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diff --git a/matita/help/C/sec_tactics.xml b/matita/help/C/sec_tactics.xml
index e658178ca..4bba23eb6 100644
--- a/matita/help/C/sec_tactics.xml
+++ b/matita/help/C/sec_tactics.xml
@@ -1,190 +1,1314 @@
-
+
Tactics
-
+
absurd <term>
- The tactic absurd
-
-
-
-
+ absurd
+ absurd P
+
+
+
+ Pre-conditions:
+
+ P must have type Prop.
+
+
+
+ Action:
+
+ It closes the current sequent by eliminating an
+ absurd term.
+
+
+
+ New sequents to prove:
+
+ It opens two new sequents of conclusion P
+ and ¬P.
+
+
+
+
+
+
apply <term>
- The tactic apply
-
-
+ apply
+ apply t
+
+
+
+ Pre-conditions:
+
+ t must have type
+ T1 â ... â
+ Tn â G
+ where G can be unified with the conclusion
+ of the current sequent.
+
+
+
+ Action:
+
+ 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
+ Ti that is not
+ instantiated by unification. Ti is
+ the conclusion of the i-th new sequent to
+ prove.
+
+
+
+
+
+
assumption
- The tactic assumption
-
-
+ assumption
+ assumption
+
+
+
+ Pre-conditions:
+
+ 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.
+
+
+
+ New sequents to prove:
+
+ None
+
+
+
+
+
+
auto [depth=<int>] [width=<int>] [paramodulation] [full]
- The tactic auto
-
-
+ auto
+ auto depth=d width=w paramodulation full
+
+
+
+ Pre-conditions:
+
+ 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
+ signature of the current sequent are considered. The signature of
+ a sequent is ...TODO
+
+
+
+ Action:
+
+ It closes the current sequent by repeated application of
+ rewriting steps (unless paramodulation is
+ omitted), hypothesis and lemmas in the library.
+
+
+
+ New sequents to prove:
+
+ None
+
+
+
+
+
+
clear <id>
- The tactic clear
-
-
+ clear
+ clear H
+
+
+
+ Pre-conditions:
+
+ H must be an hypothesis of the
+ current sequent to prove.
+
+
+
+ Action:
+
+ It hides the hypothesis H from the
+ current sequent.
+
+
+
+ New sequents to prove:
+
+ None
+
+
+
+
+
+
clearbody <id>
- The tactic clearbody
-
-
+ clearbody
+ clearbody H
+
+
+
+ Pre-conditions:
+
+ H must be an hypothesis of the
+ current sequent to prove.
+
+
+
+ Action:
+
+ It hides the definiens of a definition in the current
+ sequent context. Thus the definition becomes an hypothesis.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
change <pattern> with <term>
- The tactic change
-
-
- compare <term>
- The tactic compare
-
-
+ change
+ change patt with t
+
+
+
+ Pre-conditions:
+
+ Each subterm matched by the pattern must be convertible
+ with the term t disambiguated in the context
+ of the matched subterm.
+
+
+
+ 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:
+
+ None.
+
+
+
+
+
+
constructor <int>
- The tactic constructor
-
-
+ constructor
+ constructor n
+
+
+
+ Pre-conditions:
+
+ 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
+ inductive type of the conclusion of the current sequent.
+
+
+
+ New sequents to prove:
+
+ 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.
+
+
+
+
+
+
contradiction
- The tactic contradiction
-
-
+ contradiction
+ contradiction
+
+
+
+ Pre-conditions:
+
+ There must be in the current context an hypothesis of type
+ False.
+
+
+
+ Action:
+
+ It closes the current sequent by applying an hypothesis of
+ type False.
+
+
+
+ New sequents to prove:
+
+ None
+
+
+
+
+
+
cut <term> [as <id>]
- The tactic cut
-
-
- decide
- The tactic decide equality
-
-
+ cut
+ cut P as H
+
+
+
+ Pre-conditions:
+
+ P must have type Prop.
+
+
+
+ Action:
+
+ It closes the current sequent.
+
+
+
+ New sequents to prove:
+
+ 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
+ hypotheses the hypotheses of the current sequent to prove.
+
+
+
+
+
+
decompose [<ident list>] <ident> [<intros_spec>]
- The tactic decompose
-
-
+ decompose
+ decompose ???
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
+
+
discriminate <term>
- The tactic discriminate
-
-
+ discriminate
+ discriminate p
+
+
+
+ Pre-conditions:
+
+ 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
+ p.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
elim <term> [using <term>] [<intros_spec>]
- The tactic elim
-
-
- elimType <term> [using <term>]
- The tactic elimType
-
-
+ elim
+ elim t using th hyps
+
+
+
+ Pre-conditions:
+
+ t must inhabit an inductive type and
+ th must be an elimination principle for that
+ inductive type. If th is omitted the appropriate
+ standard elimination principle is chosen.
+
+
+
+ Action:
+
+ It proceeds by cases on the values of t,
+ according to the elimination principle th.
+
+
+
+
+ New sequents to prove:
+
+ It opens one new sequent for each case. The names of
+ the new hypotheses are picked by hyps, if
+ 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>] [<intros_spec>]
+ elimType
+ elimType T using th hyps
+
+
+
+ Pre-conditions:
+
+ T must be an inductive type.
+
+
+
+ Action:
+
+ TODO (severely bugged now).
+
+
+
+ New sequents to prove:
+
+ TODO
+
+
+
+
+
+
exact <term>
- The tactic exact
-
-
+ exact
+ exact p
+
+
+
+ Pre-conditions:
+
+ The type of p must be convertible
+ with the conclusion of the current sequent.
+
+
+
+ Action:
+
+ It closes the current sequent using p.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
exists
- The tactic exists
-
-
- fail
- The tactic fail
-
-
+ exists
+ exists
+
+
+
+ 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.
+
+
+
+
+
+
+ fail
+ failt
+ fail
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ This tactic always fail.
+
+
+
+ New sequents to prove:
+
+ N.A.
+
+
+
+
+
+
fold <reduction_kind> <term> <pattern>
- The tactic fold
-
-
+ fold
+ fold red t patt
+
+
+
+ Pre-conditions:
+
+ The pattern must not specify the wanted term.
+
+
+
+ Action:
+
+ 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
+ t every occurrence of the normal form in the
+ matched subterm.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
fourier
- The tactic fourier
-
-
+ fourier
+ fourier
+
+
+
+ Pre-conditions:
+
+ 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.
+
+
+
+ Action:
+
+ It closes the current sequent by applying the Fourier method.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
fwd <ident> [<ident list>]
- The tactic fwd
-
-
+ fwd
+ fwd ...TODO
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
+
+
generalize <pattern> [as <id>]
- The tactic generalize
-
-
+ 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
- The tactic id
-
-
+ id
+ id
+
+
+
+ Pre-conditions:
+
+ None.
+
+
+
+ Action:
+
+ This identity tactic does nothing without failing.
+
+
+
+ New sequents to prove:
+
+ None.
+
+
+
+
+
+
injection <term>
- The tactic injection
-
-
+ 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>]
- The tactic intro
-
-
+ 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>
- The tactic intros
-
-
- intros <term>
- The tactic intros
-
-
+ 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>
+ 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>]
- The tactic lapply
-
-
+ lapply
+ lapply ???
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
+
+
left
- The tactic left
-
-
+ 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>
- The tactic letin
-
-
+ 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>
- The tactic normalize
-
-
+ 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>
- The tactic paramodulation
-
-
+ paramodulation
+ paramodulation patt
+
+
+
+ Pre-conditions:
+
+ TODO.
+
+
+
+ Action:
+
+ TODO.
+
+
+
+ New sequents to prove:
+
+ TODO.
+
+
+
+
+
+
reduce <pattern>
- The tactic reduce
-
-
+ 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
- The tactic reflexivity
-
-
+ 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>
- The tactic replace
-
-
+ 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>
- The tactic rewrite
-
-
+ 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
- The tactic right
-
-
+ 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
- The tactic ring
-
-
+ 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>
- The tactic simplify
-
-
+ 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
- The tactic split
-
-
+ 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
+ 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>
- The tactic transitivity
-
-
+ 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>
- The tactic unfold
-
-
+ 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>
- The tactic whd
-
+ 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.
+
+
+
+
+
-
+