X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fhelp%2FC%2Fsec_tactics.xml;h=4bba23eb6e22c887d99171a63f6776f76dfdf4b5;hb=ddc473c05167ab7aafafcc13425a154b9b0fd57f;hp=03842a70904647081e5335656fc3b39534c5316f;hpb=a3e2307340205d6a9651d90a5342cfdd232138dd;p=helm.git diff --git a/helm/software/matita/help/C/sec_tactics.xml b/helm/software/matita/help/C/sec_tactics.xml index 03842a709..4bba23eb6 100644 --- a/helm/software/matita/help/C/sec_tactics.xml +++ b/helm/software/matita/help/C/sec_tactics.xml @@ -5,6 +5,7 @@ absurd <term> + absurd absurd P @@ -17,14 +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. @@ -33,6 +34,7 @@ apply <term> + apply apply t @@ -49,13 +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 @@ -67,26 +69,27 @@ assumption + assumption assumption 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 @@ -94,13 +97,14 @@ auto [depth=<int>] [width=<int>] [paramodulation] [full] + auto auto depth=d width=w paramodulation full 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 @@ -111,7 +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. @@ -119,7 +123,7 @@ New sequents to prove: - none + None @@ -127,6 +131,7 @@ clear <id> + clear clear H @@ -140,14 +145,14 @@ Action: - it hides the hypothesis H from the + It hides the hypothesis H from the current sequent. New sequents to prove: - none + None @@ -155,6 +160,7 @@ clearbody <id> + clearbody clearbody H @@ -168,14 +174,14 @@ 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. @@ -183,13 +189,14 @@ change <pattern> with <term> + change change patt with t 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. @@ -197,7 +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. @@ -206,7 +213,7 @@ New sequents to prove: - none. + None. @@ -214,27 +221,29 @@ constructor <int> + constructor constructor n 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. @@ -244,27 +253,28 @@ contradiction + contradiction contradiction 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 @@ -272,6 +282,7 @@ cut <term> [as <id>] + cut cut P as H @@ -284,13 +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 @@ -302,6 +313,7 @@ decompose [<ident list>] <ident> [<intros_spec>] + decompose decompose ??? @@ -328,27 +340,28 @@ discriminate <term> + discriminate discriminate p 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. @@ -356,6 +369,7 @@ its constructor takes no arguments. elim <term> [using <term>] [<intros_spec>] + elim elim t using th hyps @@ -371,7 +385,7 @@ 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. @@ -379,17 +393,21 @@ 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>] - elimType T using th + elimType <term> [using <term>] [<intros_spec>] + elimType + elimType T using th hyps @@ -415,26 +433,27 @@ its constructor takes no arguments. exact <term> + exact exact p 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. @@ -442,26 +461,28 @@ its constructor takes no arguments. exists + exists exists 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. @@ -471,19 +492,20 @@ its constructor takes no arguments. fail + failt fail Pre-conditions: - none. + None. Action: - this tactic always fail. + This tactic always fail. @@ -497,19 +519,20 @@ its constructor takes no arguments. fold <reduction_kind> <term> <pattern> + fold fold red t patt 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 @@ -520,7 +543,7 @@ its constructor takes no arguments. New sequents to prove: - none. + None. @@ -528,13 +551,14 @@ its constructor takes no arguments. fourier + fourier fourier 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. @@ -543,13 +567,13 @@ 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. @@ -557,6 +581,7 @@ its constructor takes no arguments. fwd <ident> [<ident list>] + fwd fwd ...TODO @@ -583,95 +608,706 @@ its constructor takes no arguments. 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 + ∀x.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 + 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. + + + + - intros <term> - The tactic intros + 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> ≝ <term> - The tactic letin + letin + letin x ≝ t + + + + Pre-conditions: + + None. + + + + Action: + + It adds to the context of the current sequent to prove a new + definition x ≝ 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. + + + +