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Commit 3c0b61f8 authored by MARCHE Claude's avatar MARCHE Claude
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moved set theory and its realizations into Bware repository

parent f3a91eff
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.gitignore
View file @ 3c0b61f8
......@@ -204,3 +204,4 @@ pvsbin/
/src/jessie/config.status
/src/jessie/Makefile
/src/jessie/ptests_local_config.ml
/src/jessie/tests/basic/result/*.log
\ No newline at end of file
Makefile.in
View file @ 3c0b61f8
......@@ -848,17 +848,13 @@ COQLIBS_NUMBER = $(addprefix lib/coq/number/, $(COQLIBS_NUMBER_FILES))
COQLIBS_SET_FILES = Set
COQLIBS_SET = $(addprefix lib/coq/set/, $(COQLIBS_SET_FILES))
COQLIBS_SETTHEORY_FILES = Interval PowerSet Relation Identity Image \
InverseDomRan Function
COQLIBS_SETTHEORY = $(addprefix lib/coq/settheory/, $(COQLIBS_SETTHEORY_FILES))
ifeq (@enable_coq_fp_libs@,yes)
COQLIBS_FP_FILES = Rounding SingleFormat Single DoubleFormat Double
COQLIBS_FP_ALL_FILES = GenFloat $(COQLIBS_FP_FILES)
COQLIBS_FP = $(addprefix lib/coq/floating_point/, $(COQLIBS_FP_ALL_FILES))
endif
COQLIBS_FILES = lib/coq/BuiltIn $(COQLIBS_INT) $(COQLIBS_REAL) $(COQLIBS_NUMBER) $(COQLIBS_SET) $(COQLIBS_SETTHEORY) $(COQLIBS_FP)
COQLIBS_FILES = lib/coq/BuiltIn $(COQLIBS_INT) $(COQLIBS_REAL) $(COQLIBS_NUMBER) $(COQLIBS_SET) $(COQLIBS_FP)
COQV = $(addsuffix .v, $(COQLIBS_FILES))
COQVO = $(addsuffix .vo, $(COQLIBS_FILES))
......@@ -884,8 +880,6 @@ drivers/coq-realizations.aux: Makefile
echo 'theory number.'"$$f"' meta "realized_theory" "number.'"$$f"'", "" end'; done; \
for f in $(COQLIBS_SET_FILES); do \
echo 'theory set.'"$$f"' meta "realized_theory" "set.'"$$f"'", "" end'; done; \
for f in $(COQLIBS_SETTHEORY_FILES); do \
echo 'theory settheory.'"$$f"' meta "realized_theory" "settheory.'"$$f"'", "" end'; done; \
for f in $(COQLIBS_FP_FILES); do \
echo 'theory floating_point.'"$$f"' meta "realized_theory" "floating_point.'"$$f"'", "" end'; done; \
) > $@
......@@ -903,8 +897,6 @@ install_no_local::
cp $(addsuffix .vo, $(COQLIBS_NUMBER)) $(LIBDIR)/why3/coq/number/
mkdir -p $(LIBDIR)/why3/coq/set
cp $(addsuffix .vo, $(COQLIBS_SET)) $(LIBDIR)/why3/coq/set/
mkdir -p $(LIBDIR)/why3/coq/settheory
cp $(addsuffix .vo, $(COQLIBS_SETTHEORY)) $(LIBDIR)/why3/coq/settheory/
ifeq (@enable_coq_fp_libs@,yes)
mkdir -p $(LIBDIR)/why3/coq/floating_point
cp $(addsuffix .vo, $(COQLIBS_FP)) $(LIBDIR)/why3/coq/floating_point/
......@@ -924,7 +916,7 @@ depend: $(COQVD)
clean::
rm -f $(COQVO) $(COQVD) $(addsuffix .glob, $(COQLIBS_FILES))
update-coq: update-coq-int update-coq-real update-coq-number update-coq-set update-coq-settheory update-coq-fp
update-coq: update-coq-int update-coq-real update-coq-number update-coq-set update-coq-fp
update-coq-int: bin/why3 drivers/coq-realizations.aux theories/int.why
for f in $(COQLIBS_INT_ALL_FILES); do WHY3CONFIG="" bin/why3.@OCAMLBEST@ --realize -L theories -D drivers/coq-realize.drv -T int.$$f -o lib/coq/int/; done
......@@ -938,9 +930,6 @@ update-coq-number: bin/why3 drivers/coq-realizations.aux theories/number.why
update-coq-set: bin/why3 drivers/coq-realizations.aux theories/set.why
for f in $(COQLIBS_SET_FILES); do WHY3CONFIG="" bin/why3.@OCAMLBEST@ --realize -L theories -D drivers/coq-realize.drv -T set.$$f -o lib/coq/set/; done
update-coq-settheory: bin/why3 drivers/coq-realizations.aux theories/settheory.why
for f in $(COQLIBS_SETTHEORY_FILES); do WHY3CONFIG="" bin/why3.@OCAMLBEST@ --realize -L theories -D drivers/coq-realize.drv -T settheory.$$f -o lib/coq/settheory/; done
update-coq-fp: bin/why3 drivers/coq-realizations.aux theories/floating_point.why
for f in $(COQLIBS_FP_FILES); do WHY3CONFIG="" bin/why3.@OCAMLBEST@ --realize -L theories -D drivers/coq-realize.drv -T floating_point.$$f -o lib/coq/floating_point/; done
......@@ -1293,8 +1282,7 @@ STDLIBS = algebra \
option \
real \
relations \
set \
settheory
set
# function ? sum ? tptp ?
STDMODS = arith array hashtbl impset pqueue queue random ref stack string
......
Version
View file @ 3c0b61f8
# Why version
VERSION=0.73+git
VERSION=0.80
......
doc/intro.tex
View file @ 3c0b61f8
......@@ -22,7 +22,7 @@ extended with
%\item Hilbert's epsilon construct
\end{itemize}
\todo{continue}
% \todo{continue}
\section{Organization of this document}
......
doc/macros.tex
View file @ 3c0b61f8
\newcommand{\todo}[1]{{\Huge\bfseries TODO: #1}}
% \newcommand{\todo}[1]{{\Huge\bfseries TODO: #1}}
\newcommand{\why}{\textsf{Why3}\xspace}
\newcommand{\whyml}{\textsf{WhyML}\xspace}
......
doc/manual.tex
View file @ 3c0b61f8
......@@ -91,7 +91,7 @@
\vfill
\todo{NE PAS DISTRIBUER TANT QU'IL RESTE DES TODOS}
% \todo{NE PAS DISTRIBUER TANT QU'IL RESTE DES TODOS}
%BEGIN LATEX
\begin{LARGE}
......@@ -121,15 +121,15 @@
\begin{flushleft}
\begin{tabular}{l}
$^1$ LRI, CNRS, Univ Paris-Sud, Orsay, F-91405 \\
$^2$ Inria Saclay -- \^Ile-de-France, Toccata, Palaiseau, F-91120
$^1$ LRI, CNRS \& University Paris-Sud, Orsay, F-91405 \\
$^2$ Inria Saclay -- \^Ile-de-France, Palaiseau, F-91120
\end{tabular}
%BEGIN LATEX
\bigskip
%END LATEX
\textcopyright 2010-2012 Univ Paris-Sud, CNRS, Inria
\textcopyright 2010-2012 University Paris-Sud, CNRS, Inria
\urldef{\urlutcat}{\url}{http://frama-c.cea.fr/u3cat}
\urldef{\urlhilite}{\url}{http://www.open-do.org/projects/hi-lite/}
......
lib/coq/settheory/Function.v deleted 100644 → 0
View file @ f3a91eff
(* This file is generated by Why3's Coq-realize driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require set.Set.
Require settheory.Relation.
Require settheory.InverseDomRan.
Import set.Set.
Import settheory.InverseDomRan.
(* Why3 goal *)
Definition infix_plmngt: forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, (set.Set.set a) -> (set.Set.set b) ->
(set.Set.set (set.Set.set (a* b)%type)).
intros a a_WT b b_WT s t.
exact (fun r => subset (dom r) s /\ subset (ran r) t /\
forall x y1 y2, mem (x,y1) r /\ mem (x,y2) r ->
y1 = y2).
Defined.
(* Why3 goal *)
Lemma mem_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)), (set.Set.mem f (infix_plmngt s
t)) <-> ((forall (x:a) (y:b), (set.Set.mem (x, y) f) -> ((set.Set.mem x
s) /\ (set.Set.mem y t))) /\ forall (x:a) (y1:b) (y2:b), ((set.Set.mem (x,
y1) f) /\ (set.Set.mem (x, y2) f)) -> (y1 = y2)).
intros a a_WT b b_WT f s t.
unfold infix_plmngt, mem.
unfold dom, ran.
unfold subset.
unfold mem.
split.
intros (h1 & h2 & h3).
split; auto.
intros x y h.
split.
apply h1.
now exists y.
apply h2.
now exists x.
intros (h1 & h2).
split.
intros x (y,h).
generalize (h1 x y h); tauto.
split.
intros y (x,h).
generalize (h1 x y h); tauto.
assumption.
Qed.
(* Why3 goal *)
Lemma domain_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (x:a) (y:b), (set.Set.mem f
(infix_plmngt s t)) -> ((set.Set.mem (x, y) f) -> (set.Set.mem x s)).
intros a a_WT b b_WT f s t x y (h1 & _) h4.
apply h1.
apply dom_def.
now exists y.
Qed.
(* Why3 goal *)
Lemma range_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (x:a) (y:b), (set.Set.mem f
(infix_plmngt s t)) -> ((set.Set.mem (x, y) f) -> (set.Set.mem y t)).
intros a a_WT b b_WT f s t x y h1 h2.
apply h1.
apply ran_def.
now exists x.
Qed.
(* Why3 goal *)
Lemma function_extend_range : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (u:(set.Set.set b)),
(set.Set.subset t u) -> ((set.Set.mem f (infix_plmngt s t)) ->
(set.Set.mem f (infix_plmngt s u))).
intros a a_WT b b_WT f s t u h1 h2.
unfold infix_plmngt, mem in *.
intuition.
now apply subset_trans with t.
Qed.
(* Why3 goal *)
Lemma function_reduce_range : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (u:(set.Set.set b)),
(set.Set.subset u t) -> ((set.Set.mem f (infix_plmngt s t)) ->
((forall (x:a) (y:b), (set.Set.mem (x, y) f) -> (set.Set.mem y u)) ->
(set.Set.mem f (infix_plmngt s u)))).
intros a a_WT b b_WT f s t u h1 h2 h3.
unfold infix_plmngt, mem in *.
intuition.
unfold subset.
intros y h.
destruct h as (x,h).
unfold ran, mem in h.
unfold mem; now apply h3 with (x:=x).
Qed.
(* Why3 goal *)
Definition infix_mnmngt: forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, (set.Set.set a) -> (set.Set.set b) ->
(set.Set.set (set.Set.set (a* b)%type)).
intros a a_WT b b_WT s t.
exact (fun f => mem f (infix_plmngt s t) /\ dom f = s).
Defined.
(* Why3 goal *)
Lemma mem_total_functions : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)), (set.Set.mem f (infix_mnmngt s
t)) <-> ((set.Set.mem f (infix_plmngt s t)) /\
((settheory.InverseDomRan.dom f) = s)).
intros a a_WT b b_WT f s t.
unfold infix_mnmngt, mem.
intuition.
Qed.
(* Why3 goal *)
Lemma total_function_is_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)), (set.Set.mem f (infix_mnmngt s
t)) -> (set.Set.mem f (infix_plmngt s t)).
intros a a_WT b b_WT f s t h1.
unfold infix_mnmngt, mem in h1.
unfold mem; tauto.
Qed.
(* Why3 goal *)
Lemma singleton_is_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (x:a) (y:b), (set.Set.mem (set.Set.add (
x, y) (set.Set.empty :(set.Set.set (a* b)%type)))
(infix_mnmngt (set.Set.add x (set.Set.empty :(set.Set.set a)))
(set.Set.add y (set.Set.empty :(set.Set.set b))))).
intros a a_WT b b_WT x y.
rewrite mem_total_functions.
split.
rewrite mem_function.
split.
intros x' y'.
repeat rewrite set.Set.add_def1.
(*
repeat rewrite set.Set.mem_empty.
Anomaly: unknown meta ?1230. Please report.
*)
intuition; inversion H0; auto.
intros x0 y1 y2.
repeat rewrite set.Set.add_def1.
intuition; inversion H; inversion H0; auto.
rewrite dom_add.
rewrite dom_empty; auto.
Qed.
Require Import ClassicalEpsilon.
(* Why3 goal *)
Definition apply: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(set.Set.set (a* b)%type) -> a -> b.
intros a a_WT b b_WT f x.
assert (i: inhabited b) by (apply inhabits; assumption).
exact (epsilon i (fun y:b => mem x (dom f) -> mem (x,y) f)).
Defined.
(* Why3 goal *)
Lemma apply_def0 : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (a1:a), ((set.Set.mem f
(infix_plmngt s t)) /\ (set.Set.mem a1 (settheory.InverseDomRan.dom f))) ->
(set.Set.mem (a1, (apply f a1)) f).
intros a a_WT b b_WT f s t a1 (h1,h2).
unfold apply.
unfold mem.
unfold mem in h2.
generalize h2.
apply epsilon_spec.
destruct (classic (dom f a1)).
destruct H as (x, h).
exists x.
intro.
apply h.
exists b_WT.
tauto.
Qed.
(* Why3 goal *)
Lemma apply_def1 : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (a1:a), ((set.Set.mem f
(infix_mnmngt s t)) /\ (set.Set.mem a1 s)) -> (set.Set.mem (a1, (apply f
a1)) f).
intros a a_WT b b_WT f s t a1 (h1 & h2).
eapply apply_def0.
split.
apply total_function_is_function; eauto.
destruct h1.
subst s; auto.
Qed.
(* Why3 goal *)
Lemma apply_def2 : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (f:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set b)) (a1:a) (b1:b), ((set.Set.mem f
(infix_plmngt s t)) /\ (set.Set.mem (a1, b1) f)) -> ((apply f a1) = b1).
intros a a_WT b b_WT f s t a1 b1 (h1,h2).
generalize h1; intro h1'.
rewrite mem_function in h1.
destruct h1 as (_ & h).
apply h with (x:=a1).
split; auto.
eapply apply_def0; split.
eauto.
rewrite dom_def.
exists b1; auto.
Qed.
(* Why3 goal *)
Lemma injection : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (f:(set.Set.set (a* b)%type)) (s:(set.Set.set a)) (t:(set.Set.set
b)), forall (x:a) (y:a), (set.Set.mem f (infix_mnmngt s t)) ->
((set.Set.mem (settheory.InverseDomRan.inverse f) (infix_plmngt t s)) ->
((set.Set.mem x s) -> ((set.Set.mem y s) -> (((apply f x) = (apply f y)) ->
(x = y))))).
intros a a_WT b b_WT f s t x y h1 h2 h3 h4 h5.
destruct h2 as (h6 & h7 & h8).
apply h8 with (apply f x).
split.
unfold inverse, mem.
apply apply_def1 with s t; tauto.
rewrite h5.
unfold inverse, mem.
apply apply_def1 with s t; tauto.
Qed.
(* Why3 goal *)
Lemma apply_singleton : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (x:a) (y:b), ((apply (set.Set.add (x, y)
(set.Set.empty :(set.Set.set (a* b)%type))) x) = y).
intros a a_WT b b_WT x y.
rewrite apply_def2 with (s:=(add x empty))
(t:=(add y empty)) (b1 := y); auto.
split.
apply total_function_is_function.
apply singleton_is_function.
apply add_def1; auto.
Qed.
(* Why3 goal *)
Definition semicolon: forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b} {c:Type} {c_WT:WhyType c}, (set.Set.set (a*
b)%type) -> (set.Set.set (b* c)%type) -> (set.Set.set (a* c)%type).
intros a a_WT b b_WT c c_WT f g.
exact (fun p => match p with
(x,z) => exists y:b, f (x,y) /\ g (y,z) end).
Defined.
(* Why3 goal *)
Lemma semicolon_def : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b} {c:Type} {c_WT:WhyType c}, forall (x:a) (z:c)
(p:(set.Set.set (a* b)%type)) (q:(set.Set.set (b* c)%type)), (set.Set.mem (
x, z) (semicolon p q)) <-> exists y:b, (set.Set.mem (x, y) p) /\
(set.Set.mem (y, z) q).
intros a a_WT b b_WT c c_WT x z p q.
tauto.
Qed.
(* Why3 goal *)
Lemma semicolon_is_function : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b} {c:Type} {c_WT:WhyType c}, forall (f:(set.Set.set
(a* b)%type)) (g:(set.Set.set (b* c)%type)) (s:(set.Set.set a))
(t:(set.Set.set b)) (u:(set.Set.set c)), ((set.Set.mem f (infix_plmngt s
t)) /\ (set.Set.mem g (infix_plmngt t u))) -> (set.Set.mem (semicolon f g)
(infix_plmngt s u)).
intros a a_WT b b_WT c c_WT f g s t u ((f1&f2&f3),(g1&g2&g3)).
rewrite mem_function.
split.
intros x z (y & h1 & h2).
split.
apply f1; exists y; auto.
apply g2; exists y; auto.
intros x y1 y2 ((h1&h2&h3)&(i1&i2&i3)).
eapply g3.
split. apply h3.
assert (h1=i1).
eapply f3.
split. apply h2. apply i2.
subst i1.
apply i3.
Qed.
(* Why3 goal *)
Lemma apply_compose : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b} {c:Type} {c_WT:WhyType c}, forall (x:a)
(f:(set.Set.set (a* b)%type)) (g:(set.Set.set (b* c)%type)) (s:(set.Set.set
a)) (t:(set.Set.set b)) (u:(set.Set.set c)), ((set.Set.mem f
(infix_plmngt s t)) /\ ((set.Set.mem g (infix_plmngt t u)) /\
((set.Set.mem x (settheory.InverseDomRan.dom f)) /\ (set.Set.mem (apply f
x) (settheory.InverseDomRan.dom g))))) -> ((apply (semicolon f g)
x) = (apply g (apply f x))).
intros a a_WT b b_WT c c_WT x f g s t u (h1 & h2 & h3 & h4).
eapply apply_def2.
split.
eapply semicolon_is_function.
split; eauto.
rewrite semicolon_def.
exists (apply f x).
split.
eapply apply_def0.
split.
eauto.
assumption.
eapply apply_def0.
split.
eauto.
assumption.
Qed.
lib/coq/settheory/Identity.v deleted 100644 → 0
View file @ f3a91eff
(* This file is generated by Why3's Coq-realize driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require set.Set.
Require settheory.Relation.
Require settheory.InverseDomRan.
Require settheory.Function.
Import set.Set.
(* Why3 goal *)
Definition id: forall {a:Type} {a_WT:WhyType a}, (set.Set.set a) ->
(set.Set.set (a* a)%type).
intros a a_WT s.
exact (fun p => match p with (x,y) => s x /\ x=y end).
Defined.
(* Why3 goal *)
Lemma id_def : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (y:a)
(s:(set.Set.set a)), (set.Set.mem (x, y) (id s)) <-> ((set.Set.mem x s) /\
(x = y)).
intros a a_WT x y s.
unfold id, mem; tauto.
Qed.
Import settheory.InverseDomRan.
(* Why3 goal *)
Lemma id_dom : forall {a:Type} {a_WT:WhyType a}, forall (s:(set.Set.set a)),
((settheory.InverseDomRan.dom (id s)) = s).
intros a a_WT s.
apply predicate_extensionality.
unfold id, dom, mem; split.
unfold mem.
intros (y, (h1, h2)); auto.
unfold mem.
intro h.
exists x; tauto.
Qed.
(* Why3 goal *)
Lemma id_ran : forall {a:Type} {a_WT:WhyType a}, forall (s:(set.Set.set a)),
((settheory.InverseDomRan.ran (id s)) = s).
intros a a_WT s.
apply predicate_extensionality.
unfold id, ran, mem; split.
unfold mem.
intros (y, (h1, h2)); subst; auto.
unfold mem.
intro h.
exists x; tauto.
Qed.
(* Why3 goal *)
Lemma id_fun : forall {a:Type} {a_WT:WhyType a}, forall (s:(set.Set.set a)),
(set.Set.mem (id s) (settheory.Function.infix_plmngt s s)).
intros a a_WT s.
unfold Function.infix_plmngt, mem.
split.
rewrite id_dom.
apply subset_refl.
split.
rewrite id_ran.
apply subset_refl.
unfold id; intuition.
subst; auto.
Qed.
(* Why3 goal *)
Lemma id_total_fun : forall {a:Type} {a_WT:WhyType a}, forall (s:(set.Set.set
a)), (set.Set.mem (id s) (settheory.Function.infix_mnmngt s s)).
intros a a_WT s.
unfold Function.infix_mnmngt, mem.
split.
apply id_fun.
apply id_dom.
Qed.
lib/coq/settheory/Image.v deleted 100644 → 0
View file @ f3a91eff
(* This file is generated by Why3's Coq-realize driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require set.Set.
Require settheory.Relation.
Import set.Set.
(* Why3 goal *)
Definition image: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(set.Set.set (a* b)%type) -> (set.Set.set a) -> (set.Set.set b).
intros a a_WT b b_WT r s.
exact (fun y => exists x:a, mem x s /\ mem (x,y) r).
Defined.
(* Why3 goal *)
Lemma mem_image : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (r:(set.Set.set (a* b)%type)) (dom:(set.Set.set a)) (y:b),
(set.Set.mem y (image r dom)) <-> exists x:a, (set.Set.mem x dom) /\
(set.Set.mem (x, y) r).
intros a a_WT b b_WT r dom y.
unfold image, mem; intuition.
Qed.
(* Why3 goal *)
Lemma image_union : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (r:(set.Set.set (a* b)%type))
(s:(set.Set.set a)) (t:(set.Set.set a)), ((image r (set.Set.union s
t)) = (set.Set.union (image r s) (image r t))).
intros a a_WT b b_WT r s t.
apply predicate_extensionality; intros y.
unfold image, union, mem.
split.
intros (x, ([h1 | h2] & h3)).
left; exists x; auto.
right; exists x; auto.
intros [(x, h) | (x, h)]; exists x; intuition.
Qed.
(* Why3 goal *)
Lemma image_add : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (r:(set.Set.set (a* b)%type)) (dom:(set.Set.set a)) (x:a), ((image r
(set.Set.add x dom)) = (set.Set.union (image r (set.Set.add x
(set.Set.empty :(set.Set.set a)))) (image r dom))).
intros a a_WT b b_WT r dom z.
apply predicate_extensionality; intros y.