ABBREV_TAC `x=t` |
remember t as x and replace t with x. |
ABBREV_TAC |
ABS_TAC |
extensionality in Coq.Logic.FunctionalExtensionality |
|
ACCEPT_TAC thm |
exact thm |
|
ALL_TAC |
idtac without a user message |
|
ANTS_TAC |
If the goal conclusion is (p ==> q) ==> r, this is equivalent to assert (H: p). { (subgoal) } intros HIMPLY. apply HIMPLY in H. generalize H |
|
AP_TERM_TAC |
f_equal |
|
AP_THM_TAC |
apply equal_f |
|
ARITH_TAC |
Properly apply solvers in Micromega (lia, nia, …). Sometimes nia can solve a goal that ARITH_TAC cannot (e.g., x*x-x = x*(x-1)). Note that this tactic does not utilize assumptions; to use assumptions, use ASM_ARITH_TAC. |
ARITH_TAC |
ASM_CASES_TAC tm |
Given Axiom excluded_middle_axiom = forall P, P \/ ~P, assert (H_ANON := excluded_middle_axiom tm). destruct H_ANON as [H2 | H2]; generalize H2. |
|
ASM_MESON_TAC[thm list] |
See MESON_TAC. |
|
ASM_REWRITE_TAC[thm list] |
See REWRITE_TAC. |
|
ASSUME_TAC thm |
assert (H_ANON := thm) |
|
ASSUM_LIST f |
Probably in Ltac only? |
ASSUM_LIST |
BETA_TAC |
cbv beta |
|
CHOOSE_TAC thm |
If thm is exists x. P x, do assert (HANON := thm). destruct HANON. |
|
CONJ_TAC |
split of a conjunction conclusion only |
|
CONS_11 |
destruct for list |
|
CONV_TAC NUM_REDUCE_CONV |
simpl for natural numbers |
|
CONV_TAC INT_REDUCE_CONV |
simpl for ints |
|
CHEAT_TAC |
admit |
|
DESTRUCT_TAC |
destruct, but with a slightly different syntax (see the doc.) |
|
DISCH_TAC |
intro, but moves an assumption only |
|
DISCH_THEN(LABEL_TAC "Hname") |
intro Hname |
|
DISCH_THEN(LABEL_TAC "Hname" o REWRITE_RULE[MOD_EQ_0] |
intro Hname. rewrite MOD_EQ_0 in Hname |
|
DISJ_CASES_TAC thm |
destruct for disjunction |
|
DISJ1_TAC |
left |
|
DISJ2_TAC |
right |
|
EQ_TAC |
split for an iff conclusion only |
|
EVERY_ASSUM ttac |
For each assumption thm, run ttac(thm). |
EVERY_ASSUM |
EXISTS_TAC |
exists |
|
EXPAND_TAC s |
rewrite <- H where H is t = s |
EXPAND_TAC |
FIND_ASSUM ttac term |
If assumption H:term exists, apply tactic ttac H. |
FIND_ASSUM |
FIRST_ASSUM ttac |
? |
FIRST_ASSUM |
FIRST_ASSUM CONTR_TAC |
exfalso |
CONTR_TAC |
FIX_TAC s |
No matching tactic in Coq (correct me if I am wrong); intros, but specifies the variable to introduce |
FIX_TAC |
GEN_TAC |
intro, but targets non-propositions only |
|
IMP_REWRITE_TAC[thm list] |
Given a list of theorems that look like P ==> l = r, do rewrite and add P to the goal as a conjunction. If the rewritten part is at P' of some other implication P' ==> Q', P is added as (P ==> P'[l/r]) ==> Q. Thos also works for theorems that look like P ==> l1 = r1 /\ l2 = r2 /\ .. |
IMP_REWRITE_TAC |
INDUCT_TAC |
induction on the first universal quantifier. (ex: x in forall x x2 …, P). It must have 'num' type ('nat' in Coq). |
|
INTRO_TAC |
intros + destruct of dis/conjunctions |
|
LABEL_TAC s thm |
assert (s := thm) |
|
LABEL_TAC s (SPECL [t0;t1;…] thm) |
specialize (thm t0 t1 …) as s |
|
LABEL_TAC s (ISPECL [`x:(32)word`;…] thm) |
specialize (thm t0 t1 …) as s with type instantiations |
|
LIST_INDUCT_TAC |
induction on the first universal quantifier. (ex: x in forall x x2 …, P). It must have '(ty)list' type ('list ty' in Coq). |
|
MAP_EVERY f l |
? |
MAP_EVERY |
MATCH_MP_TAC thm |
apply, but the applying thm must be of the form P ==> Q |
|
MATCH_ACCEPT_TAC thm |
apply |
|
MESON_TAC[thm list] |
firstorder with thms registered to hint databases. Unlike ASM_MESON_TAC, this does not use assumptions. |
|
MP_TAC thm |
assert (s := thm). generalize thm |
MP_TAC |
NO_TAC |
fail |
|
ONCE_REWRITE_TAC[thm list] |
rewrite but rewrites only once. |
|
tac ORELSE tac |
orelse in Ltac2? |
|
REAL_ARITH_TAC |
lra? |
|
REFL_TAC |
reflexivity, but REFL_TAC only checks syntactic equivalence (e.g. x = 0 + x cannot be proved) |
|
REMOVE_THEN s ttac |
Given an assumption whose name is s, equivalent to USE_THEN s ttac(in HOL Light) then clear s (in Coq). |
REMOVE_THEN |
REPEAT |
repeat |
|
REWRITE_TAC [thm list] |
repeat (try rewrite thm[0]; try rewrite thm[1]; …), but unlike rewrite in Coq, if the conclusion matches exactly one of thm list, the goal is immediately proved. |
|
REWRITE_TAC [GSYM thm] |
rewrite <- thm, with the characteristics described in the generic REWRITE_TAC form above |
|
RULE_ASSUM_TAC (fn:thm->thm) |
Perform fn to every assumption. |
RULE_ASSUM_TAC |
RULE_ASSUM_TAC (REWRITE_RULE[r]) |
rewrite r in * |- |
|
SIMP_TAC [thm list] |
REWRITE_TAC, but (1) applies intrinsic rewrite rules as well (basic_rewrites and basic_convs), and (2) accepts conditional rewrite rules of form c ==> l = r. The conditional rewrite rules are applied if c can be simplified into T. Different from simpl in Coq because it does not immediately look into the definitions. For example, SIMP_TAC cannot simplify 0 + x into x without additional hints. |
SIMP_TAC |
SPEC_TAC(`x:ty1`, `y:ty2`) |
generalize x as y. If x is not used in any assumption and x is y, this is equal to revert x. |
|
STRIP_TAC |
split (for conjunctions) + intro (GEN_TAC + CONJ_TAC + elaborated version of DISCH_TAC) |
|
SUBGOAL_THEN tm ASSUME_TAC |
cut tm. intros HASSUME, or assert (HASSUME: tm) with a swapped subgoal order |
|
SUBGOAL_THEN tm (LABEL_TAC "name") |
cut tm. intros name, or assert (name: tm) with a swapped subgoal order |
|
TRY tac |
try tac |
|
tac THEN tac |
tac; tac |
|
tac THENL [tac list] |
tac. { tac[0]. } { tac[1]. } … |
|
UNDISCH_TAC |
generalize for a prop. |
|
USE_THEN s ttac |
Given an assumption whose name is s, apply ttac which should be analogous to fun thm -> (*tactic*). |
USE_THEN |
X_GEN_TAC t |
intro t, but targets non-propositions only |
|
X_META_EXISTS_TAC `x:ty` |
eexists. Set the name of the meta variable to x. |
|