hcju/isabelleps huggingface dataset:
S_sound: fixes p :: "'p"   and w :: "('p, 'ba) label list"   and v :: "'n"   and c' :: "'n list multiset"   and P :: "'p multiset"   and k :: "nat" assumes "(([entry fg p], {#}), w, [v], c') \<in> trcl (trss fg)"   and "size P \<le> k"   and "{#[entry fg p]. p \<in># P#} \<subseteq># c'" shows "(v, mon_w fg w, P) \<in> S_cs fg k"
refl_le_joinable: fixes R :: "('b \<times> 'b) set" assumes "refl R" shows "R \<subseteq> joinable R"
arr_dom_iff_arr: fixes f :: "'b" shows "\<phi>.B.arr (\<phi>.B.dom f) = \<phi>.B.arr f"
simps_1: fixes c :: "'b poly"   and p :: "'b poly" shows "coefficients (CN c 0 p) = c # coefficients p"
31: shows "w \<le> f \<squnion> f\<^sup>T"
le_mid2__le12: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a"   and M' :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M' A' B'"   and "Tarski_neutral_dimensionless.Le Bet Cong A B A' B'" shows "Tarski_neutral_dimensionless.Le Bet Cong A M A' M'"
Abs_freeword_Cons: fixes x :: "'b \<times> bool"   and xs :: "('b \<times> bool) list" assumes "proper_signed_list (x # xs)" shows "Abs_freeword (x # xs) = Abs_freeword [x] + Abs_freeword xs"
get_writer_def: fixes get :: "('l \<Rightarrow> 'k) \<Rightarrow> 'k"   and f :: "'l \<Rightarrow> ('i, 'j, 'k) writerT" shows "get_writer get f = WriterT (get (\<lambda>s. run_writer (f s)))"
mult_le_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a c) (times b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
KRP_unique: fixes v :: "tm"   and x :: "tm"   and y :: "tm"   and y' :: "tm" shows "{KRP v x y, KRP v x y'} \<turnstile> y' EQ y"
mem_0_Ord: fixes k :: "V" assumes "Ord k"   and "k \<noteq> 0" shows "0 \<in> elts k"
finfun_single_apply: fixes x :: "'a"   and y :: "'a" shows "finfun_single x $ y = (x = y)"
binrelchain_Conssnoc_reduce: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "binrelchain P (x # xs @ [y])" shows "binrelchain P xs"
h_4: shows "\<exists>i<n. rec_exec (rcs ! i) xs = 1 \<and> (\<forall>j<n. j \<noteq> i \<longrightarrow> rec_exec (rcs ! j) xs = 0)"
set_pmf_finite: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and D :: "'a pmf" assumes "Finite_Markov_Decision_Process K S"   and "s \<in> S"   and "D \<in> K s" shows "finite (set_pmf D)"
cases: fixes a1 :: "Syntax.term list"   and a2 :: "Syntax.term"   and a3 :: "mode"   and a4 :: "nat"   and a5 :: "Syntax.term list"   and a6 :: "Syntax.term" assumes "\<lless>a1, a2\<ggreater> a3\<rightarrow>a4 \<lless>a5, a6\<ggreater>" obtains \<pi> :: "Syntax.term list" and e :: "Syntax.term" and uu_ :: "mode" where "a1 = \<pi>" and "a2 = e" and "a3 = uu_" and "a4 = 0" and "a5 = \<pi>" and "a6 = e"   | \<pi>\<^sub>1 :: "Syntax.term list" and e\<^sub>1 :: "Syntax.term" and m :: "mode" and i :: "nat" and \<pi>\<^sub>2 :: "Syntax.term list" and e\<^sub>2 :: "Syntax.term" and \<pi>\<^sub>3 :: "Syntax.term list" and e\<^sub>3 :: "Syntax.term" where "a1 = \<pi>\<^sub>1" and "a2 = e\<^sub>1" and "a3 = m" and "a4 = i + 1" and "a5 = \<pi>\<^sub>3" and "a6 = e\<^sub>3" and "\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater>" and "\<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>"
card_of_ordLeq_finite: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|"   and "finite B" shows "finite A"
le_fin_empty: fixes w :: "'action list" shows "[] \<preceq>\<^sub>F w"
inverse_float_interval_eq_Some_conv: fixes p :: "nat"   and X :: "float interval"   and R :: "float interval" shows "(inverse_float_interval p X = Some R) = ((0 < lower X \<or> upper X < 0) \<and> lower R = float_divl p 1 (upper X) \<and> upper R = float_divr p 1 (lower X))"
a41: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "a ((4 * real m + 1) / 2 ^ Suc n) = a ((2 * real m + 1) / 2 ^ n)"
eq_opp_is_zero: fixes x :: "'b" shows "(x = - x) = (x = (0::'b))"
CptnModWhile1: fixes P :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and b :: "'b set"   and zs :: "('b com option \<times> 'b) list" assumes "(Some P, s) # xs \<in> cptn_mod"   and "s \<in> b"   and "zs = map (lift (While b P)) xs" shows "(Some (While b P), s) # (Some (Seq P (While b P)), s) # zs \<in> cptn_mod"
S_eq_2: fixes x :: "'a" assumes "\<not> (\<forall>\<omega>\<in>streams (space M). T (x ## \<omega>) = 0)" shows "S x = L (\<lambda>\<omega>. epred (T (x ## \<omega>))) x"
rupture_surj_composed_with_pmod: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "rupture_surj K p q = rupture_surj K p (q pmod p)"
distinguishing_weak_formula_def: fixes P :: "'state"   and Q :: "'state" shows "distinguishing_weak_formula P Q \<equiv> Conj (Abs_bset {- p \<bullet> (SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True})"
uminus': shows "map of_int_hom.vec_hom fs ! i - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j = map of_int_hom.vec_hom fs ! i + - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j"
assms_2: shows "\<chi> \<in> \<P>\<^sup>2"
neg: shows "list_all3 (\<lambda>A Qi X. qtable n A P Qi X \<and> wf_set n A) A_neg Q_neg L_neg"
distinct_independent_pdevs: fixes xs :: "(real \<times> real) list" shows "distinct (independent_pdevs xs)"
no_call2_def: fixes e :: "'a expr1"   and pc :: "nat" shows "no_call2 e pc = (pc \<le> length (compE2 e) \<and> (pc < length (compE2 e) \<longrightarrow> (\<forall>M n. compE2 e ! pc \<noteq> Invoke M n)))"
ToTC_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "arr f"   and "arr g"   and "arr h" shows "T.ToTC (f, g, h) = (f \<otimes> g) \<otimes> h"
metric_tendsto_imp_tendsto: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "\<forall>\<^sub>F x in F. dist (g x) b \<le> dist (f x) a" shows "(g \<longlongrightarrow> b) F"
embed_inj_on: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "embed r r' f" shows "inj_on f (Field r)"
map_cong0: fixes x :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>z. z \<in> range x \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
nth_equalityI: fixes xs :: "'e list"   and ys :: "'e list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
sum_list_nonneg: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'b) \<le> x" shows "(0::'b) \<le> sum_list xs"
\<E>_new_actions_for_fun: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and a :: "nat"   and adal :: "'addr \<times> addr_loc"   and a' :: "nat" assumes "E \<in> \<E>"   and "a \<in> new_actions_for P E adal"   and "a' \<in> new_actions_for P E adal" shows "a = a'"
lookupb_lookupa: fixes x :: "name"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and c :: "coname"   and a :: "coname"   and P :: "trm" assumes "x \<sharp> \<theta>_c" shows "lookupb x c \<theta>_c a P = lookupa x c \<theta>_c{x:=<a>.P}"
DOWN_membI: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and x :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "x \<in> local.set a" shows "DOWN x \<^bold>\<in> ide_to_hf a"
vwb_lens_mwb: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "mwb_lens x"
4: shows "Ln z \<in> {w. - pi < Im w \<and> Im w < pi}"
card'_code_1: fixes xs :: "'a list" shows "card' (set xs) = length (remdups xs)"
polyfun_rootbound: fixes n :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "\<exists>k\<le>n. c k \<noteq> (0::'b)" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)} \<and> card {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)} \<le> n"
A2_def: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "A2 x a Q = (\<lambda>s. \<exists>x. ((\<lambda>s. \<exists>x. (x \<hookrightarrow> x) s) \<and>* (x \<hookrightarrow> x \<longrightarrow>* Q) \<and>* $ 1) s \<and> ((\<lambda>s. True) \<and>* (\<lambda>(ps, n). vars a \<subseteq> dom ps \<and> paval a ps = x)) s)"
IH: fixes x2a :: "(('b, 'c, 'a) prot_fun, 'd) Term.term"   and t :: "(('b, 'c, 'a) prot_fun, 'd) Term.term"   and u :: "(('b, 'c, 'a) prot_fun, 'd) Term.term" assumes "x2a \<in> set T"   and "timpls_transformable_to' TI x2a t"   and "timpls_transformable_to' TI t u" shows "timpls_transformable_to' TI x2a u"
convex_halfspace_le: fixes a :: "'b"   and b :: "real" shows "convex {x. a \<bullet> x \<le> b}"
Pi_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
trans_rtrancl_eq_reflcl: fixes A :: "('b \<times> 'b) set" assumes "trans A" shows "A\<^sup>* = A\<^sup>="
conversep_in_rel: fixes R :: "('e \<times> 'd) set" shows "(in_rel R)\<inverse>\<inverse> = in_rel (R\<inverse>)"
FVT_def: fixes t :: "trm" shows "FVT t = {x. \<exists>I \<nu> \<omega>. Vagree \<nu> \<omega> (- {x}) \<and> term_sem I t \<nu> \<noteq> term_sem I t \<omega>}"
corec_disc_1: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and q22 :: "'c \<Rightarrow> bool"   and g221 :: "'c \<Rightarrow> 'd llist"   and g222 :: "'c \<Rightarrow> 'c" assumes "p a" shows "lnull (corec_llist p g21 q22 g221 g222 a)"
arg_mult_real_negative: fixes k :: "real"   and z :: "complex" assumes "k < 0" shows "arg (cor k * z) = arg (- z)"
gmd_correct: fixes d :: "('a, 'b) Digit" assumes "FingerTreeStruc.is_measured_digit d" shows "FingerTreeStruc.gmd d = sum_list (map snd (FingerTreeStruc.digitToList d))"
bdd: shows "bdd_below {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
commutative_monoid_axioms: shows "commutative_monoid {0} ring0.subtraction 0"
unitor_coincidence_1: shows "\<l>\<^sub>1[\<I>\<^sub>1] = \<iota>\<^sub>1"
enter_room: fixes s :: "state"   and k :: "key"   and k' :: "key"   and g :: "guest"   and r :: "room" assumes "s \<in> reach"   and "(k, k') \<in> cards s g"   and "roomk s r \<in> {k, k'}" shows "s\<lparr>isin := (isin s)(r := isin s r \<union> {g}), roomk := (roomk s)(r := k'), safe := (safe s)(r := owns s r = \<lfloor>g\<rfloor> \<and> isin s r = {} \<and> k' = currk s r \<or> safe s r)\<rparr> \<in> reach"
star_rtc_least_eq: fixes x :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set" assumes "t_one \<union> x \<union> t_prod y y = y" shows "t_star x \<subseteq> y"
der_h: fixes y :: "(real, 'm) vec" assumes "y \<in> T" shows "(h has_derivative h' y) (at y within T)"
distrib: fixes g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "(\<Sum>x\<in>#A. g x + h x) = \<Sum>\<^sub># (image_mset g A) + \<Sum>\<^sub># (image_mset h A)"
rec: fixes f1 :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "old.rec_prod f1 (a, b) = f1 a b"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b fundef"   and b :: "'c fundef" assumes "rel_fundef R a b" obtains x1a :: "'b list" and x2a :: "nat" and y1a :: "'c list" and y2a :: "nat" where "a = Fundef x1a x2a" and "b = Fundef y1a y2a" and "list_all2 R x1a y1a" and "x2a = y2a"
pt_set_bij2: fixes x :: "'d"   and X :: "'d set"   and pi :: "('e \<times> 'e) list" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)"   and "x \<in> X" shows "pi \<bullet> x \<in> pi \<bullet> X"
MGT_body: fixes P :: "state \<Rightarrow> state \<Rightarrow> bool"   and p :: "pname"   and Q :: "state \<Rightarrow> state \<Rightarrow> bool"   and pa :: "pname"   and C :: "((state \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> (state \<Rightarrow> state \<Rightarrow> bool)) set" assumes "(P, CALL p, Q) = MGT (CALL pa)"   and "C \<tturnstile> {MGT (body p)}" shows "C \<turnstile> {P} body p {Q}"
simps_2: shows "sys' (Suc 0) = CH (ACT ''a'' (VAR (0::'a))) (PROC x)"
inverse_arrowsI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C (C g f)"   and "partial_magma.ide C (C f g)" shows "category.inverse_arrows C f g"
float_mult_l0: fixes e :: "int"   and x :: "real" shows "float (0, e) * x = float (0, 0)"
assms_4: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
ccspan_leq_using_vec: fixes A :: "'a list"   and B :: "'a list" shows "(ccspan (set A) \<le> ccspan (set B)) = is_subspace_of_vec_list (length canonical_basis) (map vec_of_basis_enum A) (map vec_of_basis_enum B)"
MkArr_in_hom': fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.is_left_adjoint f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>MkArr a b \<lbrakk>f\<rbrakk>\<^sub>B : MkIde a \<rightarrow> MkIde b\<guillemotright>"
assms: fixes s :: "'a"   and t :: "'a" assumes "s \<in> S"   and "t \<in> T" shows "is_orthogonal s t"
Sup_multiset_def: fixes A :: "'k multiset set" shows "Sup A = (if A \<noteq> {} \<and> subset_mset.bdd_above A then Abs_multiset (\<lambda>i. SUP X\<in>A. count X i) else {#})"
Step1_5_1b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Init (ImpInit p \<and> HInit rmhist p) \<and> \<box>ImpNext p \<and> \<box>[HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> \<box>(\<forall>l. $MemInv mm l) \<longrightarrow> \<box>ImpInv rmhist p"
that: fixes V :: "'a set"   and W :: "'b set" assumes "openin (top_of_set S) V"   and "x \<in> V"   and "openin (top_of_set T) W"   and "y \<in> W"   and "V \<times> W \<subseteq> U" shows "thesis"
list_of_dlist_inverse: fixes dxs :: "'c set_dlist" shows "Abs_dlist (list_of_dlist dxs) = dxs"
Rp_1: shows "integrable M (\<lambda>x. fp x * real_transfer_operator g x)"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "min (min a b) c = min a (min b c)"
span_add: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> vs2.span S"   and "y \<in> vs2.span S" shows "x + y \<in> vs2.span S"
eqvt: fixes p :: "perm"   and a :: "name list"   and aa :: "fm" shows "p \<bullet> trans_fm a aa = trans_fm (p \<bullet> a) (p \<bullet> aa)"
isometry_on_homeomorphism_3: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "X homeomorphic f ` X"
has_derivative_If_within_closures: fixes x :: "'a"   and S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "x \<in> S \<union> closure S \<inter> closure T \<Longrightarrow> (f has_derivative f' x) (at x within S \<union> closure S \<inter> closure T)"   and "x \<in> T \<union> closure S \<inter> closure T \<Longrightarrow> (g has_derivative g' x) (at x within T \<union> closure S \<inter> closure T)"   and "\<lbrakk>x \<in> closure S; x \<in> closure T\<rbrakk> \<Longrightarrow> f x = g x"   and "\<lbrakk>x \<in> closure S; x \<in> closure T\<rbrakk> \<Longrightarrow> f' x = g' x"   and "x \<in> S \<union> T" shows "((\<lambda>x. if x \<in> S then f x else g x) has_derivative (if x \<in> S then f' x else g' x)) (at x within S \<union> T)"
PPs_def: fixes X :: "'b set" shows ".[X] = {t. keys t \<subseteq> X}"
transformation_group_axioms: shows "transformation_group (left.translation ` G) G"
lemma_1_ii: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "last ` (#) s ` state_set (state_list (state_succ s a) as) = last ` state_set (state_list (state_succ s a) as)"
awalk_verts_with_proj_eq: fixes G :: "'b pair_pre_digraph" shows "pre_digraph.awalk_verts (with_proj G) = pawalk_verts"
D_not_diag_le: fixes x :: "'a"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "x \<in> {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}" shows "D m i j k \<le> x"
BSIA1: shows "BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>"
lift_Suc_mono_less_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))" shows "less (f n) (f m) = (n < m)"
f_id: fixes GH :: "'a set"   and s :: "('b, 'a, 'c) state" shows "GM_sec_model.f GH s s"
delete_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'm" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
empty_is_image: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "({} = f ` A) = (A = {})"
coeff_isolate_variable_sparse_minus_monomial: fixes mp :: "'a mpoly"   and var :: "nat"   and i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (isolate_variable_sparse mp var i) (m - monomial i var) = (if lookup m var \<le> i then MPoly_Type.coeff mp (Poly_Mapping.update var i m) else (0::'a))"
k: shows "gen_lucas_lehmer_sequence 4 (p - 2) = k * (2 ^ p - 1)"
HADelta_def: fixes HA :: "('d, 'e, 'f) hierauto" shows "HADelta HA = \<Union> (Delta ` SAs HA)"
finite_graph_def: fixes G :: "nat set \<times> nat set set" shows "finite_graph G \<equiv> finite (uverts G) \<and> finite (uedges G)"
sequence_emb_steps: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "(t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s) = (\<exists>us. us \<noteq> [] \<and> hd us = t \<and> last us = s \<and> (\<forall>i. Suc i < length us \<longrightarrow> us ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b us ! Suc i))"
infsetsum_infsetsum': fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f abs_summable_on A" shows "infsetsum f A = infsetsum' f A"
chg_map_upd: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "m a = Some b" shows "chg_map f a m = m(a \<mapsto> f b)"
isOp_castOp: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and s :: "'e"   and X :: "('b, 'c, 'd, 'a, 'f) qTerm set" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.isOp Delta s X" shows "FixSyn.wlsInp Delta (fst3 (FixSyn.castOp Delta s X)) (snd3 (FixSyn.castOp Delta s X)) \<and> FixSyn.wlsBinp Delta (fst3 (FixSyn.castOp Delta s X)) (trd3 (FixSyn.castOp Delta s X)) \<and> sortOf Delta (fst3 (FixSyn.castOp Delta s X)) = s \<and> FixVars.Op (fst3 (FixSyn.castOp Delta s X)) (snd3 (FixSyn.castOp Delta s X)) (trd3 (FixSyn.castOp Delta s X)) = X"
lincomb_isolate_1: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite A"   and "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K"   and "v \<in> A"   and "a v \<noteq> \<zero>\<^bsub>K\<^esub>"   and "lincomb a A = \<zero>\<^bsub>V\<^esub>" shows "v = lincomb (\<lambda>w. \<ominus>\<^bsub>K\<^esub> (inv\<^bsub>K\<^esub> a v) \<otimes>\<^bsub>K\<^esub> a w) (A - {v})"
canonical_proj_is_surj: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R"   and "I <+> J = carrier R" shows "canonical_proj I J ` carrier R = carrier (RDirProd (R Quot I) (R Quot J))"
add_minus_cancel: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "plus a (plus (uminus a) b) = b"
lexer_correctness_1: fixes r :: "'b rexp"   and s :: "'b list"   and v :: "'b val" shows "(lexer r s = Some v) = s \<in> r \<rightarrow> v"
removeNones_last: fixes ls :: "'a option list"   and a :: "'a" shows "removeNones (ls @ [Some a]) = removeNones ls @ [a]"
simps: fixes \<delta> :: "('c \<times> 'd \<times> 'c) set"   and q\<^sub>0 :: "'c"   and \<alpha> :: "(('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set) set"   and w :: "nat \<Rightarrow> 'd" shows "accept\<^sub>R_LTS (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. \<exists>r. LTS_is_inf_run \<delta> w r \<and> r 0 = q\<^sub>0 \<and> limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> Fin = {} \<and> limit (\<lambda>i. (r i, w i, r (Suc i))) \<inter> Inf \<noteq> {})"
zz2_def: shows "zz2 = restrict (I \<union> J) z2"
rtrancl_converseD: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> (r\<inverse>)\<^sup>*" shows "(y, x) \<in> r\<^sup>*"
prems_2: shows "rank x (n + Suc m) = Some j"
PRED_E: fixes v :: "nat"   and u :: "nat" assumes "v \<noteq> src"   and "PRED v = Some u" shows "(u, v) \<in> E"
measure_spmf_scale_spmf: fixes r :: "real"   and p :: "'b spmf" shows "measure_spmf (scale_spmf r p) = scale_measure (ennreal (min (inverse (weight_spmf p)) r)) (measure_spmf p)"
finite_graph_def: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" shows "finite_graph V E = (finite V \<and> finite E \<and> (\<forall>e\<in>E. fst e \<in> V \<and> snd e \<in> V \<and> fst e \<noteq> snd e))"
mset_factors_imp_dvd: fixes F :: "'b multiset"   and x :: "'b"   and f :: "'b" assumes "mset_factors F x"   and "f \<in># F" shows "f dvd x"
fst_comp_apfst: fixes f :: "'a \<Rightarrow> 'c" shows "(fst \<circ>\<circ> apfst) f = f \<circ> fst"
fbool: fixes X :: "nat set"   and Y :: "nat set"   and m :: "nat"   and f :: "bool \<Rightarrow> bool"   and e :: "nat" shows "finite {xs. (\<forall>i\<in>X. xs ! i) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> f (xs ! e)}"
mset_subset_eqD: fixes A :: "'b multiset"   and B :: "'b multiset"   and x :: "'b" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
arr_has_source: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "sources \<mu> \<noteq> {}"
goodBinp_def: fixes binp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option" shows "goodBinp binp \<equiv> liftAll goodAbs binp \<and> |{i. binp i \<noteq> None}| <o |UNIV|"