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open GenericLib
let gExIntro_impl (witness : coq_expr) (proof : coq_expr) : coq_expr =
gApp (gInject "ex_intro") [hole; witness; proof]
let gExIntro (x : string) (pred : var -> coq_expr) (witness : coq_expr) (proof : coq_expr) : coq_expr =
gApp (gInject "ex_intro") [(gFun [x] (fun [x] -> pred x)); witness; proof]
let gEx (x : string) (pred : var -> coq_expr) : coq_expr =
gApp (gInject "ex") [(gFun [x] (fun [x] -> pred x))]
let gConjIntro p1 p2 =
gApp (gInject "conj") [p1; p2]
let gEqType e1 e2 =
gApp (gInject "eq") [e1; e2]
let gConj p1 p2 =
gApp (gInject "and") [p1; p2]
let gProjL p =
gApp ~explicit:true (gInject "Logic.proj1") [hole; hole; p]
let gProjR p =
gApp ~explicit:true (gInject "Logic.proj2") [hole; hole; p]
let gImpl p1 p2 =
gApp (gInject "Basics.impl") [p1; p2]
let gForall typ f =
gApp ~explicit:true (gInject "Nat_util.all") [typ; f]
let gProd e1 e2 =
gApp (gInject "Coq.Init.Datatypes.prod") [e1; e2]
let gLeq e1 e2 =
gApp (gInject "leq") [e1; e2]
let gLe e1 e2 =
gApp (gInject "le") [e1; e2]
let gIsTrueLeq e1 e2 =
gApp
(gInject "is_true")
[gApp (gInject "leq") [e1; e2]]
let gOrIntroL p =
gApp (gInject "or_introl") [p]
let gOrIntroR p =
gApp (gInject "or_intror") [p]
let gEqRefl p =
gApp (gInject "Logic.eq_refl") [p]
let gI = gInject "I"
let gTrueb = gInject "true"
let gFalseb = gInject "false"
let gT = gInject "True"
let gTrue = gInject "True"
let gFalse = gInject "False"
let gTt = gInject "tt"
let gIff p1 p2 =
gApp (gInject "iff") [p1; p2]
let gIsTrue x =
gApp (gInject "is_true") [x]
let gIsTrueTrue =
gApp (gInject "is_true") [gInject "true"]
let false_ind x1 x2 =
gApp (gInject "False_ind") [x1; x2]
(* TODO extend gMatch to write the return type? *)
let discriminate h =
false_ind hole
(gMatch h [(injectCtr "Logic.eq_refl", [], fun [] -> gI)])
let rewrite pred h hin =
gApp ~explicit:true (gInject "eq_ind") [hole; hole; pred; hin; hole; h]
(* gMatch h [(injectCtr "erefl", [], fun [] -> hin)] *)
let rewrite_sym typ h hin =
gApp (gInject "eq_ind_r") [typ; hin; h]
let lt0_False hlt =
gApp (gInject "lt0_False") [hlt]
let nat_ind ret_type base_case ind_case =
gApp (gInject "nat_ind") [ret_type; base_case; ind_case]
let appn fn n arg =
gApp (gInject "appn") [fn; n; arg]
let matchDec c left right =
gMatch c [ (injectCtr "left" , ["eq" ], left)
; (injectCtr "right", ["neq"], right)
]
let matchDecOpt c left right =
gMatch c [ (injectCtr "left" , ["eq" ], left)
; (injectCtr "right", ["neq"], right)
]
let plus x y =
gApp (gInject "Nat.add") [x;y]
let plus_leq_compat_l p =
gApp ~explicit:true (gInject "plus_leq_compat_l") [hole; hole; hole; p]
let leq_addl n1 n2 =
gApp (gInject "leq_addl") [n1; n2]
(* Leo, can we have a real gProp? *)
let gProp = gInject "prop"
let succ_neq_zero x =
gApp ~explicit:true (gInject "PeanoNat.Nat.neq_succ_0") [x]
let succ_neq_zero_app x h =
gApp ~explicit:true (gInject "PeanoNat.Nat.neq_succ_0") [x; h]
let isSome x =
gApp (gInject "isSome") [x]
let isSomeSome x =
gApp ~explicit:true (gInject "isSomeSome") [hole; x]
let diff_false_true h =
gApp (gInject "Bool.diff_false_true") [h]
let gSnd x =
gApp ~explicit:true (gInject "snd") [hole; hole; x]
let gFst x =
gApp ~explicit:true (gInject "fst") [hole; hole; x]
let is_true b =
gApp ~explicit:true (gInject "is_true") [b]
let forall_nil typ =
gApp ~explicit:true (gInject "List.Forall_nil") [typ; hole]
let forall_cons typ pleq p =
gApp ~explicit:true (gInject "List.Forall_cons") [typ; hole; hole; hole; pleq; p]
let ltnOSn =
gApp ~explicit:true (gInject "ltn0Sn") [hole]
let ltnOSn_pair =
gApp ~explicit:true (gInject "ltn0Sn_pair") [hole; hole; hole]
(*
let le_S_n hleq =
gApp (gInject "le_S_n") [hole; hole; hleq]
let nle_succ_0 hleq =
gApp (gInject "PeanoNat.Nat.nle_succ_0") [hole; hleq]
*)
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