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Misc.md

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Miscellaneous Notes

Useful Custom Tactics

Rewrite assumptions

(* REWRITE_ASSUMES_TAC `x = 1` picks the `x = 1` assumption and rewrites all other assumptions using this rule. *)
let REWRITE_ASSUMES_TAC (t:term) =
    UNDISCH_THEN t (fun thm -> RULE_ASSUM_TAC (REWRITE_RULE [thm]) THEN ASSUME_TAC thm);;

note Tactic

https://cr.yp.to/2023/holhull-20230406.sage has this note tactic that is very handy when you want to add an assumption that can be concluded from a set of rewrite rules

let notetac t tac = SUBGOAL_THEN t MP_TAC THENL
[tac;
  ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th);;

let note t why = notetac t(ASM_MESON_TAC why);;
  
(* usage *)
note `1 + 2 = 2 + 1` [ADD_SYM] THEN ...

Examples

(* Get the LHS of DIVISION_SIMP which is thm `|- (!m n. m DIV n * n + m MOD n = m) /\ (!m n. n * m DIV n + m MOD n = m)`,
   and specialize it. *)
SPECL [`x:num`; `2 EXP 32:num`] (CONJUNCT1 DIVISION_SIMP);;
(* Given a theorem VAL_MOD_REFL, (1) specialize it with the `y` variable, and (2) rewrite the theorem using
   the DIMINDEX_32 theorem. *)
REWRITE_RULE [DIMINDEX_32] (ISPECL [`y:(32)word`] VAL_MOD_REFL)
(* Given n:nat, do destruct n as [ | S n'] *)
DISJ_CASES_TAC(SPECL [`x:num`] num_CASES)

(* Add the names to destruct *)
DISJ_CASES_THEN (LABEL_TAC "mcases") (SPECL [`m:num`] num_CASES)

Useful Conversions

  • NUM_REDUCE_CONV
(* Calculate 1 + 2 - 3 *)
NUM_REDUCE_CONV `1 + 2 - 3` (* Note that this is 1 because it is 1 + (2 - 3)!! *)
  • MOD_DOWN_CONV

Commands

Searching lemmas, help manuals

# search [`IMAGE`];; (* List thms that are about IMAGE prop. *)
# search [name "ASSOC"];; (* List thms that has ASSOC in its name *)
# help "search";;

Printing operators:

# infixes();; 

val it : (string * (int * string)) list = 
  [("<=>", (2, "right")); ("==>", (4, "right")); ("\\/", (6, "right")); 
  ...

Printing AST:

(* Show the AST of a term *)
#remove_printer pp_print_qterm;;
`match x with | SOME y -> 10 | NONE -> 20`;;
#install_printer pp_print_qterm;;

(* Show the AST of a type *)
#remove_printer pp_print_qtype;;
loads "Library/words.ml":;
`word 10: (32)word`;;
#install_printer pp_print_qtype;;
(* Given an OCaml string 'name' and term 'tm', make a definition `name = tm` *)
new_definition (mk_eq (mk_var (name, `:(..type..)`), tm))

Thenify

See thenify