* Abstract * Intro ** Why Gradual Typing ** Why Axiomatic Gradual Typing? ** Why These Axioms? *** beta, eta *** graduality ** Axioms to Implementation ** Inequalities to Observational Approximation * Axioms of GTT * Theorems in GTT * GTT to CBPV Contracts ** Interpreting the Dynamic Types *** Extending GTT ** Contracts as EP Pairs ** High-level Lemmas * Axiomatic CBPV to Operational CBPV ** Thunkability and Linearity ** High-level Statement * Graduality logical relation for CBPV ** Ordering Results ** Step-Indexed Logical Relation ** High-Level statement of Result * Discussion