X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fhelp%2FC%2Fsec_tactics.xml;h=cc0043724f57c33fa32f0e7bec60dfff326c8098;hb=cc16727f34114cee287e6a81b80f68657c656725;hp=fda93f0b38a8dee552ca3270520a4043c981f188;hpb=9c96424c77ddcbad9309b5cd49f9e6102bce348f;p=helm.git
diff --git a/helm/software/matita/help/C/sec_tactics.xml b/helm/software/matita/help/C/sec_tactics.xml
index fda93f0b3..cc0043724 100644
--- a/helm/software/matita/help/C/sec_tactics.xml
+++ b/helm/software/matita/help/C/sec_tactics.xml
@@ -4,7 +4,7 @@
Tactics
- absurd <term>
+ <emphasis role="bold">absurd</emphasis> &sterm;
absurd
absurd P
@@ -18,14 +18,14 @@
Action:
- it closes the current sequent by eliminating an
+ It closes the current sequent by eliminating an
absurd term.
New sequents to prove:
- it opens two new sequents of conclusion P
+ It opens two new sequents of conclusion P
and ¬P.
@@ -33,7 +33,7 @@
- apply <term>
+ <emphasis role="bold">apply</emphasis> &sterm;
apply
apply t
@@ -51,13 +51,13 @@
Action:
- it closes the current sequent by applying t to n implicit arguments (that become new sequents).
+ It closes the current sequent by applying t to n implicit arguments (that become new sequents).
New sequents to prove:
- it opens a new sequent for each premise
+ It opens a new sequent for each premise
Ti that is not
instantiated by unification. Ti is
the conclusion of the i-th new sequent to
@@ -68,7 +68,7 @@
- assumption
+ <emphasis role="bold">assumption</emphasis>
assumption
assumption
@@ -76,27 +76,27 @@
Pre-conditions:
- there must exist an hypothesis whose type can be unified with
+ There must exist an hypothesis whose type can be unified with
the conclusion of the current sequent.
Action:
- it closes the current sequent exploiting an hypothesis.
+ It closes the current sequent exploiting an hypothesis.
New sequents to prove:
- none
+ None
- auto [depth=<int>] [width=<int>] [paramodulation] [full]
+ <emphasis role="bold">auto</emphasis> [<emphasis role="bold">depth=</emphasis>&nat;] [<emphasis role="bold">width=</emphasis>&nat;] [<emphasis role="bold">paramodulation</emphasis>] [<emphasis role="bold">full</emphasis>]
auto
auto depth=d width=w paramodulation full
@@ -104,7 +104,7 @@
Pre-conditions:
- none, but the tactic may fail finding a proof if every
+ None, but the tactic may fail finding a proof if every
proof is in the search space that is pruned away. Pruning is
controlled by d and w.
Moreover, only lemmas whose type signature is a subset of the
@@ -115,7 +115,7 @@
Action:
- it closes the current sequent by repeated application of
+ It closes the current sequent by repeated application of
rewriting steps (unless paramodulation is
omitted), hypothesis and lemmas in the library.
@@ -123,14 +123,14 @@
New sequents to prove:
- none
+ None
- clear <id>
+ <emphasis role="bold">clear</emphasis> &id;
clear
clear H
@@ -145,21 +145,21 @@
Action:
- it hides the hypothesis H from the
+ It hides the hypothesis H from the
current sequent.
New sequents to prove:
- none
+ None
- clearbody <id>
+ <emphasis role="bold">clearbody</emphasis> &id;
clearbody
clearbody H
@@ -174,21 +174,21 @@
Action:
- it hides the definiens of a definition in the current
+ It hides the definiens of a definition in the current
sequent context. Thus the definition becomes an hypothesis.
New sequents to prove:
- none.
+ None.
- change <pattern> with <term>
+ <emphasis role="bold">change</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;
change
change patt with t
@@ -196,7 +196,7 @@
Pre-conditions:
- each subterm matched by the pattern must be convertible
+ Each subterm matched by the pattern must be convertible
with the term t disambiguated in the context
of the matched subterm.
@@ -204,7 +204,7 @@
Action:
- it replaces the subterms of the current sequent matched by
+ It replaces the subterms of the current sequent matched by
patt with the new term t.
For each subterm matched by the pattern, t is
disambiguated in the context of the subterm.
@@ -213,14 +213,14 @@
New sequents to prove:
- none.
+ None.
- constructor <int>
+ <emphasis role="bold">constructor</emphasis> &nat;
constructor
constructor n
@@ -228,21 +228,22 @@
Pre-conditions:
- the conclusion of the current sequent must be
- an inductive type or the application of an inductive type.
+ The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least n constructors.
Action:
- it applies the n-th constructor of the
+ It applies the n-th constructor of the
inductive type of the conclusion of the current sequent.
New sequents to prove:
- it opens a new sequent for each premise of the constructor
+ It opens a new sequent for each premise of the constructor
that can not be inferred by unification. For more details,
see the apply tactic.
@@ -251,7 +252,7 @@
- contradiction
+ <emphasis role="bold">contradiction</emphasis>
contradiction
contradiction
@@ -259,28 +260,28 @@
Pre-conditions:
- there must be in the current context an hypothesis of type
+ There must be in the current context an hypothesis of type
False.
Action:
- it closes the current sequent by applying an hypothesis of
+ It closes the current sequent by applying an hypothesis of
type False.
New sequents to prove:
- none
+ None
- cut <term> [as <id>]
+ <emphasis role="bold">cut</emphasis> &sterm; [<emphasis role="bold">as</emphasis> &id;]
cut
cut P as H
@@ -294,13 +295,13 @@
Action:
- it closes the current sequent.
+ It closes the current sequent.
New sequents to prove:
- it opens two new sequents. The first one has an extra
+ It opens two new sequents. The first one has an extra
hypothesis H:P. If H is
omitted, the name of the hypothesis is automatically generated.
The second sequent has conclusion P and
@@ -311,34 +312,75 @@
- decompose [<ident list>] <ident> [<intros_spec>]
+ <emphasis role="bold">decompose</emphasis> &id; [&id;]⦠&intros-spec;
decompose
- decompose ???
+
+ decompose (T1 ... Tn) H hips
+
Pre-conditions:
- TODO.
+
+ H must inhabit one inductive type among
+
+ T1 ... Tn
+
+ and the types of a predefined list.
+
Action:
- TODO.
+
+ Runs elim H hyps, clears H and tries to run
+ itself recursively on each new identifier introduced by
+ elim in the opened sequents.
+
New sequents to prove:
- TODO.
+
+ The ones generated by all the elim tactics run.
+
+
+
+
+
+
+
+ <emphasis role="bold">demodulation</emphasis> &pattern;
+ demodulation
+ demodulation patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ &TODO;
+
+
+
+ New sequents to prove:
+
+ None.
- discriminate <term>
+ <emphasis role="bold">discriminate</emphasis> &sterm;
discriminate
discriminate p
@@ -346,28 +388,28 @@
Pre-conditions:
- p must have type K1 t1 ... tn = K'1 t'1 ... t'm where K and K' must be different constructors of the same inductive type and each argument list can be empty if
+ p must have type K t1 ... tn = K' t'1 ... t'm where K and K' must be different constructors of the same inductive type and each argument list can be empty if
its constructor takes no arguments.
Action:
- it closes the current sequent by proving the absurdity of
+ It closes the current sequent by proving the absurdity of
p.
New sequents to prove:
- none.
+ None.
- elim <term> [using <term>] [<intros_spec>]
+ <emphasis role="bold">elim</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;
elim
elim t using th hyps
@@ -384,7 +426,7 @@ its constructor takes no arguments.
Action:
- it proceeds by cases on the values of t,
+ It proceeds by cases on the values of t,
according to the elimination principle th.
@@ -392,18 +434,21 @@ its constructor takes no arguments.
New sequents to prove:
- it opens one new sequent for each case. The names of
+ It opens one new sequent for each case. The names of
the new hypotheses are picked by hyps, if
- provided.
+ provided. If hyps specifies also a number of hypotheses that
+ is less than the number of new hypotheses for a new sequent,
+ then the exceeding hypothesis will be kept as implications in
+ the conclusion of the sequent.
- elimType <term> [using <term>]
+ <emphasis role="bold">elimType</emphasis> &sterm; [<emphasis role="bold">using</emphasis> &sterm;] &intros-spec;
elimType
- elimType T using th
+ elimType T using th hyps
@@ -428,7 +473,7 @@ its constructor takes no arguments.
- exact <term>
+ <emphasis role="bold">exact</emphasis> &sterm;
exact
exact p
@@ -436,27 +481,27 @@ its constructor takes no arguments.
Pre-conditions:
- the type of p must be convertible
+ The type of p must be convertible
with the conclusion of the current sequent.
Action:
- it closes the current sequent using p.
+ It closes the current sequent using p.
New sequents to prove:
- none.
+ None.
- exists
+ <emphasis role="bold">exists</emphasis>
exists
exists
@@ -464,20 +509,21 @@ its constructor takes no arguments.
Pre-conditions:
- the conclusion of the current sequent must be
- an inductive type or the application of an inductive type.
+ The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type
+ with at least one constructor.
Action:
- equivalent to constructor 1.
+ Equivalent to constructor 1.
New sequents to prove:
- it opens a new sequent for each premise of the first
+ It opens a new sequent for each premise of the first
constructor of the inductive type that is the conclusion of the
current sequent. For more details, see the constructor tactic.
@@ -486,21 +532,21 @@ its constructor takes no arguments.
- fail
- failt
+ <emphasis role="bold">fail</emphasis>
+ fail
fail
Pre-conditions:
- none.
+ None.
Action:
- this tactic always fail.
+ This tactic always fail.
@@ -513,7 +559,7 @@ its constructor takes no arguments.
- fold <reduction_kind> <term> <pattern>
+ <emphasis role="bold">fold</emphasis> &reduction-kind; &sterm; &pattern;
fold
fold red t patt
@@ -521,13 +567,13 @@ its constructor takes no arguments.
Pre-conditions:
- the pattern must not specify the wanted term.
+ The pattern must not specify the wanted term.
Action:
- first of all it locates all the subterms matched by
+ First of all it locates all the subterms matched by
patt. In the context of each matched subterm
it disambiguates the term t and reduces it
to its red normal form; then it replaces with
@@ -538,14 +584,14 @@ its constructor takes no arguments.
New sequents to prove:
- none.
+ None.
- fourier
+ <emphasis role="bold">fourier</emphasis>
fourier
fourier
@@ -553,7 +599,7 @@ its constructor takes no arguments.
Pre-conditions:
- the conclusion of the current sequent must be a linear
+ The conclusion of the current sequent must be a linear
inequation over real numbers taken from standard library of
Coq. Moreover the inequations in the hypotheses must imply the
inequation in the conclusion of the current sequent.
@@ -562,20 +608,20 @@ its constructor takes no arguments.
Action:
- it closes the current sequent by applying the Fourier method.
+ It closes the current sequent by applying the Fourier method.
New sequents to prove:
- none.
+ None.
- fwd <ident> [<ident list>]
+ <emphasis role="bold">fwd</emphasis> &id; [<emphasis role="bold">(</emphasis>[&id;]â¦<emphasis role="bold">)</emphasis>]
fwd
fwd ...TODO
@@ -602,119 +648,710 @@ its constructor takes no arguments.
- generalize <pattern> [as <id>]
+ <emphasis role="bold">generalize</emphasis> &pattern; [<emphasis role="bold">as</emphasis> &id;]
generalize
- The tactic generalize
+ generalize patt as H
+
+
+
+ Pre-conditions:
+
+ All the terms matched by patt must be
+ convertible and close in the context of the current sequent.
+
+
+
+ Action:
+
+ It closes the current sequent by applying a stronger
+ lemma that is proved using the new generated sequent.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent where the current sequent conclusion
+ G is generalized to
+ âx.G{x/t} where {x/t}
+ is a notation for the replacement with x of all
+ the occurrences of the term t matched by
+ patt. If patt matches no
+ subterm then t is defined as the
+ wanted part of the pattern.
+
+
+
+
- id
+ <emphasis role="bold">id</emphasis>
id
- The tactic id
+ id
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ This identity tactic does nothing without failing.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- injection <term>
- injection
- The tactic injection
+ <emphasis role="bold">injection</emphasis> &sterm;
+ injection
+ injection p
+
+
+
+ Pre-conditions:
+
+ p must have type K t1 ... tn = K t'1 ... t'n where both argument lists are empty if
+K takes no arguments.
+
+
+
+ Action:
+
+ It derives new hypotheses by injectivity of
+ K.
+
+
+
+ New sequents to prove:
+
+ The new sequent to prove is equal to the current sequent
+ with the additional hypotheses
+ t1=t'1 ... tn=t'n.
+
+
+
+
- intro [<ident>]
+ <emphasis role="bold">intro</emphasis> [&id;]
intro
- The tactic intro
+ intro H
+
+
+
+ Pre-conditions:
+
+ The conclusion of the sequent to prove must be an implication
+ or a universal quantification.
+
+
+
+ Action:
+
+ It applies the right introduction rule for implication,
+ closing the current sequent.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent to prove adding to the hypothesis
+ the antecedent of the implication and setting the conclusion
+ to the consequent of the implicaiton. The name of the new
+ hypothesis is H if provided; otherwise it
+ is automatically generated.
+
+
+
+
- intros <intros_spec>
+ <emphasis role="bold">intros</emphasis> &intros-spec;
intros
- The tactic intros
+ intros hyps
+
+
+
+ Pre-conditions:
+
+ If hyps specifies a number of hypotheses
+ to introduce, then the conclusion of the current sequent must
+ be formed by at least that number of imbricated implications
+ or universal quantifications.
+
+
+
+ Action:
+
+ It applies several times the right introduction rule for
+ implication, closing the current sequent.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent to prove adding a number of new
+ hypotheses equal to the number of new hypotheses requested.
+ If the user does not request a precise number of new hypotheses,
+ it adds as many hypotheses as possible.
+ The name of each new hypothesis is either popped from the
+ user provided list of names, or it is automatically generated when
+ the list is (or becomes) empty.
+
+
+
+
- inversion <term>
+ <emphasis role="bold">inversion</emphasis> &sterm;
inversion
- The tactic inversion
+ inversion t
+
+
+
+ Pre-conditions:
+
+ The type of the term t must be an inductive
+ type or the application of an inductive type.
+
+
+
+ Action:
+
+ It proceeds by cases on t paying attention
+ to the constraints imposed by the actual "right arguments"
+ of the inductive type.
+
+
+
+ New sequents to prove:
+
+ It opens one new sequent to prove for each case in the
+ definition of the type of t. With respect to
+ a simple elimination, each new sequent has additional hypotheses
+ that states the equalities of the "right parameters"
+ of the inductive type with terms originally present in the
+ sequent to prove.
+
+
+
+
- lapply [depth=<int>] <term> [to <term list] [using <ident>]
+ <emphasis role="bold">lapply</emphasis> [<emphasis role="bold">depth=</emphasis>&nat;] &sterm; [<emphasis role="bold">to</emphasis> &sterm; [&sterm;]â¦] [<emphasis role="bold">as</emphasis> &id;]
lapply
- The tactic lapply
+
+ lapply depth=d t
+ to t1, ..., tn as H
+
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
- left
+ <emphasis role="bold">left</emphasis>
left
- The tactic left
+ left
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type
+ with at least one constructor.
+
+
+
+ Action:
+
+ Equivalent to constructor 1.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent for each premise of the first
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the constructor tactic.
+
+
+
+
- letin <ident> â <term>
+ <emphasis role="bold">letin</emphasis> &id; <emphasis role="bold">â</emphasis> &sterm;
letin
- The tactic letin
+ letin x â t
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It adds to the context of the current sequent to prove a new
+ definition x â t.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- normalize <pattern>
+ <emphasis role="bold">normalize</emphasis> &pattern;
normalize
- The tactic normalize
+ normalize patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It replaces all the terms matched by patt
+ with their βδιζ-normal form.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- paramodulation <pattern>
+ <emphasis role="bold">paramodulation</emphasis> &pattern;
paramodulation
- The tactic paramodulation
+ paramodulation patt
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
- reduce <pattern>
+ <emphasis role="bold">reduce</emphasis> &pattern;
reduce
- The tactic reduce
+ reduce patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It replaces all the terms matched by patt
+ with their βδιζ-normal form.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- reflexivity
+ <emphasis role="bold">reflexivity</emphasis>
reflexivity
- The tactic reflexivity
+ reflexivity
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current sequent must be
+ t=t for some term t
+
+
+
+ Action:
+
+ It closes the current sequent by reflexivity
+ of equality.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- replace <pattern> with <term>
- replace
- The tactic replace
+ <emphasis role="bold">replace</emphasis> &pattern; <emphasis role="bold">with</emphasis> &sterm;
+ change
+ change patt with t
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It replaces the subterms of the current sequent matched by
+ patt with the new term t.
+ For each subterm matched by the pattern, t is
+ disambiguated in the context of the subterm.
+
+
+
+ New sequents to prove:
+
+ For each matched term t' it opens
+ a new sequent to prove whose conclusion is
+ t'=t.
+
+
+
+
- rewrite {<|>} <term> <pattern>
+ <emphasis role="bold">rewrite</emphasis> [<emphasis role="bold"><</emphasis>|<emphasis role="bold">></emphasis>] &sterm; &pattern;
rewrite
- The tactic rewrite
+ rewrite dir p patt
+
+
+
+ Pre-conditions:
+
+ p must be the proof of an equality,
+ possibly under some hypotheses.
+
+
+
+ Action:
+
+ It looks in every term matched by patt
+ for all the occurrences of the
+ left hand side of the equality that p proves
+ (resp. the right hand side if dir is
+ <). Every occurence found is replaced with
+ the opposite side of the equality.
+
+
+
+ New sequents to prove:
+
+ It opens one new sequent for each hypothesis of the
+ equality proved by p that is not closed
+ by unification.
+
+
+
+
- right
+ <emphasis role="bold">right</emphasis>
right
- The tactic right
+ right
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least two constructors.
+
+
+
+ Action:
+
+ Equivalent to constructor 2.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent for each premise of the second
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the constructor tactic.
+
+
+
+
- ring
+ <emphasis role="bold">ring</emphasis>
ring
- The tactic ring
+ ring
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current sequent must be an
+ equality over Coq's real numbers that can be proved using
+ the ring properties of the real numbers only.
+
+
+
+ Action:
+
+ It closes the current sequent veryfying the equality by
+ means of computation (i.e. this is a reflexive tactic, implemented
+ exploiting the "two level reasoning" technique).
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- simplify <pattern>
+ <emphasis role="bold">simplify</emphasis> &pattern;
simplify
- The tactic simplify
+ simplify patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It replaces all the terms matched by patt
+ with other convertible terms that are supposed to be simpler.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- split
+ <emphasis role="bold">split</emphasis>
split
- The tactic split
+ split
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current sequent must be
+ an inductive type or the application of an inductive type with
+ at least one constructor.
+
+
+
+ Action:
+
+ Equivalent to constructor 1.
+
+
+
+ New sequents to prove:
+
+ It opens a new sequent for each premise of the first
+ constructor of the inductive type that is the conclusion of the
+ current sequent. For more details, see the constructor tactic.
+
+
+
+
- symmetry
+ <emphasis role="bold">symmetry</emphasis>
symmetry
The tactic symmetry
+ symmetry
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current proof must be an equality.
+
+
+
+ Action:
+
+ It swaps the two sides of the equalityusing the symmetric
+ property.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- transitivity <term>
+ <emphasis role="bold">transitivity</emphasis> &sterm;
transitivity
- The tactic transitivity
+ transitivity t
+
+
+
+ Pre-conditions:
+
+ The conclusion of the current proof must be an equality.
+
+
+
+ Action:
+
+ It closes the current sequent by transitivity of the equality.
+
+
+
+ New sequents to prove:
+
+ It opens two new sequents l=t and
+ t=r where l and r are the left and right hand side of the equality in the conclusion of
+the current sequent to prove.
+
+
+
+
- unfold [<term>] <pattern>
+ <emphasis role="bold">unfold</emphasis> [&sterm;] &pattern;
unfold
- The tactic unfold
+ unfold t patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It finds all the occurrences of t
+ (possibly applied to arguments) in the subterms matched by
+ patt. Then it δ-expands each occurrence,
+ also performing β-reduction of the obtained term. If
+ t is omitted it defaults to each
+ subterm matched by patt.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
- whd <pattern>
+ <emphasis role="bold">whd</emphasis> &pattern;
whd
- The tactic whd
+ whd patt
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ It replaces all the terms matched by patt
+ with their βδιζ-weak-head normal form.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+