hcju/isabelleps huggingface dataset:
qTerm_templateInduct: fixes rel :: "(('d, 'e, 'f, 'g, 'h) qTerm \<times> ('d, 'e, 'f, 'g, 'h) qTerm) set"   and phi :: "('d, 'e, 'f, 'g, 'h) qTerm \<Rightarrow> bool"   and phiAbs :: "('d, 'e, 'f, 'g, 'h) qAbs \<Rightarrow> bool"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and A :: "('d, 'e, 'f, 'g, 'h) qAbs" assumes "\<And>X Y. (X, Y) \<in> rel \<Longrightarrow> qSkel Y = qSkel X"   and "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. (\<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y) \<Longrightarrow> phiAbs (qAbs xs x X)" shows "phi X \<and> phiAbs A"
move_stability_wdc: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic"   and v :: "view" shows "ts \<^bold>\<midarrow>wdc( c ) \<^bold>\<rightarrow> ts' \<longrightarrow> move ts ts' v = v"
left_total_alt_def: fixes R :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "left_total R = ((=) \<le> R OO R\<inverse>\<inverse>)"
exhaust: fixes y :: "'a ns_constraint" obtains x11 :: "linear_poly" and x12 :: "'a" where "y = LEQ_ns x11 x12"   | x21 :: "linear_poly" and x22 :: "'a" where "y = GEQ_ns x21 x22"
list_ex_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_ex P xs = Bex (set xs) P"
c1: shows "sum c {0..<length Vs + 1} = (1::'a)"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(rev_H x = rev_H y) = (x = y)"
disc_compl_opposite_ocircline: fixes H :: "ocircline" shows "disc_compl (opposite_ocircline H) = disc H"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and tllist :: "('c, 'b) tllist" shows "P (case tllist of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. tllist = TNil x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. tllist = TCons x21 x22 \<longrightarrow> P (f2 x21 x22)))"
xandy: shows "x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)"
Limit_vsingleton_in_VsetI: fixes \<alpha> :: "V"   and a :: "V" assumes "Limit \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "set {a} \<in>\<^sub>\<circ> Vset \<alpha>"
coeff_minus: fixes lp1 :: "linear_poly"   and lp2 :: "linear_poly"   and var :: "nat" shows "Abstract_Linear_Poly.coeff (lp1 - lp2) var = Abstract_Linear_Poly.coeff lp1 var - Abstract_Linear_Poly.coeff lp2 var"
profile_def: fixes A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set" shows "profile A Is P \<equiv> Is \<noteq> {} \<and> (\<forall>i\<in>Is. rpr A (P i))"
shadows_consts_sterm_simps_1: fixes t\<^sub>1 :: "sterm"   and t\<^sub>2 :: "sterm" shows "shadows_consts (t\<^sub>1 $\<^sub>s t\<^sub>2) = (shadows_consts t\<^sub>1 \<or> shadows_consts t\<^sub>2)"
language_ltln_def: fixes \<phi> :: "'b ltln" shows "language_ltln \<phi> \<equiv> {\<xi>. \<xi> \<Turnstile>\<^sub>n \<phi>}"
offs_equal_suc: fixes i :: "nat"   and ns :: "nat list"   and m :: "nat" assumes "Suc i < length ns"   and "ns ! i = 0" shows "offs ns m ! i = offs ns m ! Suc i"
lfp_trans_def: fixes T :: "(('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "lfp_trans T = Inf_trans {t. (\<forall>P. sound P \<longrightarrow> sound (t P)) \<and> le_trans (T t) t}"
simps: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and p :: "'f"   and a :: "'f" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) p a = (a = p \<or> (\<exists>q aa. a = snd aa \<and> transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) p q \<and> snd aa \<in> succ A (fst aa) q))"
rel_interior_same_affine_hull: fixes S :: "'b set" assumes "convex S" shows "affine hull rel_interior S = affine hull S"
Replace_hunion: fixes A :: "hf"   and B :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool" shows "Replace (A \<squnion> B) R = Replace A R \<squnion> Replace B R"
vlrestriction_vid_on: fixes A :: "V"   and B :: "V" shows "vid_on A \<restriction>\<^sup>l\<^sub>\<circ> B = vid_on (A \<inter>\<^sub>\<circ> B)"
obj_ty_obj_class1: fixes G :: "prog"   and obj :: "obj" assumes "wf_prog G"   and "is_type G (obj_ty obj)" shows "is_class G (obj_class obj)"
induct: fixes P :: "WS1S_Prelim.order \<Rightarrow> 'a \<Rightarrow> ('a, 'a) atomic \<Rightarrow> bool"   and a0 :: "WS1S_Prelim.order"   and a1 :: "'a"   and a2 :: "('a, 'a) atomic" assumes "\<And>i m. P FO i (Fo m)"   and "\<And>i m. P FO i (Z m)"   and "\<And>i m1 m2. P FO i (Less m1 m2)"   and "\<And>i m uu_. P FO i (In m uu_)"   and "\<And>i uv_ M. P SO i (In uv_ M)"   and "\<And>ux_ v. P SO ux_ (Fo v)"   and "\<And>ux_ v. P SO ux_ (Z v)"   and "\<And>ux_ v va. P SO ux_ (Less v va)"   and "\<And>ux_ v. P SO ux_ (Z v)"   and "\<And>ux_ v va. P SO ux_ (Less v va)" shows "P a0 a1 a2"
IH_7: shows "((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) \<in> set A) = ((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
BRBang: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel" shows "(!P, !Q) \<in> bangRel Rel"
Accessible_right_lang_eq: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M"   and "q \<in> dfa.accessible M" shows "dfa.right_lang (dfa.Accessible_dfa M) q = dfa.right_lang M q"
f: shows "index xs y = 0 \<and> index xs x = 1"
IH_2: shows "2 ^ min_height r \<le> KD_Tree.size_kdt r"
comp_null_2: fixes f :: "'t" shows "f \<cdot>\<acute> S'.null = S'.null"
real_independent_from_complex_independent: fixes B :: "'d set" assumes "cindependent B" shows "independent (B \<union> (*\<^sub>C) \<i> ` B)"
obj_simps_1: fixes a :: "'a" assumes "H.obj a" shows "V.arr a"
morphism_sheaves_of_rings_def: fixes X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<FF> :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f"   and \<FF>' :: "'e set \<Rightarrow> 'g set"   and \<rho>' :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'g \<Rightarrow> 'g"   and b' :: "'g"   and add_str' :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str' :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str' :: "'e set \<Rightarrow> 'g"   and one_str' :: "'e set \<Rightarrow> 'g"   and fam_morphisms :: "'e set \<Rightarrow> 'f \<Rightarrow> 'g" shows "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
ak0: shows "a ^ Suc k $ 0 = (a $ 0) ^ Suc k"
goal_assms: shows "\<psi> \<in> new n \<and> (\<nexists>q. \<psi> = prop\<^sub>r(q) \<or> \<psi> = nprop\<^sub>r(q)) \<and> \<psi> \<noteq> true\<^sub>r \<and> \<psi> \<noteq> false\<^sub>r \<and> (\<nexists>\<nu> \<mu>. \<psi> = \<nu> and\<^sub>r \<mu> \<or> \<psi> = X\<^sub>r \<nu>) \<and> (\<exists>\<nu> \<mu>. \<psi> = \<nu> or\<^sub>r \<mu> \<or> \<psi> = \<nu> U\<^sub>r \<mu> \<or> \<psi> = \<nu> R\<^sub>r \<mu>)"
ex1_iff_ex_Uniq: fixes P :: "'e \<Rightarrow> bool" shows "(\<exists>!x. P x) = ((\<exists>x. P x) \<and> (\<exists>\<^sub>\<le>\<^sub>1 x. P x))"
rel_pred_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_pred R = R ===> (=)"
apply_commute_cases: fixes y :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "y \<in> {f. \<forall>x y. f x y = f y x}" obtains x :: "('b, 'c) commute" where "y = apply_commute x"
u_K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
iterates_abovep_iterates_above_eq: fixes a :: "'a" shows "iterates_abovep a = (\<lambda>x. x \<in> iterates_above a)"
init_mmuaux_def: fixes args :: "Monitor.args" shows "init_mmuaux args = (0, empty_queue, 0, join_mask (args_n args) (args_L args), join_mask (args_n args) (args_R args), Mapping.empty, Mapping.update 0 Mapping.empty Mapping.empty, [], 0)"
mintree_exists: fixes bq :: "('c, 'd) BinomialTree list" shows "(bq \<noteq> []) = (getMinTree bq \<in> set bq)"
mopup_ge2: fixes x :: "nat"   and n :: "nat" shows "2 \<le> x + length (mopup n)"
assms_2: shows "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2"
principal_dchar_coprime: fixes k :: "nat"   and n :: "nat" assumes "coprime k n" shows "\<chi>\<^sub>0\<^bsub>n\<^esub> k = 1"
ofilter_AboveS_Field: fixes A :: "(nat + 'a Bonk_Schramm_extension_unfolded set) set" assumes "ofilter wo_aux A" shows "A \<union> AboveS wo_aux A = Field wo_aux"
nxs2: shows "\<langle>vcard xs\<^sub>2, x\<^sub>2\<rangle> \<notin>\<^sub>\<circ> xs\<^sub>2"
adds_lcs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus s (lcs s t)"
uminus_eq_inj: fixes a :: "'a literal"   and b :: "'a literal" shows "(- a = - b) = (a = b)"
comp_arr_ide: fixes a :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "ide a"   and "seq f a" shows "f \<cdot> a = f"
assms_1: shows "g1 +++ g2 piecewise_differentiable_on {0..1}"
iMin_in: fixes I :: "'a set" assumes "I \<noteq> {}" shows "iMin I \<in> I"
LT14: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> F \<leadsto> H"
occs_vars_subset: fixes M :: "'b trm"   and N :: "'b trm" assumes "M \<prec> N" shows "vars_of M \<subseteq> vars_of N"
Open: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"
prems_1: shows "is_valid_problem_sas_plus \<Psi>"
hull_subset: fixes s :: "'c set"   and S :: "'c set \<Rightarrow> bool" shows "s \<subseteq> S hull s"
zero_in_Ord: fixes k :: "hf" assumes "Ord k" shows "k = 0 \<or> 0 \<^bold>\<in> k"
assms_2: fixes x :: "'a" assumes "x \<in> carrier G" shows "f' x = f x"
rexp_ewp_l_ewp: fixes x :: "'a rexp" shows "l_ewp (lang x) = rexp_ewp x"
right_adjoint_determines_left_up_to_iso: fixes f :: "'a"   and g :: "'a"   and f' :: "'a" assumes "adjoint_pair f g"   and "adjoint_pair f' g" shows "f \<cong> f'"
inner_sum_right: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "inner x (sum f A) = (\<Sum>y\<in>A. inner x (f y))"
prefixI: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "ys = xs @ zs" shows "xs \<le> ys"
aux: fixes l :: "nat" shows "\<^bold>|r \<^sup>@ Suc l\<^bold>| = \<^bold>|r \<^sup>@ l\<^bold>| + \<^bold>|r\<^bold>|"
div_nonneg_neg_le0: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "b < 0" shows "a div b \<le> 0"
pseudo_proj_False_preimage': fixes g :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_False n -` g -` {g z} = g -` {g z}"
prems_1: shows "wf\<^sub>s\<^sub>t V (S @ \<langle>ac: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S' @ [x])"
equiv_partition_on: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" shows "equiv A {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p}"
gen_dfsT_def: shows "gen_dfsT \<equiv> gds_init gds \<bind> WHILE\<^sub>T gen_cond gen_step"
reachable_induced: fixes u :: "'a"   and v :: "'a" assumes "u \<in> S"   and "v \<in> S"   and "u \<rightarrow>\<^sup>* v" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
has_distI: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list" assumes "is_shortest_path G m n p" shows "has_dist G m n"
int_cases: fixes z :: "int" obtains n :: "nat" where "z = int nb__"   | n :: "nat" where "z = - int (Suc n)"
prems_2: shows "\<forall>c\<in>set (c' # cs). v c \<le> n"
cSup_eq_Sup_fin: fixes X :: "'a multiset set" assumes "finite X"   and "X \<noteq> {}" shows "\<Squnion> X = subset_mset.Sup_fin X"
D_def: fixes fs :: "int vec list" shows "fs.D fs = nat (prod (fs.d fs) {..<length fs})"
has_default_route_alt: fixes rt :: "'a routing_rule list" shows "has_default_route rt = (\<exists>r\<in>set rt. routing_prefix r = 0)"
of_bool_eq_iff: fixes one :: "'b"   and zero :: "'b"   and p :: "bool"   and q :: "bool" assumes "class.zero_neq_one one zero" shows "(zero_neq_one.of_bool one zero p = zero_neq_one.of_bool one zero q) = (p = q)"
exhaust: fixes y :: "mode_t" obtains "y = READ"   | "y = WRITE"   | "y = PROVIDE"
pdivides_imp_degree_le: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []"   and "p pdivides q" shows "degree p \<le> degree q"
endofunctor_axioms: shows "endofunctor (\<cdot>) src"
atLeast_iff: fixes i :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and k :: "'d" shows "(i \<in> ord.atLeast less_eq k) = less_eq k i"
path_shift_def: fixes \<pi> :: "'a \<Rightarrow> 'b"   and m :: "'a" shows "\<pi> \<guillemotleft> m = (\<lambda>n. \<pi> (m + n))"
is_red_plus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F (p + q)" shows "is_red F p \<or> is_red F q"
pred_True: shows "pred_pmf (\<lambda>_. True) = (\<lambda>_. True)"
cone_decomp_closed_plus: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "cone_decomp T ps"   and "a \<in> T"   and "b \<in> T" shows "a + b \<in> T"
HaddP_Zero1: fixes H :: "fm set"   and x :: "tm" assumes "H \<turnstile> OrdP x" shows "H \<turnstile> HaddP Zero x x"
assoc_simps\<^sub>A\<^sub>W\<^sub>C_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "cod \<a>[f, g, h] = f \<star> g \<star> h"
LIMSEQ_indicator_incseq: fixes A :: "nat \<Rightarrow> 'a set"   and x :: "'a" assumes "incseq A" shows "(\<lambda>i. indicator (A i) x) \<longlonglongrightarrow> indicator (\<Union> (range A)) x"
\<R>\<^sub>1_def: shows "\<R>\<^sub>1 \<equiv> \<lambda>\<Gamma>'. {(((xa, x), xb), (xd, xc), xe). \<R>\<^sub>1p \<Gamma>' xa x xb xd xc xe}"
prems_17: shows "getSATFlag state' = FALSE \<longrightarrow> \<not> satisfiable F0'"
mult_numeral_left_1: fixes v :: "num"   and w :: "num"   and z :: "'a" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
length_nths_even: fixes xs :: "'a list" assumes "even (length xs)" shows "length (nths xs (Collect even)) = length (nths xs {a. odd a})"
simps_2: fixes \<mu> :: "'a" shows "Dom \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>local.dom \<mu>\<^bold>\<rangle>"
select_convs_2: fixes lens_get :: "'d \<Rightarrow> 'e"   and lens_put :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and more :: "'f" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
Roots_ms_def: shows "Roots_ms = {X. set_mset X \<subseteq> Roots \<and> size X = fact n}"
ring_simprules_11: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
img_is_subfield_2: fixes K :: "'a set" assumes "subfield K (L\<lparr>carrier := A\<rparr>)"   and "\<one>\<^bsub>Rupt A P\<^esub> \<noteq> \<zero>\<^bsub>Rupt A P\<^esub>" shows "subfield ((rupture_surj A P \<circ> poly_of_const) ` K) (Rupt A P)"
tendsto_bot: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open a) bot"
XD_XH_gfp_F_1: shows "fst XD_XH \<subseteq> fst (gfp_F ds)"
SeqAbstFormP_imp_LstSeqP: fixes v :: "tm"   and u :: "tm"   and x :: "tm"   and x' :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqAbstFormP v u x x' s k} \<turnstile> LstSeqP s k (HPair u (HPair x x'))"
jordancurve_2: shows "frontier (inside (path_image (p +++ linepath a b))) = path_image (p +++ linepath a b)"
not_less_eq_funI: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat" assumes "\<And>c n. 0 < c \<Longrightarrow> \<exists>m>n. c * g m < f m" shows "\<not> f \<lesssim> g"
unitary_gfp: fixes P :: "'s \<Rightarrow> real" assumes "unitary P" shows "unitary (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) P)"
prems_2: shows "P,E \<turnstile> e # es [::] Ts"
regionp_region_eq: fixes X :: "'b set"   and I :: "'b \<Rightarrow> intv"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and r :: "('b \<times> 'b) set" shows "regionp (\<lambda>x. x \<in> X) I J (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> region X I J r)"
c: shows "finite (fst ` nonzero_positions (Rep_matrix A))"
paths_hits_winning_strategy_is_winning: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and v :: "'c" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.winning_strategy G p \<sigma> v"   and "v \<in> lset P" shows "ParityGame.winning_path G p P"
set_def: fixes l :: "'a list" shows "SelectionSort_Functional.set l = set_mset (mset l)"
simps_3: shows "cospan_shape.Cod cospan_shape.TT = cospan_shape.TT"
FW_zone_equiv: fixes n :: "nat"   and v :: "'d \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)" shows "[M]\<^bsub>v,n\<^esub> = [FW M n]\<^bsub>v,n\<^esub>"
powr_less_cancel: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "x powr a < x powr b"   and "1 < x" shows "a < b"
distinct_201: fixes x121 :: "expr"   and x122 :: "expr"   and x141 :: "expr"   and x142 :: "expr" shows "x121;; x122 \<noteq> while (x141) x142"
coinduct: fixes X :: "'a llist \<Rightarrow> bool"   and x :: "'a llist" assumes "X x"   and "\<And>x. X x \<Longrightarrow> x = LNil \<or> (\<exists>v. x = LCons v LNil \<and> deadend v) \<or> (\<exists>Ps v. x = LCons v Ps \<and> \<not> lnull Ps \<and> (X Ps \<or> maximal_path Ps))" shows "maximal_path x"
DA0_to_DA: shows "DA0 \<cdot> \<eta>0 = DA"
dist_nz: fixes dist :: "'g \<Rightarrow> 'g \<Rightarrow> real"   and uniformity :: "('g \<times> 'g) filter"   and "open" :: "'g set \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.metric_space dist uniformity open" shows "(x \<noteq> y) = (0 < dist x y)"
owhile_def: fixes c :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'c \<Rightarrow> 'b \<Rightarrow> 'c option"   and a :: "'c" shows "owhile c b a \<equiv> \<lambda>s. option_while (\<lambda>a. c a s) (\<lambda>a. b a s) a"
q0: shows "hom (lead_coeff q) \<noteq> (0::'b)"
index_ge_zero: shows "0 \<le> \<Lambda>\<^sub>t"
MASTER_BOUND'_def: fixes a :: "real"   and b :: "real"   and x :: "nat" shows "MASTER_BOUND' a b x = real x powr a * ln (real x) powr b"
poly_bound_exists: fixes r :: "real"   and p :: "'a poly" shows "\<exists>m>0. \<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m"
WTrtStaticCall: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and C' :: "char list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs :: "char list list"   and es :: "expr list"   and Ts' :: "ty list" assumes "P,E,h \<turnstile> e : Class C'"   and "P \<turnstile> Path C' to C unique"   and "P \<turnstile> C has least M = (Ts, T, m) via Cs"   and "P,E,h \<turnstile> es [:] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts" shows "P,E,h \<turnstile> Call e \<lfloor>C\<rfloor> M es : T"
nonzeros_Pdevs_raw_subset: fixes xs :: "('d \<times> 'e) list" shows "{i. Pdevs_raw xs i \<noteq> (0::'e)} \<subseteq> dom (map_of xs)"
has_ticket_simps_4: fixes k :: "nat"   and k' :: "nat" shows "has_ticket (REL k) k' = (k' = k)"
8: shows "set s\<^sub>2 \<inter> visible = set s\<^sub>3 \<inter> visible"
cones_map_mapsto: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" shows "cones_map f \<in> extensional (cones (A_B.cod f)) \<inter> (cones (A_B.cod f) \<rightarrow> cones (A_B.dom f))"
length_s_A: fixes n :: "nat" shows "length (s_A n) = length init"
case_eq_if: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and sum :: "'h + 'i" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
left_commute: fixes b :: "'c \<times> 'd"   and a :: "'c \<times> 'd"   and c :: "'c \<times> 'd" shows "sup_prod b (sup_prod a c) = sup_prod a (sup_prod b c)"
onl_oinvariant_sterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('i \<times> ('f, 'g, 'e, 'h) seqp, 'j) automaton"   and S :: "'i \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool"   and U :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and P :: "'i \<times> 'h \<Rightarrow> bool"   and \<sigma> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and p' :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<sigma>, l)"
inj_on_HF: shows "inj_on HF (Collect finite)"
hyps_2: shows "lead_coeff (p \<circ>\<^sub>p q) = lead_coeff p * lead_coeff q ^ degree p"
generate'_def: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "generate' A B a b = tl (Algorithm.gen2 A B a b)"
one: fixes R :: "('a \<Rightarrow> real) set"   and S :: "'a set"   and U :: "'a set"   and t0 :: "'a"   and t1 :: "'a" assumes "function_ring_on R S"   and "open U"   and "t0 \<in> S"   and "t0 \<in> U"   and "t1 \<in> S - U" shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and> (\<forall>e>0. \<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>t\<in>S \<inter> V. f t < e) \<and> (\<forall>t\<in>S - U. 1 - e < f t))"
Zn_order: fixes n :: "nat" shows "order (Z n) = n"