Who Hardened Pharaoh's Heat in Exodus 7.13?

"Pharaoh heard what had happened, and he tried to kill Moses" (Ex. 2.14). Sounds like Pharaoh hardened his heart long before. "I don’t know the LORD, and I will not let Israel go" (Ex. 5.2). "Pharaoh replied, 'Moses and Aaron, why are you distracting the people from their tasks? Get back to work! Look, there are many of your people in the land, and you are stopping them from their work'" (v.4). There is more of the same obstinacy in the rest of Exodus 5.

"Then the LORD told Moses, 'Now you will see what I will do to Pharaoh. When he feels the force of my strong hand, he will let the people go. In fact, he will force them to leave his land!'" (Ex. 6.1) "I will make you seem like God to Pharaoh" (Ex. 7.1). Seems like God is giving Pharaoh the free choice, doing much convincing with these 10 plagues. Pharaoh initially does not let them go, but then later does, only to go after them to try to stop them. Pharaoh was stubborn to the end.

I think Pharaoh trying to slay Moses and not letting the Hebrews go is a prior instance of the Pharaoh hardening his own heart first, don't you think? "Pharaoh’s heart, however, remained hard. He still refused to listen, just as the LORD had predicted" (Ex. 7.13).

John Piper did not include Ex. 7.13 as one of the instances of specifically hardening by either Pharaoh or God. I agree. "Aaron's rod swallowed up their rods, and he [Aaron] hardened Pharaoh's heart, that he hearkened not unto them [Moses and Aaron]; as the LORD had said" (Ex. 7.12,13). The first active instance mentioned specifically, though, of 'hardening' his own heart first was 8.15,32 before God first hardened his heart in 9.12.

If you count the number of instances of the Pharaoh hardened his own heart and God hardening his heart, they are about the same number. So which takes precedence? Pharaoh already seemed quite hardened before the incident of the snakes.

And why must God hardening one's heart mean by irresistibly imposing? Why can't God harden a person's heart by giving the person the free choice, and when that person freely rebels, God hardens their heart? In other words, God takes into account through His foreknowledge all possible responses, and could foresee the necessity to harden the Pharaoh's heart in His design as a response to the Pharaoh's initial choice.