hcju/isabelleps huggingface dataset:
extended_distance_triang_ineq: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded"   and z :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> extended_distance_set"   and "y \<in> extended_distance_set"   and "z \<in> extended_distance_set" shows "extended_distance x z \<le> extended_distance x y + extended_distance y z"
simps_6: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g \<Rightarrow> 'g" shows "rec_list f1 f2 [] = f1"
fst_map_prod: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and x :: "'e \<times> 'f" shows "fst (map_prod f g x) = f (fst x)"
top_higher: fixes x :: "'a" assumes "x \<in> carrier (fpl L f)" shows "x \<sqsubseteq>\<^bsub>fpl L f\<^esub> \<top>\<^bsub>fpl L f\<^esub>"
assms_5: shows "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z"
v'_def: shows "v' = \<lparr>ext = ext u', lan = lan u', own = own v\<rparr>"
constant_square_free_factorization: fixes p :: "'a poly" assumes "degree p = 0" shows "square_free_factorization p (coeff p 0, [])"
succs_iexec1: fixes c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and P :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list" assumes "c' = iexec (P !! i) (i, s, stk)"   and "0 \<le> i"   and "i < size P" shows "fst c' \<in> succs P 0"
hcomp_in_vhom: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c"   and \<nu> :: "'c"   and h :: "'c"   and k :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.in_hom V \<mu> f g"   and "partial_magma.in_hom V \<nu> h k"   and "src h = trg f" shows "partial_magma.in_hom V (H \<nu> \<mu>) (H h f) (H k g)"
mset_merge: fixes h1 :: "'b tree"   and h2 :: "'b tree" assumes "is_root h1"   and "is_root h2" shows "mset_tree (merge h1 h2) = mset_tree h1 + mset_tree h2"
coeff_map_poly_hom: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly"   and i :: "nat" assumes "zero_hom hom" shows "coeff (map_poly hom p) i = hom (coeff p i)"
simps_2: fixes X :: "binding"   and \<Gamma> :: "binding list" shows "ty_dom (X # \<Gamma>) = tyvrs_of X \<union> ty_dom \<Gamma>"
induct: fixes P :: "'a poly \<Rightarrow> bool"   and a0 :: "'a poly" assumes "\<And>n. P (poly.Bound n)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Add p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Sub p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Mul p q)"   and "\<And>p. P p \<Longrightarrow> P (Neg p)"   and "\<And>p n. P p \<Longrightarrow> P (Pw p n)"   and "\<And>c n p. P (Add c (Mul (poly.Bound n) p)) \<Longrightarrow> P (CN c n p)"   and "\<And>c. P (c)\<^sub>p" shows "P a0"
open_segment_line_iff: fixes i :: "real"   and a :: "'b"   and b :: "'b" assumes "0 < i"   and "a \<noteq> b" shows "{a<--<{a -- b}\<^bsub>i\<^esub>} = {a -- b}\<^bsub>`{0<..<i}\<^esub>"
case_4: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x41 :: "('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool"   and x42 :: "'e"   and x43 :: "('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option" shows "(case x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f4 x41 x42 x43"
fds_nth_inverse: fixes f :: "'b fds" shows "fds_nth (inverse f) = dirichlet_inverse (fds_nth f) (inverse (fds_nth f 1))"
assms_1: shows "w \<in> set_child_nodes_locs object_ptr"
semidecisive_coalitionD: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and C :: "'a set"   and x :: "'b"   and y :: "'b" assumes "semidecisive scf A Is C x y" shows "C \<subseteq> Is"
that: fixes c :: "real"   and n :: "nat" assumes "0 < c"   and "\<And>p q. 0 < q \<Longrightarrow> c / real_of_int q ^ n < \<bar>x - real_of_int p / real_of_int q\<bar>" shows "thesis"
implies_dist_implies: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> (y \<leadsto> z) \<le> x \<leadsto> y \<leadsto> (x \<leadsto> z)"
Cod_gf: shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = cd"
continuous_at_e2ennreal: fixes x :: "ereal"   and A :: "ereal set" shows "continuous (at x within A) e2ennreal"
p2rel_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "p2rel P \<equiv> {(x, y). P x y}"
f1: shows "dim_vec (l 0 \<cdot>\<^sub>v v 0 + l 1 \<cdot>\<^sub>v v 1) = dim_vec u"
P3: fixes rst :: "'a common_primitive match_expr"   and spt :: "ipt_l4_ports" assumes "P (disc2, sel2) rst" shows "P (disc2, sel2) (MatchAnd (Match (C spt)) rst)"
dim_null_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.dim (null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A"
publicKey_inject: fixes A :: "agent"   and A' :: "agent" shows "(pubK A = pubK A') = (A = A')"
card_of_empty3: fixes A :: "'d set" assumes "|A| \<le>o |{}|" shows "A = {}"
select_convs_5: fixes igWls :: "'h \<Rightarrow> 'd \<Rightarrow> bool"   and igWlsAbs :: "'g \<times> 'h \<Rightarrow> 'f \<Rightarrow> bool"   and igVar :: "'g \<Rightarrow> 'i \<Rightarrow> 'd"   and igAbs :: "'g \<Rightarrow> 'i \<Rightarrow> 'd \<Rightarrow> 'f"   and igOp :: "'b \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> ('e \<Rightarrow> 'f option) \<Rightarrow> 'd"   and igFresh :: "'g \<Rightarrow> 'i \<Rightarrow> 'd \<Rightarrow> bool"   and igFreshAbs :: "'g \<Rightarrow> 'i \<Rightarrow> 'f \<Rightarrow> bool"   and igSwap :: "'g \<Rightarrow> 'i \<Rightarrow> 'i \<Rightarrow> 'd \<Rightarrow> 'd"   and igSwapAbs :: "'g \<Rightarrow> 'i \<Rightarrow> 'i \<Rightarrow> 'f \<Rightarrow> 'f"   and igSubst :: "'g \<Rightarrow> 'd \<Rightarrow> 'i \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubstAbs :: "'g \<Rightarrow> 'd \<Rightarrow> 'i \<Rightarrow> 'f \<Rightarrow> 'f"   and more :: "'j" shows "igOp \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igOp"
phl_export2: fixes w :: "'a"   and z :: "'a"   and y :: "'a"   and x :: "'a" assumes "w \<cdot> z \<le> w \<cdot> (z \<cdot> w)"   and "y \<cdot> x \<le> z \<cdot> y" shows "w \<cdot> y \<cdot> x \<le> w \<cdot> z \<cdot> (w \<cdot> y)"
f': shows "f (a /\<^sub>R norm a) = b /\<^sub>R norm b"
open_segment_furthest_le: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and y :: "'a" assumes "x \<in> open_segment a b" shows "norm (y - x) < norm (y - a) \<or> norm (y - x) < norm (y - b)"
le: shows "\<forall>y\<in>set (sorted_list_of_set ys). x < y"
lm154: fixes X :: "'a set"   and Y :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "graph (X \<inter> Y) f = graph X f || Y"
ordered_model_resolvent_def: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.ordered_model_resolvent atom_ordering I P1 P2 C = (\<exists>L. C = P1 - {L} \<union> (P2 - {complement L}) \<and> (propositional_atoms.validated_part I P1 = {} \<and> propositional_atoms.strictly_maximal_literal atom_ordering P1 L) \<and> propositional_atoms.strictly_maximal_literal atom_ordering (propositional_atoms.validated_part I P2) (complement L))"
ba2_associative_commutative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = y \<squnion> z \<squnion> x"
WTrt_elim_cases_10: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'addr expr"   and i :: "'addr expr"   and e :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> a\<lfloor>i\<rceil> := e : T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Void" and "P,E,h \<turnstile> a : T\<lfloor>\<rceil>" and "P,E,h \<turnstile> i : Integer" and "P,E,h \<turnstile> e : T'__"   | T' :: "ty" where "Ta__ = Void" and "P,E,h \<turnstile> a : NT" and "P,E,h \<turnstile> i : Integer" and "P,E,h \<turnstile> e : T'"
no_head_gen_code: fixes C :: "'b list set"   and b :: "'b list" assumes "code C"   and "b \<in> C" shows "code {z \<cdot> b \<^sup>@ k |z k. z \<in> C \<and> z \<noteq> b}"
in_diffD: fixes a :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "a \<in># M - N" shows "a \<in># M"
E_def: shows "E = {s \<in> f. interior s = {}}"
22: shows "(\<lambda>ia. (ia div y, ia mod y)) ` {0..<x * y} \<subseteq> {0..<x} \<times> {0..<y}"
infinite_nat_iff_asc_chain: fixes S :: "nat set" assumes "S \<noteq> {}" shows "infinite S = (\<forall>m\<in>S. \<exists>n\<in>S. m < n)"
gen_ramanujan_sum_dirichlet_expr: fixes f :: "nat \<Rightarrow> complex"   and h :: "nat \<Rightarrow> complex"   and k :: "nat"   and n :: "nat" assumes "completely_multiplicative_function f"   and "multiplicative_function h"   and "\<And>p. prime p \<Longrightarrow> f p \<noteq> 0 \<and> f p \<noteq> h p"   and "0 < k"   and "0 < n" shows "s f (\<lambda>k. moebius_mu k * h k) k n = dirichlet_prod f (\<lambda>k. moebius_mu k * h k) k * (moebius_mu (k div gcd n k) * h (k div gcd n k)) / dirichlet_prod f (\<lambda>k. moebius_mu k * h k) (k div gcd n k)"
rPar2: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>P :: "name list" assumes "cP = P \<parallel> Q"   and "cRs = \<alpha> \<prec> P \<parallel> Q'"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop"
aug_minf_def: shows "Z\<^sub>-\<^sub>\<infinity> = {z. z \<noteq> \<infinity>}"
arc_regular: fixes x :: "'a" assumes "(x \<sqinter> top \<sqinter> id (x \<sqinter> top) \<le> top \<and> top \<sqinter> (x \<sqinter> top) = top) \<and> id x \<sqinter> top \<sqinter> id (id x \<sqinter> top) \<le> top \<and> top \<sqinter> (id x \<sqinter> top) = top" shows "regular x"
zero_le_floor: fixes x :: "'b" shows "(0 \<le> \<lfloor>x\<rfloor>) = ((0::'b) \<le> x)"
cball_subset_cball_iff: fixes a :: "'a"   and r :: "real"   and a' :: "'a"   and r' :: "real" shows "(cball a r \<subseteq> cball a' r') = (dist a a' + r \<le> r' \<or> r < 0)"
Sup_set_def: fixes A :: "'c set set" shows "\<mu> A = {x. \<Squnion>B\<in>A. s2p B x}"
4: shows "r\<^sup>T \<le> (v \<squnion> e\<^sup>T * top)\<^sup>T"
code_id_def: fixes m :: "nat"   and n :: "nat" shows "code_id m n \<equiv> triple_encode 2 m n"
l2_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "l2_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xpkE, xskE}), chan := {Auth A B \<langle>Number 0, epubKF (Ra $ kE)\<rangle>} \<union> chan s\<rparr>}"
assms_9: shows "\<forall>ax ax'. ax \<in> set as' \<and> sourcenode ax = sourcenode ax' \<and> valid_edge ax' \<longrightarrow> n postdominates targetnode ax'"
assms: shows "s \<in> timpl_closure_set {t \<cdot> \<delta> |\<delta>. P \<delta>} T"
ode_fa_def: fixes ode_ops :: "ode_ops"   and xs :: "floatarith list" shows "ode_operations.ode_fa ode_ops xs = map (subst_floatarith ((!) xs)) (ode_operations.ode_e ode_ops)"
intro: fixes s_ops :: "('a, 'b, 'c) set_ops_scheme"   and m_ops :: "('a, nat, 'd, 'e) map_ops_scheme" assumes "StdSet s_ops"   and "StdMap m_ops" shows "map_to_nat_loc s_ops m_ops"
erf_remainder_bigo: fixes n :: "nat" shows "erf_remainder n \<in> O(\<lambda>x. exp (- x\<^sup>2) / x ^ (2 * n + 1))"
simps_2: fixes m :: "'c"   and ms :: "'c list" shows "GSpan (m # ms) = {r \<cdot> m |r. r \<in> FG} + GSpan ms"
cases: fixes x :: "'a trm" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "'a" and Tl :: "'a trm list" where "x = Fn f Tl"
filter_def: fixes P :: "'d \<Rightarrow> bool"   and dxs :: "'d dlist" shows "Dlist.filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
rtrancl_power: fixes p :: "'c \<times> 'c"   and R :: "('c \<times> 'c) set" shows "(p \<in> R\<^sup>*) = (\<exists>n. p \<in> R ^^ n)"
IN: fixes x :: "'v" shows "d_IN f x \<le> weight \<Gamma> x"
obj_char: fixes a :: "'a" shows "obj a = (a = \<I>)"
mwb_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "mwb_lens x \<equiv> weak_lens x \<and> mwb_lens_axioms x"
dags_in_high_level: fixes t1 :: "dag"   and t2 :: "dag" assumes "t1 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high"   and "t1 \<notin> Dags (fnct ` Nodes n levellista) low high"   and "t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high"   and "t2 \<notin> Dags (fnct ` Nodes n levellista) low high" shows "P t1 t2 var"
less_eq_le_pred: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "(m < n) = (m \<le> n - Suc 0)"
less_le_not_le: fixes x :: "'a \<times> ('a, 'b) operation"   and y :: "'a \<times> ('a, 'b) operation" shows "hb x y = (weak_hb x y \<and> \<not> weak_hb y x)"
map_id: fixes t :: "('d, 'e, 'f) val" shows "\<R>\<^sub>V id id t = t"
assoc_simps_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "local.dom (assoc a b c) = local.prod (local.prod a b) c"
inject_3: fixes x3 :: "key"   and y3 :: "key" shows "(aKey x3 = aKey y3) = (x3 = y3)"
simps_2: fixes R :: "('c, 'd) ring_scheme"   and v :: "'c"   and va :: "'c list" assumes "ring R" shows "ring.dense_repr R (v # va) = (if lead_coeff (v # va) \<noteq> \<zero>\<^bsub>R\<^esub> then (lead_coeff (v # va), degree (v # va)) # ring.dense_repr R (tl (v # va)) else ring.dense_repr R (tl (v # va)))"
necessitation_1: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g" assumes "\<And>s. P s" shows "(\<box>P) \<sigma>"
Ifaces_def: shows "Ifaces = [(HasFoo, HasFooInt)]"
Sup_sup_pred: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" shows "x \<squnion> Sup {y. P y} = Sup {y. y = x \<or> P y}"
id_eq_one: shows "(\<lambda>x\<in>E. x) = \<phi> \<one>"
approximation_preproc_nat_9: fixes i :: "nat"   and n :: "nat" shows "real (i ^ n) = real i ^ n"
exhaust_sel: fixes option :: "'c option" obtains "option = None"   | "option = \<lfloor>the option\<rfloor>"
proves_eq_combination_rule_better: fixes \<Theta> :: "((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 \<Gamma> :: "Core.term set"   and f :: "Core.term"   and g :: "Core.term"   and x :: "Core.term"   and y :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq f g"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq x y"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "typ_of x = Some \<tau>"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (f $ x) (g $ y)"
funionM_commute: fixes x :: "'m"   and y :: "'m" shows "funionM x y = funionM y x"
method_name_f_def: fixes md :: "meth_def" shows "method_name_f md = (case md of meth_def_def (meth_sig_def cl meth vds) meth_body \<Rightarrow> meth)"
sim_action_cases_4: fixes ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and obs :: "('a, 'b) obs_event action" assumes "NormalAction (ReadMem ad al v) \<approx> obs" obtains v' :: "'a val" where "obs = NormalAction (ReadMem ad al v')"
B_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.B p a = (\<lambda>x. x * a) ` GAUSS.A p"
S: shows "fst (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))) = 1\<^sub>m 1 * Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)) * snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i))))"
add_mset_diff_bothsides: fixes a :: "'b"   and M :: "'b multiset"   and A :: "'b multiset" shows "add_mset a M - add_mset a A = M - A"
eval_append: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "a \<in> carrier R" shows "local.eval (p @ q) a = local.eval p a \<otimes> a [^] length q \<oplus> local.eval q a"
canonical_retraction_vertex_im: fixes A :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A" shows "canonical_retraction A C ` \<Union> X = \<Union> A"
env_restr_env_delete_other: fixes x :: "'d"   and S :: "'d set"   and m :: "'d \<Rightarrow> 'e" assumes "x \<notin> S" shows "env_delete x m f|` S = m f|` S"
fpr_mid_path: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path :: "(ctx \<times> cld) list"   and ctx' :: "ctx"   and cld' :: "cld"   and path' :: "(ctx \<times> cld) list" assumes "find_path_f P ctx cl = Some path"   and "(ctx', cld') \<in> set path"   and "find_path_f P ctx' (cl_fqn (fqn_def (class_name_f cld'))) = Some path'" shows "\<exists>path''. path = path'' @ path'"
gen_ge_dim: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "gen_set A" shows "dim \<le> card A"
remdups_remdups: fixes xs :: "'d list" shows "remdups (remdups xs) = remdups xs"
P: fixes i :: "'b" assumes "i \<in> I" shows "{x \<in> space (N i). P i x} \<in> sets (N i)"
Collect_member: fixes dxs :: "'c set_dlist" assumes "ID CEQ('c) \<noteq> None" shows "Collect (member dxs) = set (list_of_dlist dxs)"
UN_simps_5: fixes A :: "'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "(\<Union>x\<in>C. A \<inter> B x) = A \<inter> \<Union> (B ` C)"
FSwImorph_def: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme" shows "FSwImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"
fls_pow_subdegree_ge0: fixes f :: "'a fls"   and n :: "nat" assumes "0 \<le> fls_subdegree f" shows "0 \<le> fls_subdegree (f ^ n)"
is_pole_def: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b" shows "is_pole f a = filterlim f at_infinity (at a)"
strict_trans: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b < a"   and "c < b" shows "c < a"
subseqs_subset: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<in> set (subseqs xs)" shows "set ys \<subseteq> set xs"
rel_inject_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and x22 :: "'c llist"   and y21 :: "'d"   and y22 :: "'d llist" shows "lrel R (LCons x21 x22) (LCons y21 y22) = (R x21 y21 \<and> lrel R x22 y22)"
poly_lift2_def: shows "poly_lift2 = map_poly (\<lambda>a. [:[:a:]:])"
insert: fixes x :: "'a"   and A :: "'a set" shows "local.MaxR.MaxR_opt (insert x A) = (if x \<in> Field r then Some (case local.MaxR.MaxR_opt A of None \<Rightarrow> x | Some y \<Rightarrow> local.MaxR.maxR x y) else local.MaxR.MaxR_opt A)"
plus_adds_pp_0: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s + t adds\<^sub>p v" shows "s adds\<^sub>p v \<ominus> t"
vrat_uminus_uminus: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x = -\<^sub>\<rat> (-\<^sub>\<rat> x)"
drop_not_empty_conv: fixes n :: "nat"   and xs :: "'b list" shows "(xs \<up> n \<noteq> []) = (n < length xs)"
hd_in_set: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd xs \<in> list.set xs"
REFSTEP: fixes it :: "'S set"   and x :: "'S"   and x' :: "'Sa"   and it' :: "'Sa set"   and \<sigma> :: "'a"   and \<sigma>' :: "'b" assumes "\<forall>y\<in>it - {x}. RR x y"   and "x' = \<alpha> x"   and "x \<in> it"   and "x' \<in> it'"   and "it' = \<alpha> ` it"   and "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "\<Phi>'' it \<sigma> it' \<sigma>'"   and "c \<sigma>"   and "c' \<sigma>'"   and "(\<sigma>, \<sigma>') \<in> R" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). (\<sigma>, \<sigma>') \<in> R \<and> \<Phi>'' (it - {x}) \<sigma> (it' - {x'}) \<sigma>'} (f' x' \<sigma>')"
assms_1: shows "prv (imp \<phi>1 (imp \<phi>2 \<phi>3))"
S\<^sub>r_eq: shows "S\<^sub>r = S1 - S\<^sub>e"
x: shows "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
assms_2: shows "fmrel_on_fset (fmdom zm) f xm zm"
normal_imp_subgroup: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<lhd> G" shows "subgroup H G"
sp_cf_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and a :: "'a" assumes "pol_coeff S c"   and "a \<in> carrier S" shows "pol_coeff S (sp_cf S a c)"
bunch_of_lemmata_about_matches_3: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchNot MatchAny) a p = False"
assms_1: assumes "\<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)" shows "P s"
lower_cos_float_interval: fixes p :: "nat"   and x :: "float interval" shows "lower (cos_float_interval p x) = fst (bnds_cos p (lower x) (upper x))"
four_block_mat_def: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" shows "four_block_mat A B C D = (let nra = dim_row A; nrd = dim_row D; nca = dim_col A; ncd = dim_col D in mat (nra + nrd) (nca + ncd) (\<lambda>(i, j). if i < nra then if j < nca then A $$ (i, j) else B $$ (i, j - nca) else if j < nca then C $$ (i - nra, j) else D $$ (i - nra, j - nca)))"
axioms_1: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_csgrp \<alpha> \<SS>" shows "\<Z>_sgrp \<alpha> \<SS>"
eq_uptoE: fixes g :: "nat \<Rightarrow> 'a"   and B :: "(nat \<Rightarrow> 'a) set"   and f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "g \<in> eq_upto B f i" obtains "g \<in> B" and "\<And>j. j < i \<Longrightarrow> f j = g j"
Un_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
iter_unguarded: fixes guard :: "'a \<Rightarrow> bool"   and s :: "'a"   and body :: "'a \<Rightarrow> 'a spmf"   and n :: "nat" assumes "\<not> guard s" shows "loop_spmf.iter guard body n s = return_spmf s"
solution_def: shows "solution = (\<lambda>(a, b). a\<^sup>2 = (1::'b) + of_nat D * b\<^sup>2)"
size_Diff_singleton: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "size (M - {#x#}) = size M - 1"
simps_2: fixes f :: "'a \<Rightarrow> 'a fmUni" shows "map_atomUni f TrueFUni = TrueFUni"