Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
fact stringlengths 23 2.8k | type stringclasses 15
values | library stringclasses 3
values | imports listlengths 1 5 | filename stringclasses 23
values | symbolic_name stringlengths 1 55 | docstring stringclasses 1
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door : Type := left | right. | Inductive | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | door | |
door_eq_dec (d d' : door) : { d = d' } + { ~ d = d' } := ltac:(decide equality). | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | door_eq_dec | |
DOORS : interface := | IsOpen : door -> DOORS bool | Toggle : door -> DOORS unit. Generalizable All Variables. | Inductive | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | DOORS | |
is_open `{Provide ix DOORS} (d : door) : impure ix bool := request (IsOpen d). | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | is_open | |
toggle `{Provide ix DOORS} (d : door) : impure ix unit := request (Toggle d). | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | toggle | |
open_door `{Provide ix DOORS} (d : door) : impure ix unit := let* open := is_open d in when (negb open) (toggle d). | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | open_door | |
close_door `{Provide ix DOORS} (d : door) : impure ix unit := let* open := is_open d in when open (toggle d). (** ** Controller *) | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | close_door | |
CONTROLLER : interface := | Tick : CONTROLLER unit | RequestOpen (d : door) : CONTROLLER unit. | Inductive | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | CONTROLLER | |
tick `{Provide ix CONTROLLER} : impure ix unit := request Tick. | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | tick | |
request_open `{Provide ix CONTROLLER} (d : door) : impure ix unit := request (RequestOpen d). | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | request_open | |
co (d : door) : door := match d with | left => right | right => left end. | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | co | |
controller `{Provide ix DOORS, Provide ix (STORE nat)} : component CONTROLLER ix := fun _ op => match op with | Tick => let* cpt := get in when (15 <? cpt) begin close_door left;; close_door right;; put 0 end | RequestOpen d => close_door (co d);; open_door d;; put 0 end. (** * Verifying the Airlock Controller *) (** *... | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | controller | |
Ω : Type := bool * bool. | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | Ω | |
sel (d : door) : Ω -> bool := match d with | left => fst | right => snd end. | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | sel | |
tog (d : door) (ω : Ω) : Ω := match d with | left => (negb (fst ω), snd ω) | right => (fst ω, negb (snd ω)) end. | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | tog | |
tog_equ_1 (d : door) (ω : Ω) : sel d (tog d ω) = negb (sel d ω). Proof. destruct d; reflexivity. Qed. | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | tog_equ_1 | |
tog_equ_2 (d : door) (ω : Ω) : sel (co d) (tog d ω) = sel (co d) ω. Proof. destruct d; reflexivity. Qed. (** From now on, we will reason about [tog] using [tog_equ_1] and [tog_equ_2]. FreeSpec tactics rely heavily on [cbn] to simplify certain terms, so we use the <<simpl never>> options of the [Arguments] vernacular co... | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | tog_equ_2 | |
step (ω : Ω) (a : Type) (e : DOORS a) (x : a) := match e with | Toggle d => tog d ω | _ => ω end. (** *** Requirements *) | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | step | |
doors_o_caller : Ω -> forall (a : Type), DOORS a -> Prop := (** - Given the door [d] of o system [ω], it is always possible to ask for the state of [d]. *) | req_is_open (d : door) (ω : Ω) : doors_o_caller ω bool (IsOpen d) (** - Given the door [d] of o system [ω], if [d] is closed, then the second door [co d] has to b... | Inductive | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | doors_o_caller | |
doors_o_callee : Ω -> forall (a : Type), DOORS a -> a -> Prop := (** - When a system in a state [ω] reports the state of the door [d], it shall reflect the true state of [d]. *) | doors_o_callee_is_open (d : door) (ω : Ω) (x : bool) (equ : sel d ω = x) : doors_o_callee ω bool (IsOpen d) x (** - There is no particular d... | Inductive | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | doors_o_callee | |
doors_contract : contract DOORS Ω := make_contract step doors_o_caller doors_o_callee. (** ** Intermediary Lemmas *) (** Closing a door [d] in any system [ω] is always a respectful operation. *) | Definition | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | doors_contract | |
close_door_respectful `{Provide ix DOORS} (ω : Ω) (d : door) : pre (to_hoare doors_contract (close_door d)) ω. Proof. (* We use the [prove_program] tactics to erase the program monad *) prove impure with airlock; subst; constructor. (* This leaves us with one goal to prove: [sel d ω = false -> sel (co d) ω = false] Yet... | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | close_door_respectful | |
open_door_respectful `{Provide ix DOORS} (ω : Ω) (d : door) (safe : sel (co d) ω = false) : pre (to_hoare doors_contract (open_door (ix := ix) d)) ω. Proof. prove impure; repeat constructor; subst. inversion o_caller0; ssubst. now rewrite safe. Qed. #[global] Hint Resolve open_door_respectful : airlock. | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | open_door_respectful | |
close_door_run `{Provide ix DOORS} (ω : Ω) (d : door) (ω' : Ω) (x : unit) (run : post (to_hoare doors_contract (close_door d)) ω x ω') : sel d ω' = false. Proof. unroll_post run. + rewrite tog_equ_1. inversion H1; ssubst. now rewrite H5. + now inversion H1; ssubst. Qed. #[global] Hint Resolve close_door_run : airlock. ... | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | close_door_run | |
one_door_safe_all_doors_safe (ω : Ω) (d : door) (safe : sel d ω = false \/ sel (co d) ω = false) : forall (d' : door), sel d' ω = false \/ sel (co d') ω = false. Proof. intros d'. destruct d; destruct d'; auto. + cbn -[sel]. now rewrite or_comm. + cbn -[sel]. fold (co right). now rewrite or_comm. Qed. (** The objective... | Remark | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | one_door_safe_all_doors_safe | |
respectful_run_inv `{Provide ix DOORS} {a} (p : impure ix a) (ω : Ω) (safe : sel left ω = false \/ sel right ω = false) (x : a) (ω' : Ω) (hpre : pre (to_hoare doors_contract p) ω) (hpost : post (to_hoare doors_contract p) ω x ω') : sel left ω' = false \/ sel right ω' = false. (** We reason by induction on the impure co... | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | respectful_run_inv | |
controller_correct `{StrictProvide2 ix DOORS (STORE nat)} : correct_component controller (no_contract CONTROLLER) doors_contract (fun _ ω => sel left ω = false \/ sel right ω = false). Proof. intros ωc ωd pred a e req. assert (hpre : pre (to_hoare doors_contract (controller a e)) ωd) by (destruct e; prove impure with a... | Lemma | examples | [
"From Coq Require Import Arith.",
"From FreeSpec.Core Require Import Core CoreFacts."
] | examples/airlock.v | controller_correct | |
with_heap `{Monad m, MonadRefs m} : m string := let* ptr := make_ref 2 in assign ptr 3;; let* x := deref ptr in if Nat.eqb x 2 then pure "yes!" else pure "no!". (* Coq projects the [with_heap] polymorphic definition directly into [impure], thanks to its typeclass inference algorithm. *) | Definition | examples | [
"From FreeSpec.Core Require Import Core Extraction.",
"From FreeSpec.FFI Require Import FFI Refs.",
"From FreeSpec.Exec Require Import Exec.",
"From Coq Require Import String."
] | examples/heap.v | with_heap | |
with_heap_impure `{Provide ix REFS} : impure ix string := with_heap. Exec with_heap_impure. | Definition | examples | [
"From FreeSpec.Core Require Import Core Extraction.",
"From FreeSpec.FFI Require Import FFI Refs.",
"From FreeSpec.Exec Require Import Exec.",
"From Coq Require Import String."
] | examples/heap.v | with_heap_impure | |
MEMORY : interface := | ReadFrom (addr : address) : MEMORY cell | WriteTo (addr : address) (value : cell) : MEMORY unit. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | MEMORY | |
DRAM : interface := | MakeDRAM {a : Type} (e : MEMORY a) : DRAM a. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | DRAM | |
dram_read_from `{Provide ix DRAM} (addr : address) : impure ix cell := request (MakeDRAM (ReadFrom addr)). | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | dram_read_from | |
dram_write_to `{Provide ix DRAM} (addr : address) (val : cell) : impure ix unit := request (MakeDRAM (WriteTo addr val)). | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | dram_write_to | |
VGA : interface := | MakeVGA {a : Type} (e : MEMORY a) : VGA a. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | VGA | |
vga_read_from `{Provide ix VGA} (addr : address) : impure ix cell := request (MakeVGA (ReadFrom addr)). | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | vga_read_from | |
vga_write_to `{Provide ix VGA} (addr : address) (val : cell) : impure ix unit := request (MakeVGA (WriteTo addr val)). (** ** Memory Controller *) | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | vga_write_to | |
smiact := smiact_set | smiact_unset. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | smiact | |
MEMORY_CONTROLLER : interface := | Read (pin : smiact) (addr : address) : MEMORY_CONTROLLER cell | Write (pin : smiact) (addr : address) (value : cell) : MEMORY_CONTROLLER unit. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | MEMORY_CONTROLLER | |
unwrap_sumbool {A B} (x : { A } + { B }) : bool := if x then true else false. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | unwrap_sumbool | |
unwrap_sumbool : sumbool >-> bool. | Coercion | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | unwrap_sumbool | |
dispatch {a} `{Provide3 ix (STORE bool) DRAM VGA} (addr : address) (unpriv : address -> impure ix a) (priv : address -> impure ix a) : impure ix a := let* reg := get in if (andb reg (in_smram addr)) then unpriv addr else priv addr. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | dispatch | |
memory_controller `{Provide3 ix (STORE bool) DRAM VGA} : component MEMORY_CONTROLLER ix := fun _ op => match op with (** When SMIACT is set, the CPU is in SMM. According to its specification, the Memory Controller can simply forward the memory access to the DRAM. *) | Read smiact_set addr => dram_read_from addr | Write... | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | memory_controller | |
memory_view : Type := address -> cell. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | memory_view | |
update_memory_view_address (ω : memory_view) (addr : address) (content : cell) : memory_view := fun (addr' : address) => if address_eq_dec addr addr' then content else ω addr'. (** ** Specification *) (** *** DRAM Controller Specification *) | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | update_memory_view_address | |
update_dram_view (ω : memory_view) (a : Type) (p : DRAM a) (_ : a) : memory_view := match p with | MakeDRAM (WriteTo a v) => update_memory_view_address ω a v | _ => ω end. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | update_dram_view | |
dram_o_callee (ω : memory_view) : forall (a : Type), DRAM a -> a -> Prop := | read_in_smram (a : address) (v : cell) (prom : in_smram a = true -> v = ω a) : dram_o_callee ω cell (MakeDRAM (ReadFrom a)) (ω a) | write (a : address) (v : cell) (r : unit) : dram_o_callee ω unit (MakeDRAM (WriteTo a v)) r. | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | dram_o_callee | |
dram_specs : contract DRAM memory_view := {| witness_update := update_dram_view ; caller_obligation := no_caller_obligation ; callee_obligation := dram_o_callee |}. (** *** Memory Controller Specification *) | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | dram_specs | |
update_memory_controller_view (ω : memory_view) (a : Type) (p : MEMORY_CONTROLLER a) (_ : a) : memory_view := match p with | Write smiact_set a v => update_memory_view_address ω a v | _ => ω end. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | update_memory_controller_view | |
memory_controller_o_callee (ω : memory_view) : forall (a : Type) (p : MEMORY_CONTROLLER a) (x : a), Prop := | memory_controller_read_o_callee (pin : smiact) (addr : address) (content : cell) (prom : pin = smiact_set -> in_smram addr = true -> ω addr = content) : memory_controller_o_callee ω cell (Read pin addr) content... | Inductive | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | memory_controller_o_callee | |
mc_specs : contract MEMORY_CONTROLLER memory_view := {| witness_update := update_memory_controller_view ; caller_obligation := no_caller_obligation ; callee_obligation := memory_controller_o_callee |}. (** ** Main Theorem *) | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | mc_specs | |
smram_pred (ωmc : memory_view) (ωmem : memory_view * bool) : Prop := snd ωmem = true /\ forall (a : address), in_smram a = true -> ωmc a = (fst ωmem) a. | Definition | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | smram_pred | |
memory_controller_respectful `{StrictProvide3 ix (STORE bool) VGA DRAM} (a : Type) (op : MEMORY_CONTROLLER a) (ω : memory_view) : pre (to_hoare (dram_specs * store_specs bool) (memory_controller a op)) (ω, true). Proof. destruct op; destruct pin; prove impure. Qed. #[local] Open Scope semantics_scope. #[local] Open Sco... | Lemma | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | memory_controller_respectful | |
simpl_tuple := match goal with | H: (_, _) = (_, _) |- _ => inversion H; subst; clear H end. | Ltac | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | simpl_tuple | |
memory_controller_correct `{StrictProvide3 ix VGA (STORE bool) DRAM} (ω : memory_view) (sem : semantics ix) (comp : compliant_semantics (dram_specs * store_specs bool) (ω, true) sem) : compliant_semantics mc_specs ω (derive_semantics memory_controller sem). Proof. apply correct_component_derives_compliant_semantics wit... | Theorem | examples | [
"From Coq Require Import ZArith.",
"From FreeSpec Require Import Core CoreFacts."
] | examples/smram.v | memory_controller_correct | |
COUNTER : interface := | Get : COUNTER nat | Inc : COUNTER unit | Dec : COUNTER unit. | Inductive | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | COUNTER | |
counter_get `{Provide ix COUNTER} : impure ix nat := request Get. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_get | |
counter_inc `{Provide ix COUNTER} : impure ix unit := request Inc. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_inc | |
counter_dec `{Provide ix COUNTER} : impure ix unit := request Dec. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_dec | |
repeat {m : Type -> Type} `{Monad m} {a} (n : nat) (c : m a) : m unit := match n with | O => pure tt | S n => (c >>= fun _ => repeat n c) end. | Fixpoint | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | repeat | |
update_counter (x : nat) : forall (a : Type), COUNTER a -> a -> nat := fun (a : Type) (p : COUNTER a) (_ : a) => match p with | Inc => S x | Dec => Nat.pred x | _ => x end. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | update_counter | |
counter_o_caller (x : nat) : forall (a : Type), COUNTER a -> Prop := fun (a : Type) (p : COUNTER a) => match p with | Dec => 0 < x | _ => True end. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_o_caller | |
counter_o_callee (x : nat) : forall (a : Type), COUNTER a -> a -> Prop := fun (a : Type) (p : COUNTER a) (r : a) => match p, r with | Get, r => r = x | _, _ => True end. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_o_callee | |
counter_specs : contract COUNTER nat := {| witness_update := update_counter ; caller_obligation := counter_o_caller ; callee_obligation := counter_o_callee |}. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | counter_specs | |
dec_then_inc `{Provide ix COUNTER} (x y : nat) : impure ix nat := repeat x counter_dec;; repeat y counter_inc;; counter_get. | Definition | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | dec_then_inc | |
dec_then_inc_respectful `{Provide ix COUNTER} (cpt x y : nat) (init_cpt : x < cpt) : pre (to_hoare counter_specs $ dec_then_inc x y) cpt. Proof. prove impure. + revert x cpt init_cpt. induction x; intros cpt init_cpt; prove impure. ++ cbn. transitivity (S x); auto. apply PeanoNat.Nat.lt_0_succ. ++ apply IHx. now apply ... | Theorem | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | dec_then_inc_respectful | |
repeat_dec_cpt_output `(run : post (to_hoare counter_specs $ repeat x counter_dec) cpt r cpt') (init_cpt : x < cpt) : cpt' = cpt - x. Proof. revert init_cpt run; revert cpt; induction x; intros cpt init_cpt run. + unroll_post run. now rewrite PeanoNat.Nat.sub_0_r. + unroll_post run. apply IHx in run; [| lia]. subst. li... | Lemma | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | repeat_dec_cpt_output | |
repeat_inc_cpt_output `(run : post (to_hoare counter_specs $ repeat x counter_inc) cpt r cpt') : cpt' = cpt + x. Proof. revert run; revert cpt; induction x; intros cpt run. + unroll_post run. now rewrite PeanoNat.Nat.add_0_r. + unroll_post run. apply IHx in run. lia. Qed. #[local] Hint Resolve repeat_inc_cpt_output : c... | Lemma | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | repeat_inc_cpt_output | |
get_cpt_output (cpt x cpt' : nat) (run : post (to_hoare counter_specs $ counter_get) cpt x cpt') : cpt' = cpt. Proof. now unroll_post run. Qed. #[local] Hint Resolve get_cpt_output : counter. #[local] Opaque counter_get. | Lemma | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | get_cpt_output | |
dec_then_inc_cpt_output (cpt x y cpt' r : nat) (init_cpt : x < cpt) (run : post (to_hoare counter_specs $ dec_then_inc x y) cpt r cpt') : cpt' = cpt - x + y. Proof. unroll_post run. apply repeat_dec_cpt_output in run0; [| exact init_cpt ]. apply repeat_inc_cpt_output in run. apply get_cpt_output in run2. lia. Qed. #[lo... | Theorem | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | dec_then_inc_cpt_output | |
dec_then_inc_output (cpt x y cpt' r : nat) (init_cpt : x < cpt) (run : post (to_hoare counter_specs $ dec_then_inc x y) cpt r cpt') : cpt' = r. Proof. now unroll_post run. Qed. | Theorem | tests | [
"From FreeSpec.Core Require Import CoreFacts.",
"From Coq Require Import Lia."
] | tests/core_tactics.v | dec_then_inc_output | |
enum {a b ix} (p : a -> impure ix b) (l : list a) {measure (length l)} : impure ix unit := match l with | nil => pure tt | cons x rst => p x;; enum p rst end. Fail Timeout 1 Exec (enum eval [true; true; false]). (** We have provided an attribute for [Exec] which slightly changes the behavior of the command (see the doc... | Fixpoint | tests | [
"From FreeSpec.Exec Require Import Exec Eval.",
"From Coq.Program Require Import Wf.",
"From Coq Require Import List."
] | tests/program_fixpoint.v | enum | |
p1 : forall `{Provide ix}, impure ix nat. | Axiom | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | p1 | |
p2 : forall `{Provide2 ix i3 i1}, impure ix nat. | Axiom | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | p2 | |
p `{Provide4 ix i1 i4 i3 i2} : impure ix nat := p1;; p2. | Definition | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | p | |
provide_notation_test_1 {a} `{StrictProvide3 ix i1 i2 i3} (p : i2 a) : ix a := inj_p p. | Definition | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | provide_notation_test_1 | |
provide_notation_test_2 `{StrictProvide3 ix i1 i2 i3} : StrictProvide2 ix i2 i3. Proof. typeclasses eauto. Qed. | Lemma | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | provide_notation_test_2 | |
provide_notation_test_3 {a} (i1 i2 i3 : interface) (p : i2 a) : (iplus (iplus i1 i2) i3) a := inj_p p. | Definition | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | provide_notation_test_3 | |
provide_notation_test_4 (i1 i2 i3 : interface) : Provide (i1 + (i2 + i3)) i2. Proof. typeclasses eauto. Qed. | Lemma | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | provide_notation_test_4 | |
provide_notation_test_5 (i1 i2 i3 : interface) : StrictProvide2 (i1 + i2 + i3) i2 i1. Proof. typeclasses eauto. Qed. | Lemma | tests | [
"From FreeSpec.Core Require Import Core."
] | tests/provide_notation.v | provide_notation_test_5 | |
component (i j : interface) : Type := forall (α : Type), i α -> impure j α. (** The similarity between FreeSpec components and operational semantics may be confusing at first. The main difference between the two concepts is simple: operational semantics are self-contained terms which can, alone, be used to interpret im... | Definition | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Export Interface Semantics Impure."
] | theories/Core/Component.v | component | |
bootstrap {i} (c : component i iempty) : semantics i := derive_semantics c iempty_semantics. (** * In-place Primitives Handling *) (** The function [with_component] allows for locally providing an additional interface [j] within an impure computation of type [impure ix a]. The primitives of [j] will be handled by impur... | Definition | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Export Interface Semantics Impure."
] | theories/Core/Component.v | bootstrap | |
with_component_aux {ix j α} (c : component j ix) (p : impure (ix + j) α) : impure ix α := match p with | local x => local x | request_then (in_right e) f => c _ e >>= fun res => with_component_aux c (f res) | request_then (in_left e) f => request_then e (fun x => with_component_aux c (f x)) end. | Fixpoint | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Export Interface Semantics Impure."
] | theories/Core/Component.v | with_component_aux | |
with_component {ix j α} (initializer : impure ix unit) (c : component j ix) (finalizer : impure ix unit) (p : impure (ix + j) α) : impure ix α := initializer;; let* res := with_component_aux c p in finalizer;; pure res. | Definition | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Export Interface Semantics Impure."
] | theories/Core/Component.v | with_component | |
derive_semantics_equ `(c : component i j) (sem : semantics j) : (derive_semantics c sem === derive_semantics' c sem)%semantics. Proof. revert sem. cofix aux; intros sem. constructor; intros a e; cbn; unfold derive_semantics; unfold eval_effect; unfold exec_effect; unfold run_effect; unfold evalState; unfold execState; ... | Remark | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Import Contract HoareFacts SemanticsFacts InstrumentFacts.",
"From FreeSpec Require Export Component."
] | theories/Core/ComponentFacts.v | derive_semantics_equ | |
correct_component `{MayProvide jx j} `(c : component i jx) `(ci : contract i Ωi) `(cj : contract j Ωj) (pred : Ωi -> Ωj -> Prop) : Prop := forall (ωi : Ωi) (ωj : Ωj) (init : pred ωi ωj) `(e : i α) (o_caller : caller_obligation ci ωi e), pre (to_hoare cj $ c α e) ωj /\ forall (x : α) (ωj' : Ωj) (run : post (to_hoare cj ... | Definition | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Import Contract HoareFacts SemanticsFacts InstrumentFacts.",
"From FreeSpec Require Export Component."
] | theories/Core/ComponentFacts.v | correct_component | |
correct_component_derives_compliant_semantics `{MayProvide jx j} `(c : component i jx) `(ci : contract i Ωi) `(cj : contract j Ωj) (pred : Ωi -> Ωj -> Prop) (correct : correct_component c ci cj pred) (ωi : Ωi) (ωj : Ωj) (inv : pred ωi ωj) (sem : semantics jx) (comp : compliant_semantics cj ωj sem) : compliant_semantics... | Lemma | theories | [
"From ExtLib Require Import StateMonad.",
"From FreeSpec.Core Require Import Contract HoareFacts SemanticsFacts InstrumentFacts.",
"From FreeSpec Require Export Component."
] | theories/Core/ComponentFacts.v | correct_component_derives_compliant_semantics | |
contract (i : interface) (Ω : Type) : Type := make_contract { witness_update (ω : Ω) : forall (α : Type), i α -> α -> Ω ; caller_obligation (ω : Ω) : forall (α : Type), i α -> Prop ; callee_obligation (ω : Ω) : forall (α : Type), i α -> α -> Prop }. Declare Scope contract_scope. Bind Scope contract_scope with contract.... | Record | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract | |
const_witness {i} := fun (u : unit) (α : Type) (e : i α) (x : α) => u. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | const_witness | |
no_caller_obligation {i Ω} (ω : Ω) (α : Type) (e : i α) : Prop := | mk_no_caller_obligation : no_caller_obligation ω α e. #[global] Hint Constructors no_caller_obligation : freespec. | Inductive | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | no_caller_obligation | |
no_callee_obligation {i Ω} (ω : Ω) (α : Type) (e : i α) (x : α) : Prop := | mk_no_callee_obligation : no_callee_obligation ω α e x. #[global] Hint Constructors no_callee_obligation : freespec. | Inductive | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | no_callee_obligation | |
no_contract (i : interface) : contract i unit := {| witness_update := const_witness ; caller_obligation := no_caller_obligation ; callee_obligation := no_callee_obligation |}. (** A similar —and as simple— contract is the one that forbids the use of a given interface. *) | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | no_contract | |
do_no_use {i Ω} (ω : Ω) (α : Type) (e : i α) : Prop := False. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | do_no_use | |
forbid_specs (i : interface) : contract i unit := {| witness_update := const_witness ; caller_obligation := do_no_use ; callee_obligation := no_callee_obligation |}. (** * Contract Equivalence *) | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | forbid_specs | |
contract_caller_equ `(c1 : contract i Ω1) `(c2 : contract i Ω2) (f : Ω1 -> Ω2) : Prop := forall ω1 a (p : i a), caller_obligation c1 ω1 p <-> caller_obligation c2 (f ω1) p. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_caller_equ | |
contract_callee_equ `(c1 : contract i Ω1) `(c2 : contract i Ω2) (f : Ω1 -> Ω2) : Prop := forall ω1 a (p : i a) x, callee_obligation c1 ω1 p x <-> callee_obligation c2 (f ω1) p x. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_callee_equ | |
contract_witness_equ `(c1 : contract i Ω1) `(c2 : contract i Ω2) (f : Ω1 -> Ω2) : Prop := forall ω1 a (p : i a) x, f (witness_update c1 ω1 p x) = witness_update c2 (f ω1) p x. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_witness_equ | |
contract_equ `(c1 : contract i Ω1) `(c2 : contract i Ω2) : Type := | mk_contract_equ (f : Ω1 -> Ω2) (g : Ω2 -> Ω1) (iso1 : forall x, f (g x) = x) (iso2 : forall x, g (f x) = x) (caller_equ : contract_caller_equ c1 c2 f) (callee_equ : contract_callee_equ c1 c2 f) (witness_equ : contract_witness_equ c1 c2 f) : contract_e... | Inductive | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_equ | |
contract_iso_lr `(c1 : contract i Ω1) `(c2 : contract i Ω2) (equ : contract_equ c1 c2) (ω1 : Ω1) : Ω2 := match equ with | @mk_contract_equ _ _ _ _ _ f _ _ _ _ _ _ => f ω1 end. | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_iso_lr | |
contract_iso_rl `(c1 : contract i Ω1) `(c2 : contract i Ω2) (equ : contract_equ c1 c2) (ω2 : Ω2) : Ω1 := match equ with | @mk_contract_equ _ _ _ _ _ _ g _ _ _ _ _ => g ω2 end. Arguments contract_iso_lr {i Ω1 c1 Ω2 c2} (equ ω1). Arguments contract_iso_rl {i Ω1 c1 Ω2 c2} (equ ω2). | Definition | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_iso_rl | |
contract_equ_refl `(c : contract i Ω) : contract_equ c c. Proof. apply mk_contract_equ with (f:=fun x => x) (g:=fun x => x); auto. + now intros ω α p. + now intros ω α p x. + now intros ω α p x. Defined. | Lemma | theories | [
"From Coq Require Import Setoid Morphisms.",
"From ExtLib Require Import StateMonad MonadState MonadTrans.",
"From FreeSpec.Core Require Import Interface Impure Semantics Component."
] | theories/Core/Contract.v | contract_equ_refl |
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Coq-FreeSpec
Structured dataset from FreeSpec — Modular verification of effectful programs.
279 declarations extracted from Coq source files.
Applications
- Training language models on formal proofs
- Fine-tuning theorem provers
- Retrieval-augmented generation for proof assistants
- Learning proof embeddings and representations
Source
- Repository: https://github.com/lthms/FreeSpec
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Declaration body |
| type | string | Lemma, Definition, Theorem, etc. |
| library | string | Source module |
| imports | list | Required imports |
| filename | string | Source file path |
| symbolic_name | string | Identifier |
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