hcju/isabelleps huggingface dataset:
exec: shows "steps0 (Suc 0, l, r) B stp = (0, l', r')"
irbb_irbd_demorgan: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "bb\<^sup>-\<^sub>\<R> \<phi> = (bd\<^sup>-\<^sub>\<R> \<circ> \<partial>\<^sub>F) \<phi>"
All_less_Suc: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>i<Suc n. P i) = (P n \<and> (\<forall>i<n. P i))"
filtercomap: fixes f' :: "'b \<Rightarrow> 'c"   and F'' :: "'b filter"   and g' :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b" assumes "f' \<in> \<Omega>[F''](g')" shows "(\<lambda>x. f' (h' x)) \<in> \<Omega>[filtercomap h' F''](\<lambda>x. g' (h' x))"
eval: shows "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"
gen_lossless_gpv_parametric': fixes C :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'k \<Rightarrow> bool" shows "((=) ===> rel_\<I> C R ===> rel_gpv'' A C R ===> (=)) gen_lossless_gpv gen_lossless_gpv"
lunit_simps_2: fixes f :: "'a" assumes "S.ide f" shows "S.src (S.lunit f) = S.src f"
linear_continuous_within: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set" assumes "bounded_linear f" shows "continuous (at x within s) f"
carrier_dsumag: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "carrier (a\<Oplus>\<^bsub>I\<^esub> A) = carr_dsumag I A"
runit'_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.arr V (category.inv V (bicategory.runit V H \<a> \<i> src f))"
C_def: shows "C = {b \<in> B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
leadsTo_induct: fixes F :: "'d program"   and za :: "'d set"   and zb :: "'d set"   and P :: "'d set \<Rightarrow> 'd set \<Rightarrow> bool" assumes "F \<in> za \<longmapsto> zb"   and "\<And>A B. F \<in> A ensures B \<Longrightarrow> P A B"   and "\<And>A B C. \<lbrakk>F \<in> A \<longmapsto> B; P A B; F \<in> B \<longmapsto> C; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>B S. \<forall>A\<in>S. F \<in> A \<longmapsto> B \<and> P A B \<Longrightarrow> P (\<Union> S) B" shows "P za zb"
code_constn: fixes n :: "nat"   and c :: "nat" shows "code_constn (Suc n) c = encode (r_constn n c)"
delta_pf_val_process: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "val_process Mkt (delta_pf N der matur) n w = geom_proc n w * delta_price N der matur n w"
B: shows "[(p - y mod p)\<^sup>2 = x] (mod p)"
drlex_fun_lin: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "drlex_fun s t \<or> drlex_fun t s"
CSP2_ok_a: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "CSP2 P (a, b\<lparr>ok := True\<rparr>)" shows "P (a, b\<lparr>ok := True\<rparr>) \<or> P (a, b\<lparr>ok := False\<rparr>)"
Separation_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" shows "Separation G v0 v1 S \<equiv> v0_v1_Digraph G v0 v1 \<and> Separation_axioms G v0 v1 S"
alwaysE: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g"   and i :: "nat" assumes "(\<box>P) \<sigma>" obtains "P (\<sigma> |\<^sub>s i)"
simps_1: fixes access :: "acc_modi"   and more :: "'a" shows "access \<lparr>access = access, \<dots> = more\<rparr> = access"
delta_minimal_delta: fixes xys :: "('b \<times> 'b) list" assumes "\<And>x y. (x, y) \<in> set xys \<Longrightarrow> y < x" shows "\<exists>\<delta>>0::'b. \<forall>x y. (x, y) \<in> set xys \<longrightarrow> delta_gt \<delta> x y"
poch'_def: shows "poch' = deriv (\<lambda>z. pochhammer z (Suc (2 * n)))"
verit_minus_simplify_4: fixes b :: "'d" shows "- (- b) = b"
that: fixes \<H> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set"   and \<K> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set set" assumes "subgroup \<H> (homology_group p X)"   and "subgroup \<K> (homology_group p X)"   and "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>) (homology_group p X)"   and "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)"   and "hom_induced p X {} X S id \<in> Group.iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)" shows "thesis"
T_def: fixes P :: "'b process" shows "T P = TRACES (Rep_process P)"
im_sheaf_def: fixes V :: "'e set" shows "im0.im_sheaf V \<equiv> {0::'d}"
g_set_xy_defs_loc_def: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" shows "g_set_xy_defs_loc ops1 ops2 \<equiv> StdSet ops1 \<and> StdSet ops2"
pdevs_val_pdevs_domain: fixes e :: "nat \<Rightarrow> real"   and X :: "'b pdevs" shows "pdevs_val e X = (\<Sum>i\<in>pdevs_domain X. e i *\<^sub>R pdevs_apply X i)"
finite_conj: fixes E :: "(('c \<Rightarrow> 'd) \<Rightarrow> bool) set" assumes "finite E"   and "\<forall>e\<in>E. finite (Bexp.vars e)" shows "finite (Bexp.vars (conjunct E))"
vsubstAbs_substAbs_compose2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "goodAbs A"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "A $[Y / y]_ys $[z1 // z]_zs = A $[z1 // z]_zs $[Y #[z1 // z]_zs / y]_ys"
y_to_z: shows "from y to z = \<lparr>tdir = tval z - tval y, xdir = 0::'a, ydir = 0::'a, zdir = 0::'a\<rparr>"
divide_by_norm_iarray_def: fixes A :: "real iarray iarray" shows "divide_by_norm_iarray A = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>a b. ((1 / norm_iarray (column_iarray b A)) *\<^sub>R column_iarray b A) !! a)"
assms_3: shows "s\<^sub>2 r' = Some (\<sigma>, \<tau>, \<E> [Rfork e])"
wf_dbtm_iff_is_tm: fixes x :: "dbtm" shows "wf_dbtm x = (\<exists>t. x = trans_tm [] t)"
lim_1_over_complex_power: fixes s :: "complex" assumes "0 < Re s" shows "(\<lambda>n. 1 / of_nat n powr s) \<longlonglongrightarrow> 0"
assms_4: shows "set (p\<^sub>0 # ps) = ps\<^sub>L \<union> ps\<^sub>R"
complex_normed_vector_axioms: shows "class.complex_normed_vector (*\<^sub>R) (*\<^sub>C) (+) (0::'a) (-) uminus dist norm sgn uniformity open"
eval_tm_ORD_OF: fixes e :: "(name, hf) finfun"   and n :: "nat" shows "\<lbrakk>ORD_OF n\<rbrakk>e = ord_of n"
exhaust: fixes y :: "bit" obtains "y = 0"   | "y = 1"
simps_2: fixes p :: "linear_poly"   and c :: "rat" shows "lec_of_constraint (LT p c) = Le_Constraint Lt_Rel p c"
inf_absorb1: fixes x :: "'d multiset"   and y :: "'d multiset" assumes "x \<subseteq># y" shows "x \<inter># y = x"
induct: fixes P :: "('c \<times> nat) option \<Rightarrow> 'c \<Rightarrow> lock_action list \<Rightarrow> bool"   and a0 :: "('c \<times> nat) option"   and a1 :: "'c"   and a2 :: "lock_action list" assumes "\<And>l t. P l t []"   and "\<And>l t L Ls. P (upd_lock l t L) t Ls \<Longrightarrow> P l t (L # Ls)" shows "P a0 a1 a2"
set_cruel: fixes s :: "'b list"   and A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and "is" :: "'c"   and n :: "nat" assumes "s \<noteq> []" shows "set (cruel A (s, is) n) \<subseteq> set s"
glbH_is_fixp: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and H :: "'b set" assumes "CLF cl f"   and "H = {x. (f x, x) \<in> order cl \<and> x \<in> pset cl}" shows "glb H cl \<in> fix f (pset cl)"
lex_fun_lin: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "rlex_fun t s \<or> rlex_fun s t"
PLemmas_15: fixes src_net :: "'a set set"   and dest_net :: "'a set set" shows "deny_all_from_to src_net dest_net = {pa. src pa \<sqsubset> src_net \<and> dest pa \<sqsubset> dest_net} \<triangleleft> D\<^sub>U"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
qp_close_under_unitary_operator: fixes U :: "complex mat"   and P :: "complex mat" assumes "U \<in> carrier_mat d d"   and "unitary U"   and "is_quantum_predicate P" shows "is_quantum_predicate (adjoint U * P * U)"
all_zero_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(\<forall>u. inner x u = 0) = (x = zero)"
cprod_cinfinite_bound: fixes p :: "('c \<times> 'c) set"   and r :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p \<le>o r"   and "q \<le>o r"   and "Card_order p"   and "Card_order q"   and "Cinfinite r" shows "p *c q \<le>o r"
range_subsetD: fixes f :: "'b \<Rightarrow> 'a"   and B :: "'a set"   and i :: "'b" assumes "range f \<subseteq> B" shows "f i \<in> B"
InvId: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "cinv\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X id\<^bsub>C\<^esub> X"
kildall_properties: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and ss0 :: "'b list" assumes "Semilat A r f"   and "acc r"   and "pres_type step n A"   and "SemilatAlg.mono r step n A"   and "bounded step n"   and "ss0 \<in> list n A" shows "kildall r f step ss0 \<in> list n A \<and> stables r step (kildall r f step ss0) \<and> ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] kildall r f step ss0 \<and> (\<forall>ts\<in>list n A. ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ts \<and> stables r step ts \<longrightarrow> kildall r f step ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ts)"
lm033: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "Domain P \<subseteq> Domain Q" shows "P +< Q = Q"
med_a0iim1_ri_def: fixes o1 :: "m1_state" shows "med_a0iim1_ri o1 \<equiv> \<lparr>signals = ri_runs2sigs (runs o1), corrupted = {}\<rparr>"
propa: shows "propa f (step (hd w) (ss ! hd w)) ss (tl w) = (ss', w')"
less_eq_Suc_le: fixes n :: "nat"   and m :: "nat" shows "order.greater m n = order.greater_eq m (Suc n)"
permutation_orbit_step: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" assumes "permutation f" shows "orbit f (f s) = orbit f s"
distinct_lists_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "local.a_distinct_lists h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
check_monom_strict_mono: fixes m :: "'c monom"   and v :: "'c"   and \<alpha> :: "'c \<Rightarrow> 'a"   and \<beta> :: "'c \<Rightarrow> 'a" assumes "check_monom_strict_mono power_mono m v"   and "\<alpha> v \<succ> \<beta> v"   and "(0::'a) \<le> \<beta> v" shows "eval_monom \<alpha> m \<succ> eval_monom \<beta> m"
select_convs_3: fixes nodes :: "'d set"   and edges :: "('d \<times> 'e \<times> 'd) set"   and more :: "'c" shows "more \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = more"
exec_Normal_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> \<Rightarrow> t" obtains "t = Stuck"
monom_decomp: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p" shows "p = UP.poly_of_dense (UP.dense_repr p)"
partitions_increase1: fixes p :: "nat \<Rightarrow> nat"   and m :: "nat"   and k :: "nat" assumes "p partitions m - k"   and "sum p {..m - k} = k" shows "(\<lambda>i. p (i - 1)) partitions m"
run_asset_ii_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_asset_ii n s))) \<notin> used s"
ndFalse1: fixes A :: "('b set set \<times> 'c) set"   and B :: "('b set set \<times> 'd) set"   and D :: "'b" assumes "\<forall>a b c d. (a, b) \<in> A \<and> (c, d) \<in> B \<longrightarrow> netsDistinct a c"   and "\<exists>(a, b)\<in>A. a \<in> subnetsOfAdr D"   and "\<exists>(a, b)\<in>B. a \<in> subnetsOfAdr D" shows "False"
shift_simps_1: fixes xs :: "'b list"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(xs @- \<sigma>) 0 = (if xs = [] then \<sigma> 0 else hd xs)"
elem_set_implies_incl_in: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "a \<in> local.set b" shows "incl_in a b"
that: fixes K' :: "'a" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> K'\<lparr>N\<rparr> \<prec> R'"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K'"   and "A\<^sub>R \<sharp>* K'" shows "thesis"
closed_Collect_all: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.topological_space open"   and "\<And>i. topological_space.closed open {x. P i x}" shows "topological_space.closed open {x. \<forall>i. P i x}"
sphere_eq_empty: fixes a :: "'a"   and r :: "real" shows "(sphere a r = {}) = (r < 0)"
semiring_simprules_10: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
circuit_dep: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep_system.circuit carrier indep X" shows "\<not> indep X"
inf_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f \<sqinter> g) x = f x \<sqinter> g x"
prod_eq_iff: fixes s :: "'d \<times> 'e"   and t :: "'d \<times> 'e" shows "(s = t) = (fst s = fst t \<and> snd s = snd t)"
igWlsInp_def: fixes MOD :: "('q, 'r, 's, 'sort, 't, 'u, 'v, 'w, 'x) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'v option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
simps_5: fixes f :: "'c \<Rightarrow> 'b"   and Inv :: "'c"   and b :: "bexp"   and c :: "'c acom"   and P :: "'c" shows "ACom.map_acom f ({Inv} WHILE b DO c {P}) = {f Inv} WHILE b DO ACom.map_acom f c {f P}"
ex_iter': fixes R :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "(\<exists>n. R (iter g a n)) = (R a \<or> (\<exists>n. R (iter g (g a) n)))"
blinfun_apply_embed2_blinfun: fixes x :: "'c" shows "blinfun_apply embed2_blinfun x = (0::'b, x)"
m2r_rel_def: fixes r :: "('d \<times> 'd option) set" shows "m2r_rel r \<equiv> {(Some a, b) |a b. (a, b) \<in> r}"
split_mod: fixes P :: "nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" shows "P (m mod n) = ((n = 0 \<longrightarrow> P m) \<and> (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<longrightarrow> m = n *n i + j \<longrightarrow> P j)))"
R\<^sub>0_def: shows "TfHgh.R\<^sub>0 = \<lbrakk>\<lbrakk>tab\<^sub>0 f\<rbrakk>\<rbrakk>"
rec_accum3_def: fixes f :: "recf" shows "rec_accum3 f = PR (CN f [CN Z [recf.Id 3 0], recf.Id 3 0, recf.Id 3 1, recf.Id 3 2]) (CN rec_mult [recf.Id 5 1, CN f [CN S [recf.Id 5 0], recf.Id 5 2, recf.Id 5 3, recf.Id 5 4]])"
nat_pow_Suc2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "[Suc n] \<cdot> x = x \<oplus> [n] \<cdot> x"
Un_Union_image: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>x\<in>C. A x \<union> B x) = \<Union> (A ` C) \<union> \<Union> (B ` C)"
fundamental_lemma: fixes bad1 :: "'b \<Rightarrow> bool"   and bad2 :: "'c \<Rightarrow> bool"   and A :: "'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'c spmf" assumes "rel_spmf (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) p q" shows "\<bar>measure (measure_spmf p) (_Collect x (A x)) - measure (measure_spmf q) (_Collect y (B y))\<bar> \<le> measure (measure_spmf p) (_Collect x (bad1 x))"
bounded_clinear_right: fixes a :: "'a" shows "bounded_clinear ((**) a)"
gcs_minus_gcs: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "class.ulcs_powerprod plus minus zero lcs" shows "lcs_powerprod.gcs plus minus lcs (minus s (lcs_powerprod.gcs plus minus lcs s t)) (minus t (lcs_powerprod.gcs plus minus lcs s t)) = zero"
span_mat_mul_subset: fixes A :: "'b mat"   and n :: "nat"   and d :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat n d"   and "B \<in> carrier_mat d nc" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (cols (A * B))) \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (cols A))"
assms: shows "R \<in> carrier (relative_homology_group (int p) X S)"
simps_1: fixes n :: "int"   and s :: "char list \<Rightarrow> int option" shows "paval (N n) s = n"
Class_equivalence: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>"   and "b \<in> Sigma I \<FF>" shows "(rel.Class a = rel.Class b) = ((a, b) \<in> {(x, y). x \<sim> y})"
LImp: fixes z :: "name"   and N :: "trm"   and y :: "name"   and P :: "trm"   and x :: "name"   and M :: "trm"   and b :: "coname"   and a :: "coname"   and c :: "coname" assumes "z \<sharp> (N, [y].P, [x].M, y, x)"   and "b \<sharp> ([a].M, [c].N, P, c, a)"   and "x \<sharp> (N, [y].P, y)"   and "c \<sharp> (P, [a].M, b, a)"   and "a \<sharp> ([c].N, P)"   and "y \<sharp> (N, [x].M)" shows "Cut <b>.ImpR x.<a>.M b z.ImpL <c>.N y.P z \<longrightarrow>\<^sub>l Cut <a>.Cut <c>.N x.M y.P"
Sup_least: fixes A :: "V set"   and z :: "V" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion> A \<le> z"
rect: fixes i :: "nat" assumes "i < length xs" shows "length (xs ! i) = n"
assms_1: shows "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"
frontier_subset_eq: fixes S :: "'a set" shows "(frontier S \<subseteq> S) = closed S"
mset_sorted_list_of_multiset: fixes M :: "'d multiset" shows "mset (sorted_list_of_multiset M) = M"
iT_Mult_empty_iff: fixes I :: "nat set"   and k :: "nat" shows "(I \<otimes> k = {}) = (I = {})"
redT_updTs_Some1: fixes ts :: "'f \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and t :: "'f"   and xw :: "'d \<times> 'e \<Rightarrow>f nat"   and tas :: "('f, 'd, 'g) new_thread_action list" assumes "ts t = \<lfloor>xw\<rfloor>" shows "\<exists>xw. redT_updTs ts tas t = \<lfloor>xw\<rfloor>"
bst_insert: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Set2_Join join inv"   and "Tree2.bst t" shows "Tree2.bst (Set2_Join.insert join x t)"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(LDCFG_node.Node x1 = LDCFG_node.Node y1) = (x1 = y1)"
abs_float_def: fixes a :: "('a, 'b) IEEE.float" shows "\<bar>a\<bar> = (if sign a = 0 then a else - a)"
connected_component_idemp: fixes S :: "'b set"   and x :: "'b" shows "connected_component_set (connected_component_set S x) x = connected_component_set S x"
pOp_assocTr42: fixes A :: "('h, 'i) aGroup_scheme"   and a :: "'h"   and b :: "'h"   and c :: "'h"   and d :: "'h" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c \<plusminus>\<^bsub>A\<^esub> d = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c) \<plusminus>\<^bsub>A\<^esub> d"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> real set" assumes "range A \<subseteq> sets M"   and "decseq A" shows "(\<lambda>n. Sigma_Algebra.measure M (A n)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Inter> (range A))"
open_lessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.order_topology less_eq less open" shows "open (ord.lessThan less a)"
vconn_implies_hpar_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "trg \<mu> = trg f"
mapping_of_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" shows "mapping_of (monom m a) = monomial a m"
unat_plus_if': fixes a :: "'b word"   and b :: "'b word" shows "unat (a + b) = (if unat a + unat b < 2 ^ LENGTH('b) then unat a + unat b else unat a + unat b - 2 ^ LENGTH('b))"
A_FactGroup_def': fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = (<+>\<^bsub>G\<^esub>), one = H\<rparr>"
rel_open_def: fixes S :: "'a set" shows "rel_open S = (rel_interior S = S)"
maddux_3_19_pp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> y \<squnion> - - x \<sqinter> z = (- - x \<squnion> y) \<sqinter> (- x \<squnion> z)"
oreturn_bind: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> 'e option" shows "oreturn x |>> f = f x"
assms: fixes u :: "'a"   and p :: "('a \<times> 'a) list" assumes "path g u p u"   and "p \<noteq> []"   and "simple p" shows "False"
l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs Shadow_DOM.remove ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel"
le_degree: fixes p :: "'b poly"   and n :: "nat" assumes "coeff p n \<noteq> (0::'b)" shows "n \<le> degree p"
RelStay: fixes R :: "pi"   and S :: "pi"   and c :: "name"   and d :: "name" assumes "(R, S) \<in> Rel"   and "c \<noteq> d" shows "([c\<noteq>d]R, S) \<in> Rel"
G2_def: shows "G2 = {x \<in> G. real_cond_exp M Invariants F2 x < epsilon}"