statement stringlengths 6 506 | proof stringlengths 0 2.66k | type stringclasses 17
values | symbolic_name stringlengths 1 55 | library stringclasses 5
values | filename stringclasses 22
values | imports listlengths 2 12 | deps listlengths 0 20 | docstring stringlengths 0 1.35k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
interp i | := (state (semantics i)). | Notation | interp | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"semantics"
] | A term of type [impure a] describes an impure computation expected to return
a term of type [a]. Interpreting this term means actually realizing the
computation and producing the result. This requires to provide an
operational semantics for the interfaces used by the computation.
Some operational sem... | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
interface_to_state {i} : i ~> interp i | :=
fun a e => mkState (fun sem => run_effect sem e). | Definition | interface_to_state | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"interp",
"run_effect"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
to_state {i} : impure i ~> state (semantics i) | :=
impure_lift interface_to_state. | Definition | to_state | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"impure",
"impure_lift",
"interface_to_state",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_impure {i a} (sem : semantics i) (p : impure i a) : a * semantics i | :=
runState (to_state p) sem. | Definition | run_impure | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"impure",
"semantics",
"to_state"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_impure {i a} (sem : semantics i) (p : impure i a) : a | :=
fst (run_impure sem p). | Definition | eval_impure | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"impure",
"run_impure",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_impure {i a} (sem : semantics i) (p : impure i a) : semantics i | :=
snd (run_impure sem p). | Definition | exec_impure | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"impure",
"run_impure",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
with_semantics {ix j α} (sem : semantics j) (p : impure (ix + j) α)
: impure ix α | :=
match p with
| local x => local x
| request_then (in_right e) f =>
let (res, next) := run_effect sem e in
with_semantics next (f res)
| request_then (in_left e) f =>
request_then e (fun x => with_semantics sem (f x))
end. | Fixpoint | with_semantics | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"impure",
"run_effect",
"semantics"
] | * In-place Primitives Handling | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
with_store {ix s a} (x : s) (p : impure (ix + STORE s) a)
: impure ix a | :=
with_semantics (store x) p. | Definition | with_store | Core | theories/Core/Semantics.v | [
"Coq",
"Program",
"Setoid",
"Morphisms",
"ExtLib",
"Monad",
"StateMonad",
"FreeSpec.Core",
"Interface",
"Impure"
] | [
"STORE",
"impure",
"store",
"with_semantics"
] | We provide [with_store], a helper function to locally provide a mutable
variable. | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
compliant_semantics `{MayProvide ix i} `(c : contract i Ω)
: Ω -> semantics ix -> Prop | :=
| compliant_semantics_rec
(sem : semantics ix) (ω : Ω)
(o_callee : forall {α} (e : ix α),
gen_caller_obligation c ω e -> gen_callee_obligation c ω e (eval_effect sem e))
(next : forall {α} (e : ix α),
gen_caller_obligation c ω e
-> compliant_semantics c (gen_witness_update c ω e (... | CoInductive | compliant_semantics | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"MayProvide",
"contract",
"eval_effect",
"exec_effect",
"gen_callee_obligation",
"gen_caller_obligation",
"gen_witness_update",
"semantics"
] | Given a semantics [sem], and a witness [ω] if [sem] computes results which
satisfies [c] callee obligations for effects which satisfy [c] caller
obligations and if the resulting semantics produces after interpreting [e]
complies to [c] in accordance to [ω'], where [ω'] is the new witness state
after [e]... | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
store_complies_to_store_specs {s} (st : s)
: compliant_semantics (store_specs s) st (store st). | Proof.
revert st; cofix compliant_semantics_rec; intros st.
constructor.
+ intros a [| st'] _.
++ now constructor.
++ now constructor.
+ intros a e req.
cbn.
replace (exec_effect (store st) e)
with (store (store_update s st a e (eval_effect (store st) e)))
by now destruct e as [| st'... | Lemma | store_complies_to_store_specs | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"compliant_semantics",
"eval_effect",
"exec_effect",
"store",
"store_specs",
"store_update"
] | Proving that semantics obtained from the [store] function are compliant with
[store_specs] is relatively simple. | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
compliant_semantics_caller_obligation_callee_obligation `{MayProvide ix i}
`(c : contract i Ω) (ω : Ω)
`(e : ix α) (o_caller : gen_caller_obligation c ω e)
(sem : semantics ix) (comp : compliant_semantics c ω sem)
: gen_callee_obligation c ω e (eval_effect sem e). | Proof.
inversion comp; ssubst.
now apply o_callee.
Qed. | Lemma | compliant_semantics_caller_obligation_callee_obligation | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"MayProvide",
"compliant_semantics",
"contract",
"eval_effect",
"gen_callee_obligation",
"gen_caller_obligation",
"semantics",
"ssubst"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
compliant_semantics_caller_obligation_compliant `{MayProvide ix i} {Ω α} (c : contract i Ω) (ω : Ω)
(e : ix α) (o_caller : gen_caller_obligation c ω e)
(sem : semantics ix) (comp : compliant_semantics c ω sem)
: compliant_semantics c (gen_witness_update c ω e (eval_effect sem e)) (exec_effect sem e). | Proof.
inversion comp; ssubst.
now apply next.
Qed. | Lemma | compliant_semantics_caller_obligation_compliant | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"MayProvide",
"compliant_semantics",
"contract",
"eval_effect",
"exec_effect",
"gen_caller_obligation",
"gen_witness_update",
"semantics",
"ssubst"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
no_contract_compliant_semantics `{MayProvide ix i} (sem : semantics ix) (u : unit)
: compliant_semantics (no_contract i) u sem. | Proof.
revert sem.
cofix no_contract_compliant_semantics; intros sem.
constructor.
+ intros α e req.
unfold gen_caller_obligation, gen_callee_obligation.
destruct proj_p; constructor.
+ intros α e req.
unfold gen_witness_update.
destruct (proj_p e); [ apply no_contract_compliant_semantics | au... | Lemma | no_contract_compliant_semantics | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"MayProvide",
"compliant_semantics",
"gen_callee_obligation",
"gen_caller_obligation",
"gen_witness_update",
"no_contract",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
semantics_eq {i} : semantics i -> semantics i -> Prop | :=
| semantics_eq_rec
(sem sem' : semantics i)
(res_eq : forall {a} (e : i a), eval_effect sem e = eval_effect sem' e)
(next_eq : forall {a} (e : i a),
semantics_eq (exec_effect sem e) (exec_effect sem' e))
: semantics_eq sem sem'. | CoInductive | semantics_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"exec_effect",
"semantics"
] | We say two semantics are equivalent when they produce equivalent outputs
given the same primitive. | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
semantics_Equivalence (i : interface) : Equivalence (@semantics_eq i). | Next Obligation.
cofix semantics_eq_refl.
intros sem.
constructor; intros α e.
+ reflexivity.
+ apply semantics_eq_refl.
Qed. | Instance | semantics_Equivalence | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"interface",
"semantics_eq"
] | We prove [semantics_eq] is indeed an equivalence. | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
run_effect_eq `(x : α * semantics i) (y : α * semantics i) : Prop | :=
fst x = fst y /\ (snd x === snd y)%semantics. | Definition | run_effect_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"semantics"
] | ** Interpretation Results | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
run_effect_Equivalence a i
: @Equivalence (a * semantics i) run_effect_eq. | Next Obligation.
intros [x next]; now split.
Qed. | Instance | run_effect_Equivalence | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"run_effect_eq",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
fst_Proper i α
: Proper (run_effect_eq ==> eq) (@fst α (semantics i)). | Next Obligation.
add_morphism_tactic.
intros [x next] [y next'] [equ1 equ2].
apply equ1.
Qed. | Instance | fst_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"run_effect_eq",
"semantics"
] | ** Proper Instances | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
snd_Proper i α
: Proper (run_effect_eq ==> semantics_eq) (@snd α (semantics i)). | Next Obligation.
add_morphism_tactic.
intros [x next] [y next'] [equ1 equ2].
apply equ2.
Qed. | Instance | snd_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"run_effect_eq",
"semantics",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
prod_Proper i α
: Proper (eq ==> semantics_eq ==> run_effect_eq) (@pair α (semantics i)). | Next Obligation.
add_morphism_tactic.
intros x sem sem' equ.
split.
+ reflexivity.
+ apply equ.
Qed. | Instance | prod_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"run_effect_eq",
"semantics",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_effect_Proper i α
: Proper (semantics_eq ==> eq ==> run_effect_eq) (@run_effect i α). | Proof.
add_morphism_tactic.
intros sem sem' equ e.
rewrite run_effect_equation.
inversion equ; subst.
split.
+ apply res_eq.
+ apply next_eq.
Qed. | Instance | run_effect_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"run_effect",
"run_effect_eq",
"run_effect_equation",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_effect_Proper i α
: Proper (semantics_eq ==> eq ==> eq) (@eval_effect i α). | Next Obligation.
add_morphism_tactic.
intros o o' equ e.
unfold eval_effect.
now rewrite equ.
Qed. | Instance | eval_effect_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_effect_Proper i α
: Proper (semantics_eq ==> eq ==> semantics_eq) (@exec_effect i α). | Next Obligation.
add_morphism_tactic.
intros o o' equ e.
unfold exec_effect.
now rewrite equ.
Qed. | Instance | exec_effect_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"exec_effect",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_semprod_in_left_eq `(semi : semantics i) `(semj : semantics j) `(e : i α)
: eval_effect (semi * semj) (in_left e) = eval_effect semi e. | Proof.
unfold eval_effect; cbn.
destruct semi as [semi].
cbn.
now destruct (semi α e).
Qed. | Remark | eval_semprod_in_left_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_semprod_in_right_eq `(semi : semantics i) `(semj : semantics j) `(e : j α)
: eval_effect (semi * semj) (in_right e) = eval_effect semj e. | Proof.
unfold eval_effect; cbn.
destruct semj as [semj].
cbn.
now destruct (semj α e).
Qed. | Remark | eval_semprod_in_right_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_semprod_in_left_eq `(semi : semantics i) `(semj : semantics j) `(e : i α)
: exec_effect (semi * semj) (in_left e) = (semprod (exec_effect semi e) semj). | Proof.
unfold exec_effect; cbn.
destruct semi as [semi].
cbn.
now destruct (semi _ e).
Qed. | Remark | exec_semprod_in_left_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"exec_effect",
"semantics",
"semprod"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_semprod_in_right_eq `(semi : semantics i) `(semj : semantics j) `(e : j α)
: exec_effect (semi * semj) (in_right e) = (semprod semi (exec_effect semj e)). | Proof.
unfold exec_effect; cbn.
destruct semj as [semj].
cbn.
now destruct (semj _ e).
Qed. | Remark | exec_semprod_in_right_eq | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"exec_effect",
"semantics",
"semprod"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
semprod_Proper i j
: Proper (semantics_eq ==> semantics_eq ==> semantics_eq) (@semprod i j). | Next Obligation.
add_morphism_tactic.
cofix semprod_Proper.
intros semi semi' equi semj semj' equj.
constructor; intros α e; destruct e.
+ repeat rewrite eval_semprod_in_left_eq.
now inversion equi.
+ repeat rewrite eval_semprod_in_right_eq.
now inversion equj.
+ repeat rewrite exec_semprod_in_lef... | Instance | semprod_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_semprod_in_left_eq",
"eval_semprod_in_right_eq",
"exec_semprod_in_left_eq",
"exec_semprod_in_right_eq",
"repeat",
"semantics_eq",
"semprod"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
compliant_semantics_Proper `{MayProvide ix i} `(c : contract i Ω)
: Proper (eq ==> semantics_eq ==> Basics.impl) (compliant_semantics c). | Next Obligation.
add_morphism_tactic.
unfold Basics.impl.
cofix proper.
intros ω sem sem' equ comp.
inversion equ; subst.
inversion comp; subst.
constructor; intros a e pre.
+ rewrite <- res_eq.
now apply o_callee.
+ specialize next_eq with a e.
eapply proper.
exact next_eq.
rewrite <-... | Instance | compliant_semantics_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"MayProvide",
"compliant_semantics",
"contract",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_impure_equation {i a} (sem : semantics i) (p : impure i a)
: run_impure sem p = (eval_impure sem p, exec_impure sem p). | Proof.
unfold eval_impure, exec_impure.
destruct run_impure; reflexivity.
Qed. | Lemma | run_impure_equation | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_impure",
"exec_impure",
"impure",
"run_impure",
"semantics"
] | We provide several lemmas and the necessary [Proper] instances to use these
functions in conjunction with [semantics_equiv] and [impure_equiv]. | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
run_impure_request_then_assoc {i a b}
(sem : semantics i) (e : i a) (f : a -> impure i b)
: run_impure sem (request_then e f)
= run_impure (exec_effect sem e) (f (eval_effect sem e)). | Proof.
cbn; now rewrite run_effect_equation.
Qed. | Lemma | run_impure_request_then_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"exec_effect",
"impure",
"run_effect_equation",
"run_impure",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_impure_request_then_assoc {i a b}
(sem : semantics i) (e : i a) (f : a -> impure i b)
: eval_impure sem (request_then e f)
= eval_impure (exec_effect sem e) (f (eval_effect sem e)). | Proof.
unfold eval_impure.
now rewrite run_impure_request_then_assoc.
Qed. | Lemma | eval_impure_request_then_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"eval_impure",
"exec_effect",
"impure",
"run_impure_request_then_assoc",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_impure_request_then_assoc {i a b}
(sem : semantics i) (e : i a) (f : a -> impure i b)
: exec_impure sem (request_then e f)
= exec_impure (exec_effect sem e) (f (eval_effect sem e)). | Proof.
unfold exec_impure.
now rewrite run_impure_request_then_assoc.
Qed. | Lemma | exec_impure_request_then_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"exec_effect",
"exec_impure",
"impure",
"run_impure_request_then_assoc",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_impure_bind_assoc {i a b}
(sem : semantics i) (p : impure i a) (f : a -> impure i b)
: run_impure sem (p >>= f)
= run_impure (exec_impure sem p) (f (eval_impure sem p)). | Proof.
revert sem f.
induction p; intros sem g.
+ reflexivity.
+ rewrite bind_request_then_assoc.
rewrite run_impure_request_then_assoc.
rewrite H.
rewrite exec_impure_request_then_assoc.
rewrite eval_impure_request_then_assoc.
reflexivity.
Qed. | Lemma | run_impure_bind_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"bind_request_then_assoc",
"eval_impure",
"eval_impure_request_then_assoc",
"exec_impure",
"exec_impure_request_then_assoc",
"impure",
"run_impure",
"run_impure_request_then_assoc",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_impure_bind_assoc {i a b}
(sem : semantics i) (p : impure i a) (f : a -> impure i b)
: eval_impure sem (p >>= f)
= eval_impure (exec_impure sem p) (f (eval_impure sem p)). | Proof.
unfold eval_impure.
now rewrite run_impure_bind_assoc.
Qed. | Lemma | eval_impure_bind_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_impure",
"exec_impure",
"impure",
"run_impure_bind_assoc",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_impure_bind_assoc {i a b}
(sem : semantics i) (p : impure i a) (f : a -> impure i b)
: exec_impure sem (p >>= f)
= exec_impure (exec_impure sem p) (f (eval_impure sem p)). | Proof.
unfold exec_impure.
now rewrite run_impure_bind_assoc.
Qed. | Lemma | exec_impure_bind_assoc | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_impure",
"exec_impure",
"impure",
"run_impure_bind_assoc",
"semantics"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_impure_Proper_1 (i : interface) (a : Type)
: Proper (semantics_eq ==> eq ==> run_effect_eq) (@run_impure i a). | Next Obligation.
add_morphism_tactic.
intros sem sem' equ p.
revert sem sem' equ.
induction p; intros sem sem' equ.
+ cbn.
now rewrite equ.
+ repeat rewrite eval_impure_request_then_assoc.
repeat rewrite run_impure_request_then_assoc.
specialize H
with (eval_effect sem e) (exec_effect sem ... | Instance | run_impure_Proper_1 | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_effect",
"eval_impure_request_then_assoc",
"exec_effect",
"interface",
"repeat",
"run_effect_eq",
"run_impure",
"run_impure_request_then_assoc",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
run_impure_Proper_2 (i : interface) (a : Type)
: Proper (eq ==> impure_eq ==> run_effect_eq) (@run_impure i a). | Next Obligation.
add_morphism_tactic.
intros sem p q equ.
revert sem.
induction equ; intros sem.
+ reflexivity.
+ repeat rewrite run_impure_request_then_assoc.
apply H.
Qed. | Instance | run_impure_Proper_2 | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"impure_eq",
"interface",
"repeat",
"run_effect_eq",
"run_impure",
"run_impure_request_then_assoc"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval_impure_Proper (i : interface) (a : Type)
: Proper (semantics_eq ==> impure_eq ==> eq) (@eval_impure i a). | Next Obligation.
add_morphism_tactic.
intros sem sem' equ1 p q equ2.
unfold eval_impure.
now rewrite equ1, equ2.
Qed. | Instance | eval_impure_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"eval_impure",
"impure_eq",
"interface",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
exec_impure_Proper (i : interface) (a : Type)
: Proper (semantics_eq ==> impure_eq ==> semantics_eq) (@exec_impure i a). | Next Obligation.
add_morphism_tactic.
intros sem sem' equ1 p q equ2.
unfold exec_impure.
now rewrite equ1, equ2.
Qed. | Instance | exec_impure_Proper | Core | theories/Core/SemanticsFacts.v | [
"ExtLib",
"StateMonad",
"FreeSpec.Core",
"ImpureFacts",
"Contract",
"Semantics"
] | [
"exec_impure",
"impure_eq",
"interface",
"semantics_eq"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
destruct_if_when | :=
let equ_cond := fresh "equ_cond" in
match goal with
| |- context[when (negb ?B) _] => case_eq B; intros equ_cond; cbn
| |- context[when ?B _] => case_eq B; intros equ_cond; cbn
| |- context[if (negb ?B) then _ else _] => case_eq B; intros equ_cond; cbn
| |- context[if ?B then _ else _] => case_eq B; intr... | Ltac | destruct_if_when | Core | theories/Core/Tactics.v | [
"ExtLib",
"Monad",
"FreeSpec.Core",
"Contract",
"ImpureFacts",
"HoareFacts"
] | [
"when"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
destruct_if_when_in hyp | :=
let equ_cond := fresh "equ" in
match type of hyp with
| context[when (negb ?B) _] => case_eq B;
intro equ_cond;
rewrite equ_cond in hyp
| context[when ?B _] => case_eq B;
intro equ_cond;
rewr... | Ltac | destruct_if_when_in | Core | theories/Core/Tactics.v | [
"ExtLib",
"Monad",
"FreeSpec.Core",
"Contract",
"ImpureFacts",
"HoareFacts"
] | [
"when"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
simplify_gens | :=
repeat match goal with
| H : True |- _ =>
clear H
| H: _ /\ _ |- _ =>
destruct H
| |- context[@proj_p ?ix ?ix (refl_MayProvide ?ix) _ ?e] =>
change (@proj_p ix ix (refl_MayProvide ix) _ e) with (Some e);
cbn match;
cbn beta
... | Ltac | simplify_gens | Core | theories/Core/Tactics.v | [
"ExtLib",
"Monad",
"FreeSpec.Core",
"Contract",
"ImpureFacts",
"HoareFacts"
] | [
"gen_callee_obligation",
"gen_caller_obligation",
"gen_witness_update",
"refl_MayProvide",
"repeat"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
prove_impure | :=
repeat (cbn -[
to_hoare
gen_caller_obligation
gen_callee_obligation
gen_witness_update
] in *;
simplify_gens;
destruct_if_when);
lazymatch goal with
| |- _ /\ _ =>
split;
prove_impure
| |- pre (to_hoare ?c (impure... | Ltac | prove_impure | Core | theories/Core/Tactics.v | [
"ExtLib",
"Monad",
"FreeSpec.Core",
"Contract",
"ImpureFacts",
"HoareFacts"
] | [
"destruct_if_when",
"eval",
"gen_callee_obligation",
"gen_caller_obligation",
"gen_witness_update",
"impure_apply",
"impure_bind",
"impure_bind_assoc",
"impure_map",
"repeat",
"simplify_gens",
"to_hoare",
"to_hoare_pre_bind_assoc"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
unroll_post run | :=
repeat (cbn -[
to_hoare
gen_caller_obligation
gen_callee_obligation
gen_witness_update
] in *;
simplify_gens;
destruct_if_when_in run);
lazymatch type of run with
| post (to_hoare ?c ?p) ?ω ?x ?ω' =>
let p := (eval hnf i... | Ltac | unroll_post | Core | theories/Core/Tactics.v | [
"ExtLib",
"Monad",
"FreeSpec.Core",
"Contract",
"ImpureFacts",
"HoareFacts"
] | [
"destruct_if_when_in",
"eval",
"gen_callee_obligation",
"gen_caller_obligation",
"gen_witness_update",
"impure_bind",
"interface_to_hoare",
"repeat",
"simplify_gens",
"ssubst",
"to_hoare",
"to_hoare_post_bind_assoc"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
EVAL (a : Type) : Type | :=
| Eval (x : a) : EVAL a. | Inductive | EVAL | Exec | theories/Exec/Eval.v | [
"FreeSpec",
"Core"
] | [] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
eval `{Provide ix EVAL} {a} (x : a) : impure ix a | :=
request (Eval x). | Definition | eval | Exec | theories/Exec/Eval.v | [
"FreeSpec",
"Core"
] | [
"EVAL",
"Provide",
"impure",
"request"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 | |
FreeSpec_Inject `(H : Provide ix i) : Inject i (impure ix) | :=
{ inject := @request ix i _ H }. | Instance | FreeSpec_Inject | FFI | theories/FFI/FFI.v | [
"FreeSpec",
"Core",
"Extraction",
"CoqFFI",
"Interface"
] | [
"Provide",
"impure",
"request"
] | https://github.com/lthms/FreeSpec | d4e2f3a3fc7e82effddca202a8b0210dbbcf3663 |
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