hcju/isabelleps huggingface dataset:
trim_in_R: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "shd (trim rs s) \<in> R"
inv: fixes a :: "'a" assumes "A.ide a" shows "inv.B.inverse_arrows (\<sigma>' a) (\<sigma> a)"
prems: fixes x :: "'b" shows "(0::'a) \<le> f x"
preallocated_def: fixes empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a" shows "heap_base.preallocated empty_heap allocate typeof_addr h \<equiv> \<forall>C\<in>sys_xcpts. typeof_addr h (heap_base.addr_of_sys_xcpt empty_heap allocate C) = \<lfloor>Class_type C\<rfloor>"
simps: fixes P1 :: "pol"   and mps :: "(mon \<times> pol) list"   and m :: "nat"   and n :: "nat" shows "pnsubstl P1 mps m n = (case psubstl P1 mps n of None \<Rightarrow> P1 | Some P3 \<Rightarrow> if m = 0 then P3 else pnsubstl P3 mps (m - 1) n)"
subterms\<^sub>s\<^sub>e\<^sub>t_mono: fixes M :: "('c, 'd) Term.term set"   and N :: "('c, 'd) Term.term set" assumes "M \<subseteq> N" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t N"
sup_commute: fixes x :: "'c"   and y :: "'c" shows "x + y = y + x"
agree_sub: fixes A :: "('d + 'd) set"   and B :: "('d + 'd) set"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "A \<subseteq> B"   and "Vagree \<nu> \<omega> B" shows "Vagree \<nu> \<omega> A"
con_comp_failures_traces: fixes xs :: "'e list"   and X :: "'e set"   and P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" assumes "(xs, X) \<in> con_comp_failures P Q p q" shows "map (inv p) (filter (\<lambda>x. x \<in> range p) xs) \<in> traces P \<and> map (inv q) (filter (\<lambda>x. x \<in> range q) xs) \<in> traces Q"
strategy_winning_by_Even_def: fixes A :: "'b llist set"   and N :: "enat"   and \<sigma>\<^sub>e :: "'b list \<Rightarrow> 'b"   and p :: "'b list" assumes "GSgame A N" shows "GSgame.strategy_winning_by_Even A N \<sigma>\<^sub>e p \<equiv> \<forall>\<sigma>\<^sub>o. GSgame.induced_play N (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p \<in> A"
Min_le: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "x \<in> A" shows "signed.Min A \<le>s x"
hermitian_is_normal: fixes A :: "'a mat" assumes "hermitian A" shows "A * adjoint A = adjoint A * A"
degree_neville_aitken: fixes x :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "degree (neville_aitken x f n) \<le> n"
span_sum: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and S :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S" shows "sum f A \<in> span S"
pickFT_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "pickFT cl \<equiv> SOME n. pickFT_pred cl n"
prob_space_rT: fixes K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c" shows "prob_space (MC_syntax.rT K x)"
commute: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k" assumes "abel_semigroup f" shows "f a b = f b a"
valid_plains_def: fixes msg1 :: "bool list"   and msg2 :: "bool list" shows "valid_plains msg1 msg2 = (length msg1 = len_plain \<and> length msg2 = len_plain)"
wt_tpOf_subst: fixes \<pi> :: "Preliminaries.var \<Rightarrow> ('fsym, 'tp) efsym trm"   and T :: "('fsym, 'tp) efsym trm" assumes "GE.wtSB \<pi>"   and "GE.wt T" shows "GE.wt (subst \<pi> T) \<and> GE.tpOf (subst \<pi> T) = GE.tpOf T"
path_join_is_path: fixes p :: "nat list"   and q :: "nat list"   and n :: "nat"   and S :: "(nat \<times> nat) set" assumes "joinable p q"   and "is_path n S p"   and "is_path n S q" shows "is_path n S (path_join p q)"
inner_scaleR_pdevs_def: fixes b :: "'b"   and x :: "'b pdevs" shows "inner_scaleR_pdevs b x = unop_pdevs (\<lambda>x. (b \<bullet> x) *\<^sub>R x) x"
assms_13: shows "forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h"
atLeast0_lessThan_Suc_eq_insert_0: fixes n :: "nat" shows "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
continuous_map_image_subset_topspace: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
prj_simps_13: shows "C.dom fgh.Prj\<^sub>1 = HfHgh.chine"
set_iterator_genord_def: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" shows "set_iterator_genord iti S0 R \<equiv> \<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
Subobjs_subclass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs P C Cs" shows "P \<turnstile> C \<preceq>\<^sup>* last Cs"
inj_enum_nxt: fixes N :: "nat set"   and a :: "nat"   and A :: "nat set" assumes "infinite N" shows "inj_on (enum (nxt N a)) A"
zip_eq_Nil_iff: fixes xs :: "'d list"   and ys :: "'e list" shows "(xs || ys = []) = (xs = [] \<or> ys = [])"
adjoint_col: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_row A" shows "col (adjoint A) i = conjugate (row A i)"
dflt_None_set_def: fixes S :: "'b set" shows "dflt_None_set S \<equiv> if S = {} then None else Some S"
FAssThrow1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
unit_disc_fix_moebius_add: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "unit_disc_fix M1"   and "unit_disc_fix M2" shows "unit_disc_fix (M1 + M2)"
Limit_add_Sup_distrib: fixes \<alpha> :: "V"   and x :: "V"   and f :: "V \<Rightarrow> V" assumes "Limit \<alpha>" shows "x + \<Squnion> (f ` elts \<alpha>) = (SUP z\<in>elts \<alpha>. x + f z)"
SETId: fixes x :: "ZF"   and X :: "ZF" assumes "x |\<in>| X" shows "id\<^bsub>SET\<^esub> X |@| x = x"
ys_def: shows "ys = permute_list p [0..<n]"
gauss_cnj_dvd_left_iff: fixes a :: "gauss_int"   and b :: "gauss_int" shows "(gauss_cnj a dvd b) = (a dvd gauss_cnj b)"
preserves_reflects_arr: fixes f :: "'b" shows "C.arr (F f) = B.arr f"
int: shows "integrable (stock_measure t) (\<lambda>x. extract_real (cexpr_sem (case_nat x \<sigma>) e))"
slice1_eq_of_bl: fixes n :: "nat"   and w :: "'d word" shows "slice1 n w = of_bl (takefill False n (to_bl w))"
fp_defI: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and ctxclds_opt :: "(ctx \<times> cld) list option" assumes "find_path_rec P ctx cl [] ctxclds_opt" shows "find_path P ctx cl ctxclds_opt"
vec_scalar_mult_fun: fixes n :: "nat"   and c :: "'c"   and f :: "nat \<Rightarrow> 'c" shows "vec n (\<lambda>x. c * f x) = c \<cdot>\<^sub>v vec n f"
conf_fs_shupd': fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and C\<^sub>0 :: "char list"   and M :: "char list"   and n :: "nat"   and T :: "ty"   and frs :: "frame list"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields" assumes "conf_fs P h sh \<Phi> C\<^sub>0 M n T frs"   and "sh C = \<lfloor>(sfs, i)\<rfloor>" shows "conf_fs P h (sh(C \<mapsto> (sfs', i))) \<Phi> C\<^sub>0 M n T frs"
sWELL: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "Well_order s"
simps_5: fixes r :: "'b mrexp2"   and s :: "'b mrexp2"   and uw :: "bool"   and ux :: "bool" shows "strip2 (Times2 r s uw ux) = Times (strip2 r) (strip2 s)"
ba_mp_1: fixes uminus :: "'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra_mp uminus sup" shows "sup (uminus (sup (uminus x) y)) x = x"
order_iff_strict: fixes b :: "'b::order"   and a :: "'b::order" shows "((b::'b::order) \<le> (a::'b::order)) = (b < a \<or> a = b)"
is_RefI: fixes P :: "'b prog"   and h :: "'heap"   and v :: "'addr val"   and T :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "is_refT T" shows "is_Ref v"
inf_pred_def: fixes P :: "'a pred"   and Q :: "'a pred" shows "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>l. P l \<langle>\<rangle>"   and "\<And>l1 a1 r1 l2 a2 r2. (a1 \<le> a2 \<Longrightarrow> P l1 r1) \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> \<langle>l2, a2, r2\<rangle>"   and "\<And>v va vb. P \<langle>\<rangle> \<langle>v, va, vb\<rangle>" shows "P a0 a1"
\<Phi>_def: shows "\<Phi> = map (\<lambda>i. fut_derivs i \<psi>) (rev [0..<k])"
D: shows "enat (nat X) \<le> enat (nat (X * X))"
distinct_remdups: fixes xs :: "'k list" shows "distinct (remdups xs)"
distinct_67: fixes x8 :: "'a" shows "String \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
uint8_div_def: fixes x :: "uint8"   and y :: "uint8" shows "uint8_div x y = fst (uint8_divmod x y)"
G_nested_finite: fixes \<phi> :: "'b ltl" shows "finite (\<^bold>G \<phi>)"
induct: fixes P :: "Syntax.term \<Rightarrow> (Syntax.var, Syntax.term) fmap \<Rightarrow> bool"   and a0 :: "Syntax.term"   and a1 :: "(Syntax.var, Syntax.term) fmap" assumes "\<And>f. P Unit f"   and "\<And>y f. P (Var y) f"   and "\<And>y f t. \<lbrakk>atom y \<sharp> f; P t f\<rbrakk> \<Longrightarrow> P (Lam y t) f"   and "\<And>y f t. \<lbrakk>atom y \<sharp> f; P t f\<rbrakk> \<Longrightarrow> P (Rec y t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Inj1 t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Inj2 t) f"   and "\<And>t1 t2 f. \<lbrakk>P t1 f; P t2 f\<rbrakk> \<Longrightarrow> P (Syntax.Pair t1 t2) f"   and "\<And>t f. P t f \<Longrightarrow> P (Roll t) f"   and "\<And>y f t1 t2. \<lbrakk>atom y \<sharp> f; P t1 f; P t2 f\<rbrakk> \<Longrightarrow> P (Syntax.Let t1 y t2) f"   and "\<And>t1 t2 f. \<lbrakk>P t1 f; P t2 f\<rbrakk> \<Longrightarrow> P (App t1 t2) f"   and "\<And>t1 t2 t3 f. \<lbrakk>P t1 f; P t2 f; P t3 f\<rbrakk> \<Longrightarrow> P (Case t1 t2 t3) f"   and "\<And>t f. P t f \<Longrightarrow> P (Prj1 t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Prj2 t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Unroll t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Auth t) f"   and "\<And>t f. P t f \<Longrightarrow> P (Unauth t) f"   and "\<And>h f. P (Hash h) f"   and "\<And>h t f. P t f \<Longrightarrow> P (Hashed h t) f" shows "P a0 a1"
eff_cont: fixes n :: "nat"   and I :: "nat set"   and i :: "nat"   and j :: "nat" assumes "n < length cl"   and "I \<in> P n"   and "i \<in> I"   and "j \<in> F n I" shows "eff (cl ! n) s i \<approx> eff (dl ! n) t j \<and> local.cont (cl ! n) s i \<approx>s local.cont (dl ! n) t j"
igSwapAllIPresIGWlsAll_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapAllIPresIGWlsAll MOD \<equiv> igSwapIPresIGWls MOD \<and> igSwapAbsIPresIGWlsAbs MOD"
BIT_step_def: fixes s :: "'c list \<times> bool list \<times> 'c list"   and q :: "'c" shows "BIT_step s q = (let a = (if fst (snd s) ! index (snd (snd s)) q then 0 else length (fst s), []) in return_pmf (a, flip (index (snd (snd s)) q) (fst (snd s)), snd (snd s)))"
ltrm_decomp: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "0 < deg R f" obtains g :: "nat \<Rightarrow> 'a" where "g \<in> carrier P \<and> f = g \<oplus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f) \<and> deg R g < deg R f"
hom_of_fst: fixes H :: "('b, 'c) monoid_scheme"   and f :: "'d \<Rightarrow> 'e"   and G :: "('d, 'g) monoid_scheme"   and K :: "('e, 'f) monoid_scheme" assumes "Group.group H" shows "(f \<circ> fst \<in> hom (G \<times>\<times> H) K) = (f \<in> hom G K)"
SeqSubstFormP_Disj: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, x, y, x', y', v, i)"   and "atom k \<sharp> (s1, s2, k1, k2, x, y, x', y', v, i)" shows "{SeqSubstFormP v i x x' s1 k1, SeqSubstFormP v i y y' s2 k2, TermP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i (Q_Disj x y) (Q_Disj x' y') (Var s) (Var k)))"
cInf_lower2: fixes x :: "'a multiset"   and X :: "'a multiset set"   and y :: "'a multiset" assumes "x \<in> X"   and "x \<subseteq># y"   and "subset_mset.bdd_below X" shows "Inf X \<subseteq># y"
of_complex_add_inf: fixes x :: "complex" shows "of_complex x +\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
is_pure_conv: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "is_pure R = (\<exists>R'. R = pure R')"
5: shows "map snd r = take n (map snd r) @ [q]"
3: shows "fmlookup (action_needed_asses a s) v = fmlookup (fst a) v"
intro: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme" assumes "Ring R"   and "\<And>a m. \<lbrakk>a \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<in> carrier M"   and "\<And>a b m. \<lbrakk>a \<in> carrier R; b \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> b \<cdot>\<^sub>s\<^bsub>M\<^esub> m"   and "\<And>a m n. \<lbrakk>a \<in> carrier R; m \<in> carrier M; n \<in> carrier M\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> (m \<plusminus>\<^bsub>M\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> n"   and "\<And>a b m. \<lbrakk>a \<in> carrier R; b \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> (b \<cdot>\<^sub>s\<^bsub>M\<^esub> m)"   and "\<And>m. m \<in> carrier M \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = m" shows "Module_axioms M R"
monic_gen: fixes bs :: "('a poly \<times> nat) set" assumes "bs \<subseteq> as" shows "monic (\<Prod>(a, i)\<in>bs. a)"
Suc: shows "matpow (A + mat (1::'a)) n $h i $h j \<le> matpow (A + mat (1::'a)) (n + k) $h i $h j"
adjoint_matrix: fixes A :: "((real, 'b) vec, 'a) vec" shows "adjoint ((*v) A) = (*v) (Finite_Cartesian_Product.transpose A)"
rBang: fixes P :: "('a, 'b, 'c) psi" assumes "cP = !P"   and "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> cRs"   and "guarded P" shows "Prop"
simps_2: fixes x21 :: "'b list set"   and x22 :: "'b rexp"   and y21 :: "'b list set"   and y22 :: "'b rexp" shows "(Trn x21 x22 = Trn y21 y22) = (x21 = y21 \<and> x22 = y22)"
bottomD_2: fixes P :: "'a set"   and x :: "'a" assumes "has_bottom P"   and "x \<in> P" shows "x \<succeq> bottom P"
abs_eq: shows "RBT_Mapping2.empty = Mapping_RBT rbt.Empty"
adj_map': fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<sim> D"   and "f ` C \<noteq> f ` D" shows "f ` C \<sim> f ` D"
disjnt_subset1: fixes X :: "'c set"   and Y :: "'c set"   and Z :: "'c set" assumes "disjnt X Y"   and "Z \<subseteq> X" shows "disjnt Z Y"
changes_itv_smods_ext_def: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_itv_smods_ext a b p q = (let ps = smods_ext p q in changes_poly_at ps a - changes_poly_at ps b)"
abs_test_def: shows "abs_test \<equiv> let u = per_init' 5; u = per_union u 1 2; u = per_union u 2 3 in RETURN (per_compare u 1 3)"
iprefix_same_length_le: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and ys :: "'b list" assumes "xs \<sqsubseteq> f"   and "ys \<sqsubseteq> f"   and "length xs \<le> length ys" shows "prefix xs ys"
extend_def: fixes forms :: "'a sequent"   and seq :: "'a sequent" shows "extend forms seq \<equiv>  antec forms + antec seq \<Rightarrow>* succ forms + succ seq"
Rep_perm_ext: fixes p1 :: "perm"   and p2 :: "perm" assumes "Rep_perm p1 = Rep_perm p2" shows "p1 = p2"
replay_sound_pre_hyps: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and P :: "proofterm"   and res :: "Core.term" assumes "wf_theory thy"   and "replay thy P = Some res" shows "thy,set (hyps P) \<turnstile> res"
poly_eval_one: fixes a :: "'d \<Rightarrow> 'c" shows "poly_eval a 1 = (1::'c)"
poly_pcompose: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "poly (p \<circ>\<^sub>p q) x = poly p (poly q x)"
intro: fixes get :: "'e \<Rightarrow> 'c \<Rightarrow> 'd option"   and empty :: "'e"   and add :: "'e \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e"   and to_list :: "'e \<Rightarrow> ('c \<times> 'd) list" assumes "\<And>x. get empty x = None"   and "\<And>e x v. get (add e x v) x = Some v"   and "\<And>x y e v. x \<noteq> y \<Longrightarrow> get (add e x v) y = get e y"   and "\<And>e. map_of (to_list e) = get e" shows "env empty get add to_list"
quasiconvex_of_geodesic: fixes G :: "'b set" assumes "geodesic_segment G" shows "quasiconvex 0 G"
map_graph_minus: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "map_graph (f -- g) = map_graph f - map_graph g"
is_monom_vars_not_empty: fixes l :: "linear_poly" assumes "is_monom l" shows "vars l \<noteq> {}"
Df_borel: shows "(\<lambda>x. if x \<in> S then \<bar>det (matrix (g' x))\<bar> * f (g x) else 0) \<in> borel_measurable lebesgue"
fBallI: fixes A :: "'d fset"   and P :: "'d \<Rightarrow> bool" assumes "\<And>x. x |\<in>| A \<Longrightarrow> P x" shows "fBall A P"
map_tailrec_def: fixes f :: "'b \<Rightarrow> 'a"   and as :: "'b list" shows "map_tailrec f as = rev (map_tailrec_rev f as [])"
map_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "identity_functor C"   and "partial_magma.arr C f" shows "identity_functor.map C f = f"
in_face_cycle_setsD: fixes x :: "'b set" assumes "x \<in> face_cycle_sets" shows "x \<subseteq> arcs G"
l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs \<and> l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
Int_Collect: fixes x :: "'d"   and A :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
elem_index_less_varnum: fixes x :: "'a"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "x \<in> keys t" obtains "x \<in> X"   | "elem_index x < varnum X t"
cond_spmf_fst_eq_return_None: fixes p :: "('e \<times> 'd) spmf"   and x :: "'e" shows "(cond_spmf_fst p x = return_pmf None) = (x \<notin> fst ` set_spmf p)"
well_def_subsets2: shows "all_list_constr subsets2 (length qs2)"
F_nonneg: fixes x :: "real" shows "0 \<le> F x"
A_7: shows "v \<notin> \<lbrace>GT x c\<rbrace>"
Units_r_inv_ex: fixes x :: "'c" assumes "x \<in> Units H" shows "\<exists>y\<in>carrier H. x \<otimes>\<^bsub>H\<^esub> y = \<one>\<^bsub>H\<^esub>"
ladder_cut_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" shows "ladder_cut L n = (let i = length L - 1 in L[i := (n, snd (L ! i))])"
assms_2: shows "eval c t t' n [\<phi>]\<^sub>b"
aform_err_acc_err_aform_to_aform_errI: fixes err :: "real"   and x :: "real"   and e :: "nat \<Rightarrow> real"   and d :: "nat"   and X1 :: "real \<times> real pdevs"   and e1 :: "real"   and p :: "nat" assumes "- 1 \<le> err"   and "err \<le> 1"   and "x \<in> aform_err (e(d := err)) (X1, e1)" shows "x \<in> aform_err e (acc_err p (aform_to_aform_err X1 d) e1)"
vsv_vinsert: fixes a :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "vsv (vinsert \<langle>a, b\<rangle> r)"
conj_subset_def: fixes A :: "'g set"   and P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
comp_cod_arr: fixes f :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" assumes "arr f"   and "cod f = b" shows "b \<cdot> f = f"
Hp: shows "Gmod3 = G\<lparr>carrier := K \<inter> H\<rparr> Mod (K \<inter> H1 <#> K1 \<inter> H)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) A.map A.map A.map"
bij_lens_axioms_def: fixes x :: "'b \<Longrightarrow> 'c" shows "bij_lens_axioms x \<equiv> \<forall>\<sigma> \<rho>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>"
mult2_s_cancel: fixes s :: "('b \<times> 'b) set"   and ns :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Z :: "'b multiset"   and Y :: "'b multiset" assumes "s O ns \<subseteq> s"   and "ns O s \<subseteq> s"   and "refl ns"   and "trans ns"   and "irrefl s"   and "trans s"   and "(X + Z, Y + Z) \<in> mult2_s ns s" shows "(X, Y) \<in> mult2_s ns s"
sets_bind_measurable: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and B :: "'b measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets B"
curryE: fixes f :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "curry f a b" obtains "f (a, b)"
extf_irrefl: fixes xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> extf f gt xs xs"
Abrupt: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term" shows "G,A\<turnstile>{P\<leftarrow>undefined3 t \<and>. Not \<circ> normal} t\<succ> {P}"
cl_inv'B: fixes t :: "('a, 'b) rbt" assumes "cl_inv' t"   and "cl t = B" shows "cl_inv t"