hcju/isabelleps huggingface dataset:
steps_hd_progress: fixes f :: "nat \<Rightarrow> ('s, 'p, 'f) com list \<times> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list \<times> ('s, 'f) xstate"   and c :: "('s, 'p, 'f) com"   and cs :: "('s, 'p, 'f) com list"   and css :: "(('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and s :: "('s, 'f) xstate"   and \<Gamma> :: "'p \<Rightarrow> ('s, 'p, 'f) com option"   and k :: "nat" assumes "f 0 = (c # cs, css, s)"   and "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "\<forall>i<k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)" shows "\<forall>i\<le>k. \<exists>pcs pcss. if pcss = [] then CSS (f i) = css \<and> CS (f i) = pcs @ cs else CS (f i) = pcs \<and> CSS (f i) = butlast pcss @ [(fst (last pcss) @ cs, snd (last pcss) @ cs)] @ css"
red_external_ta_bisim21: fixes wf_md :: "'addr J1_prog \<Rightarrow> String.literal \<Rightarrow> 'addr expr1 mdecl \<Rightarrow> bool"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "wf_prog wf_md P"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "hconf h'"   and "preallocated h'" shows "ta_bisim wbisim1 (extTA2J1 P ta) (extTA2JVM (compP2 P) ta)"
face_of_convex_hull_affine_independent: fixes S :: "'b set"   and T :: "'b set" assumes "\<not> affine_dependent S" shows "(T face_of convex hull S) = (\<exists>c\<subseteq>S. T = convex hull c)"
iMODb_plus_divisor_mult: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat"   and k :: "nat" assumes "x \<in> [ r, mod m, c ]"   and "k * m \<le> r + m * c - x" shows "x + k * m \<in> [ r, mod m, c ]"
Un_least: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
Q2'_def: shows "Q2' = (\<lambda>t t' X. \<forall>f T. Fun f T \<in> subterms t \<union> subterms t' \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X))"
NSBseqD2: fixes X :: "nat \<Rightarrow> 'a"   and N :: "nat star" assumes "NSBseq X" shows "(*f* X) N \<in> HFinite"
isoD_2: fixes f :: "'a1 \<times> 'a2" assumes "local.iso f" shows "C2.iso (snd f)"
simps_4: fixes \<A> :: "'a ltln set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> or\<^sub>n \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<or> \<A> \<Turnstile>\<^sub>P \<psi>)"
coeff_sdiv_poly: fixes f :: "'a poly"   and a :: "'a"   and n :: "nat" shows "poly.coeff (sdiv_poly f a) n = poly.coeff f n div a"
wellformed_\<R>: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "finite (reach \<Sigma> (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0) (semi_mojmir_def.initial q\<^sub>0))"
override_on_apply_notin: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "a \<notin> A" shows "(f ++\<^bsub>A\<^esub> g) a = f a"
in_set_of_round_interval: fixes x :: "real"   and X :: "float interval"   and prec :: "nat" assumes "x \<in>\<^sub>r X" shows "x \<in>\<^sub>r round_interval prec X"
icard_1_imp_singleton: fixes A :: "'a set" assumes "icard A = eSuc 0" shows "\<exists>a. A = {a}"
row_echelon_form_imp_1_or_0_row: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "row_echelon_form A" shows "A = 1\<^sub>m n \<or> 0 < n \<and> row A (n - 1) = 0\<^sub>v n"
apply_matrix_add: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b matrix"   and b :: "'b matrix" assumes "\<forall>x y. f (x + y) = f x + f y"   and "f (0::'b) = (0::'b)" shows "apply_matrix f (a + b) = apply_matrix f a + apply_matrix f b"
pq_def: shows "pq = endpoint_in_S (Rep_hyp2 b) (Rep_hyp2 a)"
distances: shows "max (dist S X) (dist X T) = dist S M + dist X M \<and> min (dist S X) (dist X T) = dist S M - dist X M"
ffold_empty: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "ffold f z {||} = z"
finsert_fminus_single: fixes a :: "'b"   and A :: "'b fset" shows "finsert a (A |-| {|a|}) = finsert a A"
approx_form_spec_def: fixes ea :: "form"   and E :: "real list set" shows "approx_form_spec ea E = SPEC (\<lambda>r. r \<longrightarrow> (\<forall>e\<in>E. interpret_form ea e))"
dep3_event_def: fixes Q :: "'a set"   and R :: "'a set"   and S :: "'a set"   and x :: "'a" shows "dep3_event Q R S x \<equiv> Q \<noteq> R \<and> Q \<noteq> S \<and> R \<noteq> S \<and> Q \<in> SPRAY x \<and> R \<in> SPRAY x \<and> S \<in> SPRAY x \<and> (\<exists>T\<in>\<P>. T \<notin> SPRAY x \<and> (\<exists>y\<in>Q. y \<in> T) \<and> (\<exists>y\<in>R. y \<in> T) \<and> (\<exists>y\<in>S. y \<in> T))"
S_alt_def: shows "S A = {r} \<union> fst ` A"
C: shows "\<forall>X. (X \<in> refusals P xs) = (X \<inter> next_events P xs = {})"
signof_inv: fixes S :: "nat set"   and p :: "nat \<Rightarrow> nat" assumes "finite S"   and "p permutes S" shows "signof (inv p) = signof p"
frrestriction_vsingleton_nin: fixes b :: "V"   and A :: "V"   and a :: "V" assumes "b \<notin>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} \<restriction>\<^sup>r\<^sub>\<bullet> A = []\<^sub>\<circ>"
ex_def: fixes l :: "'b list"   and f :: "'b \<Rightarrow> bool" shows "ex l f = list_ex f l"
sr_max: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "r x \<le> sr"
real_polynomial_function_eq: fixes p :: "'b \<Rightarrow> real" shows "real_polynomial_function p = polynomial_function p"
qSwapAll_sym: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "X #[[x \<and> y]]_zs = X #[[y \<and> x]]_zs \<and> A $[[x \<and> y]]_zs = A $[[y \<and> x]]_zs"
bij_swap_compose: fixes x :: "'b"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "bij (x \<rightleftharpoons>\<^sub>F y \<circ> f) = bij f"
div_reals: fixes a :: "complex"   and b :: "complex" assumes "is_real a"   and "is_real b" shows "is_real (a / b)"
vTeT: fixes v :: "'a"   and e :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T" shows "v[[e\<^sup>T]] = bot"
invertible_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "invertible A = (\<exists>A'. A ** A' = Finite_Cartesian_Product.mat (1::'a) \<and> A' ** A = Finite_Cartesian_Product.mat (1::'a))"
intros_1: fixes T :: "'a set"   and A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "dep3_event T A B x" shows "dep_path T {A, B} x"
prime_1_mod_4_sum_of_squares_unique_aux: fixes p :: "nat"   and x :: "nat"   and y :: "nat" assumes "prime p"   and "[p = 1] (mod 4)"   and "x\<^sup>2 + y\<^sup>2 = p" shows "0 < x \<and> 0 < y \<and> x \<noteq> y"
left_totalE: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "left_total R" obtains "\<And>x. \<exists>y. R x y"
par_rreq_rrep_sn_quality_increases: fixes i :: "nat" shows "opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). quality_increases (\<sigma> i) (\<sigma>' i))"
monad_fail_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind" shows "monad_fail (return_option return) (bind_option return bind) (fail_option return)"
simps_9: fixes x3 :: "'b formula"   and x1 :: "'b" shows "\<^bold>\<not> x3 \<noteq> Atom x1"
TunerMode_CTRL_def: shows "TunerMode_CTRL = Abs_seqauto (TunerMode_CTRL_States, TunerMode_CTRL_Init, TunerMode_CTRL_Labels, TunerMode_CTRL_Delta)"
before_id: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "xs ! i < xs ! j in xs = (i < j)"
simps_3: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_pair (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va) = (let p\<^sub>1 = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va) in if dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then find_closest_pair (c\<^sub>0, c\<^sub>1) (v # va) else find_closest_pair (p\<^sub>0, p\<^sub>1) (v # va))"
vcgaDefinitenessVariant: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "card (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b)) = 1"
H_rcosets_aux: fixes x :: "nat" assumes "euler_witness (int x) p"   and "x \<in> carrier G" shows "{H, H #>\<^bsub>G\<^esub> x} \<subseteq> rcosets\<^bsub>G\<^esub> H"
z_facts_4: shows "z \<notin> Fvars \<phi>"
add_le_imp_le_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a + c \<le> b + c" shows "a \<le> b"
mult: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> H"   and "h2 \<in> H" shows "(\<lambda>x. h1 x * h2 x) \<in> H"
bind_tell: fixes w :: "'w"   and m :: "'m"   and f :: "'a option \<Rightarrow> 'm" shows "bind (tell w m) f = tell w (bind m f)"
nabla_fusion2: fixes X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rel_nabla_bin X Q \<subseteq> rel_nabla X \<union> relfdia (X\<^sup>*) Q"
less_and_segment: fixes b :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "b \<in> carrier D" shows "(\<forall>a. a \<prec> b \<and> a \<in> carrier D \<longrightarrow> Q a) = (\<forall>a\<in>carrier (Iod D (segment D b)). Q a)"
scale: fixes r :: "real"   and b :: "'a" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
subformulas\<^sub>\<nu>_list_length: fixes \<phi> :: "'b ltln" shows "length (subformulas\<^sub>\<nu>_list \<phi>) = card (subformulas\<^sub>\<nu> \<phi>)"
fact_prod: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (\<Prod>{1..n})"
height_full_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "height (full_node k a h) = h"
mtf2_def: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "mtf2 n x xs = (if x \<in> set xs then swaps [index xs x - n..<index xs x] xs else xs)"
select_convs_2: fixes newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "newAID \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = newAID"
noAbrupt_startn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<forall>t'. t \<noteq> Abrupt t'" shows "s \<noteq> Abrupt s'"
rep_nat_term_prod_def: fixes u :: "'a \<times> 'b" shows "rep_nat_term u = (rep_nat_pp (fst u), rep_nat (snd u))"
inf_compl_bot_left2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf (uminus x) y) = bot"
chine_composite_2: shows "Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>) = cods.\<mu>.leg0 \<down>\<down> cods.\<nu>.leg1 \<cdot> cods.\<nu>\<pi>.prj\<^sub>1"
binomial_fact_lemma: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "fact k * fact (n - k) * (n choose k) = fact n"
inf_commute: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "inf_prod x y = inf_prod y x"
induct: fixes P :: "('d, 'e, 'f) cntxt \<Rightarrow> bool"   and cntxt :: "('d, 'e, 'f) cntxt" assumes "P \<box>"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (ApplyL\<^sub>\<E> x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (ApplyR\<^sub>\<E> x1 x2)"   and "\<And>x1 x2 x3. P x1 \<Longrightarrow> P (Ite\<^sub>\<E> x1 x2 x3)"   and "\<And>x. P x \<Longrightarrow> P (Ref\<^sub>\<E> x)"   and "\<And>x. P x \<Longrightarrow> P (Read\<^sub>\<E> x)"   and "\<And>x1 x2. P x1 \<Longrightarrow> P (AssignL\<^sub>\<E> x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (AssignR\<^sub>\<E> x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (Rjoin\<^sub>\<E> x)" shows "P cntxt"
inv_insert: fixes t :: "('a \<times> color) tree"   and x :: "'a" assumes "invc t \<and> invh t" shows "invc (insert_rbt x t) \<and> invh (insert_rbt x t)"
continuous_at_real_range: fixes x :: "'b"   and f :: "'b \<Rightarrow> real" shows "isCont f x = (\<forall>e>0. \<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e)"
iIN_iMODb_neq2: fixes m :: "nat"   and d :: "nat"   and n :: "nat"   and r :: "nat"   and c :: "nat" assumes "m \<noteq> Suc 0"   and "0 < d" shows "[n\<dots>,d] \<noteq> [ r, mod m, c ]"
snoc_1: assumes "insert_ops xs"   and "(oid, ref) \<in> set xs" shows "\<exists>pre suf. xs = pre @ [(oid, ref)] @ suf \<and> (\<forall>i\<in>set (map fst pre). i < oid) \<and> (\<forall>a\<in>set (map fst suf). oid < a)"
neq_if_length_neq: fixes xs :: "'a list"   and ys :: "'a list" assumes "|xs| \<noteq> |ys|" shows "xs = ys \<equiv> False"
homeomorphic_connected_space: fixes X :: "'c topology"   and Y :: "'d topology" assumes "X homeomorphic_space Y" shows "connected_space X = connected_space Y"
simps_58: fixes x8 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type" shows "Sequence x8 \<noteq> Collection x4"
extrp_extr_eq: fixes T :: "atom set"   and H :: "chmsg set" shows "extrp (\<lambda>x. x \<in> T) (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> extr T H)"
converse_Int: fixes r :: "('c \<times> 'b) set"   and s :: "('c \<times> 'b) set" shows "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
ball_def: fixes x :: "'a"   and e :: "real" shows "ball x e = {y. dist x y < e}"
F_Inf_def: shows "F_Inf = {uu_. \<exists>CAs DA AAs As \<sigma> E. uu_ = Infer (CAs @ [DA]) E \<and> ord_resolve_rename S CAs DA AAs As \<sigma> E}"
7: shows "sset (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p))) \<subseteq> gunodes A w"
empty_inside: fixes S :: "complex set" assumes "SC_Chain S"   and "connected S"   and "\<And>C. C \<in> components (- S) \<Longrightarrow> \<not> bounded C" shows "inside S = {}"
l3_inv8I: fixes x :: "l3_state" assumes "\<And>A B M. implAuth A B M \<in> synth (analz (ik x)) \<and> M \<in> payload \<Longrightarrow> implAuth A B M \<in> ik x \<or> M \<in> synth (analz (ik x)) \<and> (A \<in> l3_state.bad x \<or> B \<in> l3_state.bad x)" shows "x \<in> l3_inv8"
emb_lfinite_antimono: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "emb xs ys"   and "lfinite ys" shows "lfinite xs"
pars__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "Coplanar A B C D"
create_graph_aimpl_refine: fixes \<phi> :: "'a ltlr" shows "create_graph_aimpl \<phi> \<le> \<Down> Id (create_graph\<^sub>T \<phi>)"
implSecure_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags"   and "asym.implSecure A B M \<in> synth (analz H)" shows "asym.implSecure A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
intros: fixes p :: "'b \<times> 'c" shows "fstsp p (fst p)"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<ll>.map a = \<l>[a]"
order_eq_iff: fixes x :: "'d"   and y :: "'d" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
higher_differentiable_on_congI: fixes S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and n :: "nat"   and f :: "'c \<Rightarrow> 'd" assumes "open S"   and "higher_differentiable_on S g n"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "higher_differentiable_on S f n"
list_of_eucl_def: fixes x :: "'a" shows "list_of_eucl x = map ((\<bullet>) x) Basis_list"
assms_3: shows "\<FF>' = \<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>"
simps_2: fixes uv :: "int \<times> int"   and p :: "int \<times> int" shows "find_closest_bf uv [p] = p"
card_vimage_inj: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "inj f"   and "A \<subseteq> range f" shows "card (f -` A) = card A"
reflp_on_restrict_to_rtranclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "reflp_on P A"   and "transp_on P A"   and "x \<in> A"   and "y \<in> A" shows "(restrict_to P A)\<^sup>*\<^sup>* x y = P x y"
extremum: fixes a :: "'b" shows "(0::'b) \<le> a"
is_expansion_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and abort :: "bool"   and n :: "nat" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "trimmed F"   and "is_expansion F basis" shows "is_expansion (power_expansion abort F n) basis"
arg_complex_of_real_negative: fixes k :: "real" assumes "k < 0" shows "arg (cor k) = pi"
cases: fixes x :: "value" obtains uu_ :: "int" where "x = Num uu_"   | uv_ :: "String.literal" where "x = Str uv_"
that: assumes "r = Array.length h' a"   and "h' = h" shows "thesis"
lookup_prod: shows "lookup (tensors_from_net (Pool m1 m2) $ j) is = lookup (tensors_from_net m1 $ j) is1 * lookup (tensors_from_net m2 $ j) is2"
coeff_minus: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "coeff P (p \<ominus>\<^bsub>P\<^esub> q) n = coeff P p n \<ominus> coeff P q n"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>L 'b" shows "blinfun_apply (x + xa) = (\<lambda>xb. blinfun_apply x xb + blinfun_apply xa xb)"
RestrpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and a :: "'c"   and b :: "'c" assumes "(r \<restriction> A) a b" obtains "a \<in> A" and "b \<in> A" and "r a b"
arr_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "arr f = (f = Zero \<or> f = One \<or> f \<in> fromArr ` {False, True})"
LIM_zero_iff: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'d)) F = (f \<longlongrightarrow> l) F"
cSup_subset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq B"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
flip_flip: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "flip (flip f) = f"
\<S>_L: fixes l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "(l, R) \<in> \<S>" shows "l \<in> L"
simps_15: fixes x3 :: "'e" shows "Stuck \<noteq> Fault x3"
induct: fixes x1 :: "dB"   and x2 :: "dB"   and P :: "dB \<Rightarrow> dB \<Rightarrow> bool" assumes "x1 => x2"   and "\<And>n. P (Var n) (Var n)"   and "\<And>s t. \<lbrakk>s => t; P s t\<rbrakk> \<Longrightarrow> P (Abs s) (Abs t)"   and "\<And>s s' t t'. \<lbrakk>s => s'; P s s'; t => t'; P t t'\<rbrakk> \<Longrightarrow> P (s \<degree> t) (s' \<degree> t')"   and "\<And>s s' t t'. \<lbrakk>s => s'; P s s'; t => t'; P t t'\<rbrakk> \<Longrightarrow> P (Abs s \<degree> t) (s'[t'/0])" shows "P x1 x2"
M: shows "partn_lst {..<M} \<alpha> \<gamma>"
difficult_direction: shows "consistent_sign_vectors_R qs UNIV \<subseteq> consistent_sign_vectors_R qs (set S)"
Main_no_call_target: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and f :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>Main\<^esub>f" shows "False"
rsp: shows "eq_onp has_ell2_norm (\<lambda>_. 0) (\<lambda>_. 0)"
finprod_reindex: fixes f :: "'c \<Rightarrow> 'a set"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "f \<in> h ` A \<rightarrow> carrier (ideals_set R)"   and "inj_on h A" shows "finprod (ideals_set R) f (h ` A) = (\<Otimes>\<^bsub>ideals_set R\<^esub>x\<in>A. f (h x))"
map_scale_uminus_right: fixes k :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "k \<cdot> - p = - (k \<cdot> p)"
integrable_const: fixes M :: "'b measure"   and a :: "'c" assumes "finite_measure M" shows "integrable M (\<lambda>x. a)"
inject: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex"   and y31 :: "'d"   and y32 :: "'d ifex"   and y33 :: "'d ifex" shows "(IF x31 x32 x33 = IF y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
epred_Sup: fixes A :: "enat set" shows "epred (\<Squnion> A) = \<Squnion> (epred ` A)"
mat_of_cblinfun_uminus: fixes M :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun (- M) = - mat_of_cblinfun M"
NF_not_suc: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> r\<^sup>*"   and "x \<in> NF r" shows "x = y"
le_transI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> t P \<tturnstile> u P" shows "le_trans t u"
Liminf_filtermap_le: fixes f :: "'e \<Rightarrow> 'd"   and F :: "'e filter"   and g :: "'d \<Rightarrow> 'b" shows "Liminf (filtermap f F) g \<le> Liminf F (\<lambda>x. g (f x))"
set_intros_2: fixes y :: "'a"   and x22 :: "'a list"   and x21 :: "'a" assumes "y mem x22" shows "y mem (x21 # x22)"
simps_5: fixes v :: "term"   and va :: "term" shows "unfree (v $ va) = None"
Goal: fixes P :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "depth P + depth Q \<le> Suc n"   and "valid P"   and "valid Q"   and "uhnf P"   and "uhnf Q"   and "P \<leadsto>[bisim] Q"   and "Q' \<in> summands Q" shows "\<exists>P'\<in>summands P. Q' \<equiv>\<^sub>e P'"