;
 
 | Name | Value |
 | - | - | 
-| `MAX_SHARDS` | `2**10` (= 1024) |
-| `INITIAL_ACTIVE_SHARDS` | `2**6` (= 64) |
-| `LIGHT_CLIENT_COMMITTEE_SIZE` | `2**7` (= 128) |
-| `GASPRICE_ADJUSTMENT_COEFFICIENT` | `2**3` (= 8) | 
+| `MAX_SHARDS` | `uint64(2**10)` (= 1024) |
+| `INITIAL_ACTIVE_SHARDS` | `uint64(2**6)` (= 64) |
+| `LIGHT_CLIENT_COMMITTEE_SIZE` | `uint64(2**7)` (= 128) |
+| `GASPRICE_ADJUSTMENT_COEFFICIENT` | `uint64(2**3)` (= 8) | 
 
 ### Shard block configs
 
 | Name | Value | Unit |
 | - | - | - |
-| `MAX_SHARD_BLOCK_SIZE` | `2**20` (= 1,048,576) | bytes |
-| `TARGET_SHARD_BLOCK_SIZE` | `2**18` (= 262,144) |  bytes |
-| `SHARD_BLOCK_OFFSETS` | `[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233]` | - |
+| `MAX_SHARD_BLOCK_SIZE` | `uint64(2**20)` (= 1,048,576) | bytes |
+| `TARGET_SHARD_BLOCK_SIZE` | `uint64(2**18)` (= 262,144) |  bytes |
+| `SHARD_BLOCK_OFFSETS` | `List[uint64, 12]([1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233])` | - |
 | `MAX_SHARD_BLOCKS_PER_ATTESTATION` | `len(SHARD_BLOCK_OFFSETS)` | - |
-| `BYTES_PER_CUSTODY_CHUNK` | `2**12` (= 4,096) | bytes |
+| `BYTES_PER_CUSTODY_CHUNK` | `uint64(2**12)` (= 4,096) | bytes |
 | `CUSTODY_RESPONSE_DEPTH` | `ceillog2(MAX_SHARD_BLOCK_SIZE // BYTES_PER_CUSTODY_CHUNK)` | - | 
 
 ### Gwei values
@@ -504,7 +504,7 @@ def compute_committee_source_epoch(epoch: Epoch, period: uint64) -> Epoch:
     """
     Return the source epoch for computing the committee.
     """
-    source_epoch = epoch - epoch % period
+    source_epoch = Epoch(epoch - epoch % period)
     if source_epoch >= period:
         source_epoch -= period  # `period` epochs lookahead
     return source_epoch
@@ -575,7 +575,7 @@ def get_light_client_committee(beacon_state: BeaconState, epoch: Epoch) -> Seque
     return compute_committee(
         indices=active_validator_indices,
         seed=seed,
-        index=0,
+        index=uint64(0),
         count=get_active_shard_count(beacon_state),
     )[:LIGHT_CLIENT_COMMITTEE_SIZE]
 ```
@@ -601,10 +601,10 @@ def get_committee_count_delta(state: BeaconState, start_slot: Slot, stop_slot: S
     """
     Return the sum of committee counts in range ``[start_slot, stop_slot)``.
     """
-    return sum(
+    return uint64(sum(
         get_committee_count_per_slot(state, compute_epoch_at_slot(Slot(slot)))
         for slot in range(start_slot, stop_slot)
-    )
+    ))
 ```
 
 #### `get_start_shard`
diff --git a/specs/phase1/custody-game.md b/specs/phase1/custody-game.md
index 6746c26e2..82eb35d86 100644
--- a/specs/phase1/custody-game.md
+++ b/specs/phase1/custody-game.md
@@ -56,10 +56,10 @@ This document details the beacon chain additions and changes in Phase 1 of Ether
 
 | Name | Value | Unit |
 | - | - | - |
-| `CUSTODY_PRIME` | `2 ** 256 - 189` | - |
-| `CUSTODY_SECRETS` | `3` | - |
-| `BYTES_PER_CUSTODY_ATOM` | `32` | bytes |
-| `CUSTODY_PROBABILITY_EXPONENT` | `10` | - |
+| `CUSTODY_PRIME` | `int(2 ** 256 - 189)` | - |
+| `CUSTODY_SECRETS` | `uint64(3)` | - |
+| `BYTES_PER_CUSTODY_ATOM` | `uint64(32)` | bytes |
+| `CUSTODY_PROBABILITY_EXPONENT` | `uint64(10)` | - |
 
 ## Configuration
 
@@ -67,29 +67,29 @@ This document details the beacon chain additions and changes in Phase 1 of Ether
 
 | Name | Value | Unit | Duration |
 | - | - | :-: | :-: |
-| `RANDAO_PENALTY_EPOCHS` | `2**1` (= 2) | epochs | 12.8 minutes |
-| `EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS` | `2**15` (= 32,768) | epochs | ~146 days |
-| `EPOCHS_PER_CUSTODY_PERIOD` | `2**14` (= 16,384) | epochs | ~73 days |
-| `CUSTODY_PERIOD_TO_RANDAO_PADDING` | `2**11` (= 2,048) | epochs | ~9 days |
-| `MAX_CHUNK_CHALLENGE_DELAY` | `2**15` (= 32,768) | epochs | ~146 days |
+| `RANDAO_PENALTY_EPOCHS` | `uint64(2**1)` (= 2) | epochs | 12.8 minutes |
+| `EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS` | `uint64(2**15)` (= 32,768) | epochs | ~146 days |
+| `EPOCHS_PER_CUSTODY_PERIOD` | `uint64(2**14)` (= 16,384) | epochs | ~73 days |
+| `CUSTODY_PERIOD_TO_RANDAO_PADDING` | `uint64(2**11)` (= 2,048) | epochs | ~9 days |
+| `MAX_CHUNK_CHALLENGE_DELAY` | `uint64(2**15)` (= 32,768) | epochs | ~146 days |
 
 ### Max operations per block
 
 | Name | Value |
 | - | - |
-| `MAX_CUSTODY_CHUNK_CHALLENGE_RECORDS` | `2**20` (= 1,048,576) |
-| `MAX_CUSTODY_KEY_REVEALS` | `2**8` (= 256) |
-| `MAX_EARLY_DERIVED_SECRET_REVEALS` | `2**0` (= 1) |
-| `MAX_CUSTODY_CHUNK_CHALLENGES` | `2**2` (= 4) |
-| `MAX_CUSTODY_CHUNK_CHALLENGE_RESPONSES` | `2**4` (= 16) |
-| `MAX_CUSTODY_SLASHINGS` | `2**0` (= 1) |
+| `MAX_CUSTODY_CHUNK_CHALLENGE_RECORDS` | `uint64(2**20)` (= 1,048,576) |
+| `MAX_CUSTODY_KEY_REVEALS` | `uint64(2**8)` (= 256) |
+| `MAX_EARLY_DERIVED_SECRET_REVEALS` | `uint64(2**0)` (= 1) |
+| `MAX_CUSTODY_CHUNK_CHALLENGES` | `uint64(2**2)` (= 4) |
+| `MAX_CUSTODY_CHUNK_CHALLENGE_RESPONSES` | `uint64(2**4)` (= 16) |
+| `MAX_CUSTODY_SLASHINGS` | `uint64(2**0)` (= 1) |
 
 ### Reward and penalty quotients
 
 | Name | Value |
 | - | - |
-| `EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE` | `2**1` (= 2) |
-| `MINOR_REWARD_QUOTIENT` | `2**8` (= 256) |
+| `EARLY_DERIVED_SECRET_REVEAL_SLOT_REWARD_MULTIPLE` | `uint64(2**1)` (= 2) |
+| `MINOR_REWARD_QUOTIENT` | `uint64(2**8)` (= 256) |
 
 ## Data structures
 
@@ -265,12 +265,12 @@ def universal_hash_function(data_chunks: Sequence[bytes], secrets: Sequence[int]
 ### `compute_custody_bit`
 
 ```python
-def compute_custody_bit(key: BLSSignature, data: ByteList[MAX_SHARD_BLOCK_SIZE]) -> bit:
+def compute_custody_bit(key: BLSSignature, data: ByteList) -> bit:
     custody_atoms = get_custody_atoms(data)
     secrets = get_custody_secrets(key)
     uhf = universal_hash_function(custody_atoms, secrets)
     legendre_bits = [legendre_bit(uhf + secrets[0] + i, CUSTODY_PRIME) for i in range(CUSTODY_PROBABILITY_EXPONENT)]
-    return all(legendre_bits)
+    return bit(all(legendre_bits))
 ```
 
 ### `get_randao_epoch_for_custody_period`
@@ -316,7 +316,7 @@ def process_chunk_challenge(state: BeaconState, challenge: CustodyChunkChallenge
     # Verify the attestation
     assert is_valid_indexed_attestation(state, get_indexed_attestation(state, challenge.attestation))
     # Verify it is not too late to challenge the attestation
-    max_attestation_challenge_epoch = challenge.attestation.data.target.epoch + MAX_CHUNK_CHALLENGE_DELAY
+    max_attestation_challenge_epoch = Epoch(challenge.attestation.data.target.epoch + MAX_CHUNK_CHALLENGE_DELAY)
     assert get_current_epoch(state) <= max_attestation_challenge_epoch
     # Verify it is not too late to challenge the responder
     responder = state.validators[challenge.responder_index]
@@ -438,7 +438,7 @@ def process_early_derived_secret_reveal(state: BeaconState, reveal: EarlyDerived
     Note that this function mutates ``state``.
     """
     revealed_validator = state.validators[reveal.revealed_index]
-    derived_secret_location = reveal.epoch % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
+    derived_secret_location = uint64(reveal.epoch % EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS)
 
     assert reveal.epoch >= get_current_epoch(state) + RANDAO_PENALTY_EPOCHS
     assert reveal.epoch < get_current_epoch(state) + EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS
diff --git a/specs/phase1/validator.md b/specs/phase1/validator.md
index e6f01177b..2bfea017e 100644
--- a/specs/phase1/validator.md
+++ b/specs/phase1/validator.md
@@ -281,9 +281,9 @@ def get_shard_transition_fields(
     shard: Shard,
     shard_blocks: Sequence[SignedShardBlock],
 ) -> Tuple[Sequence[uint64], Sequence[Root], Sequence[ShardState]]:
-    shard_states = []
-    shard_data_roots = []
-    shard_block_lengths = []
+    shard_block_lengths = []  # type: PyList[uint64]
+    shard_data_roots = []  # type: PyList[Root]
+    shard_states = []  # type: PyList[ShardState]
 
     shard_state = beacon_state.shard_states[shard]
     shard_block_slots = [shard_block.message.slot for shard_block in shard_blocks]
@@ -301,7 +301,7 @@ def get_shard_transition_fields(
         shard_state = shard_state.copy()
         process_shard_block(shard_state, shard_block.message)
         shard_states.append(shard_state)
-        shard_block_lengths.append(len(shard_block.message.body))
+        shard_block_lengths.append(uint64(len(shard_block.message.body)))
 
     return shard_block_lengths, shard_data_roots, shard_states
 ```
diff --git a/ssz/simple-serialize.md b/ssz/simple-serialize.md
index 4f0c5c9fa..607dd0946 100644
--- a/ssz/simple-serialize.md
+++ b/ssz/simple-serialize.md
@@ -165,7 +165,7 @@ variable_lengths = [len(part) for part in variable_parts]
 assert sum(fixed_lengths + variable_lengths) < 2**(BYTES_PER_LENGTH_OFFSET * BITS_PER_BYTE)
 
 # Interleave offsets of variable-size parts with fixed-size parts
-variable_offsets = [serialize(sum(fixed_lengths + variable_lengths[:i])) for i in range(len(value))]
+variable_offsets = [serialize(uint32(sum(fixed_lengths + variable_lengths[:i]))) for i in range(len(value))]
 fixed_parts = [part if part != None else variable_offsets[i] for i, part in enumerate(fixed_parts)]
 
 # Return the concatenation of the fixed-size parts (offsets interleaved) with the variable-size parts