Section Bounds.
Variables variable function constructor : Type.
Variable variable_eq_dec : forall (x1 x2 : variable), {x1=x2}+{x1<>x2}.
Variable function_eq_dec : forall (x1 x2 : function), {x1=x2}+{x1<>x2}.
Variable constructor_eq_dec : forall (x1 x2 : constructor), {x1=x2}+{x1<>x2}.
Variable max_arity : nat.
Notation rule := (Syntax.rule variable function constructor).
Notation term := (Syntax.term variable function constructor).
Notation value := (Syntax.value constructor).
Notation cbv := (CBV_cache.cbv variable function constructor).
Variable prog : list rule.
Variable rule_default : rule.
Variable subst_default : variable -> value.
Notation assoc_cache := (assoc (term_beq variable_eq_dec function_eq_dec constructor_eq_dec)).
Notation wf :=
(CBV_cache.wf variable_eq_dec function_eq_dec constructor_eq_dec rule_default prog max_arity).
Variable rank: function -> nat.
Variable max_rank : nat.
Hypothesis pos_max_rank : 0 < max_rank.
Hypothesis max_rank_is_max_rank : forall f, rank f <= max_rank.
Notation PPO_prog := (Ordering.PPO_prog prog rank).
(* Proposition 7 *)
Definition activations_at_rank (rk: nat) (p : cbv) : list cbv :=
filter (fun p => beq_nat rk (activation_rank rank p)) (activations p).
(* Computes A_T, with a given maximal *)
Definition A_T (proof_tree: cbv) : nat :=
maxl (map (fun p => length (activations_at_rank (activation_rank rank p) p)) (activations proof_tree)).
(* B_i, with i = rank *)
Definition activations_at_rank_bound (A_T: nat) (rank: nat) :=
Nat.pow max_rank (max_rank - rank) *
Nat.pow (max 1 (max_nb_rhs_functions prog)) (max_rank - rank) *
Nat.pow A_T (S (max_rank - rank)).
Definition gary_seven_poly (A_T:nat) :=
suml (map (activations_at_rank_bound A_T) (seq 0 (S max_rank))).

Lemma PPO_activation_rank_decreasing i s p c1 f lt c2 v:
let proof_tree := cbv_update i s p c1 (fapply f lt) c2 v in
PPO_prog ->
wf proof_tree ->
forall p, In p (activations proof_tree) ->
activation_rank rank p <= rank f.