hcju/isabelleps huggingface dataset:
RID\<^sub>L_def: fixes s :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" shows "RID\<^sub>L s \<equiv> case s of (\<sigma>, \<tau>, e) \<Rightarrow> RID\<^sub>S \<sigma> \<union> RID\<^sub>S \<tau> \<union> RID\<^sub>E e"
genPrefix_def: shows "genPrefix \<equiv> \<lambda>r. {(xa, x). genPrefixp (\<lambda>x xa. (x, xa) \<in> r) xa x}"
rotater1_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys" shows "rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)"
sel_2: fixes even :: "bool"   and t :: "'a tree" shows "left (even_odd_mirror even t) = even_odd_mirror (\<not> even) (if even then right t else left t)"
s_def: fixes n :: "nat" shows "s n = integral\<^sup>N (count_space (Y n)) (weight \<Gamma>)"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<le> max a b"
vconn_implies_hpar_3: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "src g = src f"
last_prog_append_Prog\<^sub>s\<^sub>b: fixes x :: "'b"   and sb :: "'b memref list"   and p :: "'b"   and p' :: "'b"   and mis :: "instr list" shows "last_prog x (sb @ [Prog\<^sub>s\<^sub>b p p' mis]) = p'"
nth_root_nat_1: fixes k :: "nat" assumes "0 < k" shows "nth_root_nat k 1 = 1"
unit_divides: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c"   and a :: "'c" assumes "Group.monoid G"   and "u \<in> Units G"   and "a \<in> carrier G" shows "u divides\<^bsub>G\<^esub> a"
assms_4: shows "cross_ratio z u v w = cross_ratio z' u v w"
simps_17: fixes x1 :: "'a" shows "vars_term (Var x1) = {x1}"
assms: fixes P :: "'a llist" assumes "vmc_path G P v0 p \<sigma>" shows "winning_path p P"
2: shows "(1 - (cmod nu / cmod du)\<^sup>2) * (1 - (cmod nv / cmod dv)\<^sup>2) = cmod (ddu * ddv * kk * kk / (du * cnj du * dv * cnj dv))"
prob_le_1: fixes M :: "'a measure"   and A :: "'a set" assumes "prob_space M" shows "Sigma_Algebra.measure M A \<le> 1"
yes: fixes xs :: "'c list"   and x :: "'c" assumes "index xs x < length xs" shows "xs ! index xs x = x"
Diff_Int_distrib: fixes C :: "'e set"   and A :: "'e set"   and B :: "'e set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
assms_1: shows "nprv F (ldsj \<psi>s)"
monom_0: fixes a :: "'a" shows "Polynomial.monom a 0 = [:a:]"
AssertAllState''_unsat: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "\<U> (assert_bound_loop ats (init t)) \<longrightarrow> minimal_unsat_state_core (assert_bound_loop ats (init t))"
refl_ext: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "f x x = (if \<upsilon> x then true else invalid endif)"
Cauchy_convergent: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.complete_space dist uniformity open"   and "uniform_space.Cauchy uniformity X" shows "topological_space.convergent open X"
\<tau>mRedT_preserves_final_thread: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "\<tau>mRed2 s s'"   and "r2.final_thread s t" shows "r2.final_thread s' t"
cases: 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 a1 :: "ty"   and a2 :: "val"   and a3 :: "val" assumes "P \<turnstile> a1 casts a2 to a3 " obtains T :: "ty" and v :: "val" where "a1 = T" and "a2 = v" and "a3 = v" and "\<forall>C. T \<noteq> Class C"   | C :: "char list" where "a1 = Class Ca__" and "a2 = Null" and "a3 = Null"   | Cs :: "char list list" and C :: "char list" and Cs' :: "char list list" and Ds :: "char list list" and a :: "nat" where "a1 = Class C" and "a2 = Ref (a, Cs)" and "a3 = Ref (a, Ds)" and "P \<turnstile> Path last Cs to C via Cs' " and "Ds = Cs @\<^sub>p Cs'"
mru_vote_implies_safe: fixes s :: "v_state"   and Q :: "process set"   and v :: "val" assumes "s \<in> SV_inv4"   and "s \<in> Vinv1"   and "mru_guard s Q v"   and "Q \<in> Quorum" shows "safe s (v_state.next_round s) v"
cases: fixes x :: "'tp list" obtains "x = []"   | \<sigma> :: "'tp" and \<sigma>l :: "'tp list" where "x = \<sigma> # \<sigma>l"
IH_2: assumes "wbt r" shows "wbt (Weight_Balanced_Trees.insert x r)"
distinct_5: fixes x41 :: "'a rexp"   and x42 :: "'a rexp" shows "Zero \<noteq> Plus x41 x42"
multiplicative_function_ind: fixes P :: "nat \<Rightarrow> bool" assumes "P 1"   and "\<And>a b. \<lbrakk>1 < a; 1 < b; coprime a b\<rbrakk> \<Longrightarrow> P (a * b) = (P a \<and> P b)" shows "multiplicative_function (ind P)"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "sup (sup a b) c = sup a (sup b c)"
emeasure_lebesgue_affine: fixes m :: "real"   and \<delta> :: "'b"   and S :: "'b set" shows "emeasure lebesgue ((\<lambda>x. m *\<^sub>R x + \<delta>) ` S) = ennreal (\<bar>m\<bar> ^ DIM('b)) * emeasure lebesgue S"
eval_G1: fixes s :: "complex" shows "eval_fps G1 (s - 1) = (if s = 1 then - Ln 2 else (2 powr (1 - s) - 1) / (s - 1))"
fmadd_drop_left_dom: fixes n :: "'e \<rightharpoonup>\<^sub>f 'f"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdrop_fset (fmdom n) m ++\<^sub>f n = m ++\<^sub>f n"
lfilter_llist_of: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "lfilter P (llist_of xs) = llist_of (filter P xs)"
code: fixes x :: "quat"   and y :: "quat" shows "x + y = Quat (quat.Re x + quat.Re y) (Im1 x + Im1 y) (Im2 x + Im2 y) (Im3 x + Im3 y)"
open_contains_ball_eq: fixes S :: "'b set"   and x :: "'b" assumes "open S" shows "(x \<in> S) = (\<exists>e>0. ball x e \<subseteq> S)"
C_sees: shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"
domo_afilter: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and test_num' :: "int \<Rightarrow> 'b \<Rightarrow> bool"   and filter_plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and filter_less' :: "bool \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and a :: "aexp"   and X :: "char list set"   and S :: "'b Abs_State.st option"   and i :: "'b" assumes "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"   and "vars a \<subseteq> X"   and "domo S \<subseteq> X" shows "domo (Abs_Int1.afilter num' plus' test_num' filter_plus' a i S) \<subseteq> X"
\<phi>_local_bij: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "bij_betw inD (D.hom b a) (HomD.set (b, a))"
Un_Cinfinite_bound: fixes A :: "'d set"   and r :: "('e \<times> 'e) set"   and B :: "'d set" assumes "|A| \<le>o r"   and "|B| \<le>o r"   and "Cinfinite r" shows "|A \<union> B| \<le>o r"
reduction_deletes_bad_sign_conds: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "set (characterize_consistent_signs_at_roots_copr p qs) = set (reduction_signs p qs signs subsets (M_mat signs subsets))"
comp_coretract_closed: fixes G :: "('d, 'e, unit, unit) galcon"   and F :: "('f, 'd, unit, unit) galcon" assumes "coretract G"   and "coretract F"   and "\<Y>\<^bsub>F\<^esub> = \<X>\<^bsub>G\<^esub>" shows "coretract (G \<circ>\<^sub>g F)"
ext_inject: fixes nothing :: "unit"   and more :: "'m"   and nothing' :: "unit"   and more' :: "'m" shows "(\<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing', \<dots> = more'\<rparr>) = (nothing = nothing' \<and> more = more')"
smallomegaI_filterlim_at_top_norm: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "LIM x F. norm (f x / g x) :> at_top" shows "f \<in> \<omega>[F](g)"
simps_3: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and v :: "nat" shows "C M2 M1 \<Omega> V m (Suc (Suc v)) = append_set (C M2 M1 \<Omega> V m (Suc v) - RM M2 M1 \<Omega> V m (Suc v)) (inputs M2) - TS M2 M1 \<Omega> V m (Suc v)"
uint_word_of_int_eq: fixes k :: "int" shows "uint (word_of_int k) = take_bit LENGTH('a) k"
a: shows "\<forall>y. y \<in> Y \<longrightarrow> atom y"
heap_is_wellformed_disc_nodes_in_heap: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node :: "(_) node_ptr" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"   and "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node \<in> set disc_nodes" shows "node |\<in>| node_ptr_kinds h"
ts''_def_1: shows "\<C> \<turnstile> [C v] : ts _> ts''"
elims: fixes x :: "'a bdd"   and xa :: "bool list"   and y :: "'a" assumes "bdd_lookup x xa = y" obtains xb :: "'a" and bs :: "bool list" where "x = Leaf xb" and "xa = bs__" and "y = xb"   | l :: "'a bdd" and r :: "'a bdd" and b :: "bool" and bs :: "bool list" where "x = Branch l r" and "xa = b # bs" and "y = bdd_lookup (if b then r else l) bs"   | v :: "'a bdd" and va :: "'a bdd" where "x = Branch v va" and "xa = []" and "y = undefined"
monofun_cfun: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "f \<sqsubseteq> g"   and "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> g\<cdot>y"
collinear_def: fixes S :: "'a set" shows "collinear S = (\<exists>u. \<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u)"
Eq\<^sub>3I: fixes F :: "\<Pi>\<^sub>3"   and G :: "\<Pi>\<^sub>3"   and v :: "i" assumes "F = G" shows "[F \<^bold>=\<^sub>3 G in v]"
pred_ss_V: fixes u :: "nat" shows "cf.E\<inverse> `` {u} \<subseteq> cf.V"
sim_rI: fixes M :: "('b, 'e, 'c) KripkeStructure"   and f :: "'c \<Rightarrow> 'd"   and M' :: "('b, 'e, 'd) KripkeStructure" assumes "\<And>a u v'. \<lbrakk>u \<in> worlds M; (f u, v') \<in> relations M' a\<rbrakk> \<Longrightarrow> \<exists>v. (u, v) \<in> relations M a \<and> f v = v'" shows "sim_r M M' f"
cl_def: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "matroid carrier indep" shows "matroid.cl carrier indep X \<equiv> {x \<in> carrier. indep_system.lower_rank_of indep (insert x X) = indep_system.lower_rank_of indep X}"
distinct_3: fixes x1 :: "int"   and x3 :: "bool" shows "A x1 \<noteq> C x3"
simps: fixes C :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" shows "lexmin C i = min_elt (ith (eq_upto C (lexmin C) i) i)"
trans_Int: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "trans r"   and "trans s" shows "trans (r \<inter> s)"
IH: assumes "op_history xs" shows "apply_operations xs initial_state \<noteq> None"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lookup (swapPoly\<^sub>0 x xa xb) = (\<lambda>mon. lookup xb (swap0 x xa mon))"
assms_1: shows "\<exists>n w. a n w \<noteq> 0"
simps_1: fixes \<AA> :: "interp"   and m :: "nat" shows "satisfies0 \<AA> (Fo m) = (\<exists>x. m\<^bsup>\<AA>\<^esup>FO = {|x|})"
integrable': shows "set_integrable lborel D (\<lambda>(x, y). (1 - y) ^ n * (x * y * w) ^ n * (1 - x) ^ n / (1 - (1 - x * y) * w) ^ (n + 1))"
hsize_size: fixes t :: "('a, 'b) tm" shows "hsize t * 2 = size t + 1"
absorb_iff2: fixes b :: "'b upred"   and a :: "'b upred" shows "(a \<sqsubseteq> b) = ((a \<and> b) = b)"
prod_list_dvd: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "x dvd prod_list xs"
induct: fixes P :: "('a \<times> nat) list \<Rightarrow> bool"   and a0 :: "('a \<times> nat) list" assumes "P []"   and "\<And>x n xs. P xs \<Longrightarrow> P ((x, n) # xs)" shows "P a0"
psubst_change: fixes \<rho> :: "(nat \<times> exp) list"   and \<rho>' :: "(nat \<times> exp) list"   and e :: "exp" assumes "equiv_env \<rho> \<rho>'" shows "psubst \<rho> e = psubst \<rho>' e"
le: fixes h :: "'a"   and n :: "nat" assumes "h \<noteq> (0::'a)"   and "norm h < k" shows "norm (g h n) \<le> f n * norm h"
smooth_compat_commute: fixes k :: "enat"   and x :: "('b, 'c) chart"   and y :: "('b, 'c) chart" shows "k-smooth_compat x y = k-smooth_compat y x"
fst_image_times: fixes A :: "'a set"   and B :: "'b set" shows "fst ` (A \<times> B) = (if B = {} then {} else A)"
fresh_quote_all: fixes V :: "name set"   and i :: "atom"   and p :: "perm" assumes "finite V" shows "i \<sharp> quote_all p V = (i \<sharp> V \<and> i \<sharp> p \<bullet> V)"
rel_fun_comm: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and g :: "'c \<times> 'd \<Rightarrow> 'e" shows "ps_prod.rel_partial_semigroup.times_rel_fun f g = ps_prod.rel_partial_semigroup.times_rel_fun g f"
lproject_ltake: fixes k :: "nat"   and A :: "'a set"   and w :: "'a llist" assumes "enat k \<le> llength (lproject A w)" shows "lproject A (ltake (enat (nth_least (lift (liset A w)) k)) w) = ltake (enat k) (lproject A w)"
finite_subset: fixes M :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "finite M" shows "finite {x \<in> M. P x}"
rel_fun_eq_conversep: fixes A :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun A\<inverse>\<inverse> (=) = (rel_fun A (=))\<inverse>\<inverse>"
F3: shows "set_mset K \<subseteq> r \<down>m J"
equal_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "equal_class.equal i1 i2 = (i1 \<le> i2 \<and> i2 \<le> i1)"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastLessThan c d) = (a \<subseteq># b \<longrightarrow> c \<subseteq># a \<and> b \<subset># d)"
rep_eq: shows "compare default = (\<lambda>x y. if x < y then Less else if y < x then Greater else Equiv)"
embed_spmf_def: fixes f :: "'c \<Rightarrow> real" shows "embed_spmf f = embed_pmf (\<lambda>x. case x of None \<Rightarrow> 1 - enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV) | Some x' \<Rightarrow> max 0 (f x'))"
restriction_to_subgroup_is_hom: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and H :: "'a set" assumes "FGModuleHom G smult V smult' T"   and "Group H \<and> H \<subseteq> G" shows "FGModuleHom H smult V smult' T"
map_of_map_Pair_key: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and x :: "'d" shows "map_of (map (\<lambda>k. (k, f k)) xs) x = (if x \<in> set xs then Some (f x) else None)"
stirling_integral_bound_aux': fixes n :: "nat"   and \<alpha> :: "real" assumes "1 < n"   and "\<alpha> \<in> {0<..<pi}" obtains c :: "real" where "\<And>s. s \<in> complex_cone' \<alpha> - {0} \<Longrightarrow> cmod (stirling_integral n s) \<le> c / cmod s ^ (n - 1)"
us_facts_3: shows "set us \<inter> \<Union> (FvarsT ` fst ` set txs) = {}"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and LVQ :: "char list set"   and LVE :: "char list set" shows "vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q LVQ LVE = (vc C\<^sub>1 Q LVQ LVE \<and> vc C\<^sub>2 Q LVQ LVE)"
Up_To_Technique: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set"   and d :: "'f"   and PP :: "nat set"   and R' :: "('a list \<times> 'a list) set"   and R :: "('a list \<times> 'a list) set" assumes "WHATWHERE.disj_dlHPP_Bisimulation_Up_To_R' SR E pp DA lH d PP R' R"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP (R \<union> R')"
RSpan_contains_RSpans_append_left: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ns :: "'b list"   and ms :: "'b list" assumes "RModule R smult M"   and "set ns \<subseteq> M" shows "R_scalar_mult.RSpan R smult ms \<subseteq> R_scalar_mult.RSpan R smult (ms @ ns)"
h_def: fixes i :: "nat" shows "h i = (case i of 0 \<Rightarrow> (Catch c1 c2 # cs, css, Normal s) | Suc j \<Rightarrow> f j)"
invariantsHoldInValidRunsFromInitialState: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars" shows "invariantsHoldInState state F0 decisionVars"
dvd1_eq1: fixes x :: "int" assumes "0 < x" shows "is_unit x = (x = 1)"
sup_ge1: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod x (sup_prod x y)"
zero_smult: fixes m :: "'v" assumes "m \<in> V" shows "0 \<cdot> m = (0::'v)"
arr_eqI: fixes \<mu> :: "'a arrow_of_spans_data"   and \<mu>' :: "'a arrow_of_spans_data" assumes "par \<mu> \<mu>'"   and "Chn \<mu> = Chn \<mu>'" shows "\<mu> = \<mu>'"
polyfun_eq_const: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "'a" shows "(\<forall>z. (\<Sum>i\<le>n. c i * z ^ i) = k) = (c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = (0::'a)))"
mod_minus_minus: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus a) (uminus b) = uminus (modulo a b)"
nlistToTree_inv_1: fixes nl :: "('a, 'b) Node list" assumes "\<forall>x\<in>set nl. FingerTreeStruc.is_measured_node x" shows "FingerTreeStruc.is_measured_ftree (FingerTreeStruc.nlistToTree nl)"
is_suffix_eq_proper_suffix: fixes a :: "'a list"   and b :: "'a list" shows "is_suffix a b = (a = b \<or> is_proper_suffix a b)"
funpow_add: fixes m :: "nat"   and n :: "nat"   and f :: "'g \<Rightarrow> 'g" shows "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
recvmsg_simps_2: fixes P :: "'d \<Rightarrow> bool"   and ips :: "nat set"   and m :: "'d" shows "recvmsg P (groupcast ips m) = True"
preserves_reflects_arr: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>" shows "partial_magma.arr B (\<tau> f) = partial_magma.arr A f"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Right_a.hom a b = hom a b \<inter> Collect (right a)"
EM_remainder_converges_of_real: fixes n :: "nat"   and f :: "real \<Rightarrow> real"   and a :: "int" assumes "EM_remainder_converges n f a"   and "continuous_on {real_of_int a..} f" shows "EM_remainder_converges n (\<lambda>x. of_real (f x)) a"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.in_hom f a b" shows "S.in_hom (M.\<ll> f) (M\<^sub>\<omega>.tensor M.unity a) (S.map b)"
abs_summable_on_altdef: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(f abs_summable_on A) = set_integrable (count_space UNIV) A f"
concat_map_maps: fixes f :: "'d \<Rightarrow> 'c list"   and xs :: "'d list" shows "concat (map f xs) = List.maps f xs"
upt_raw_def: fixes n :: "nat"   and m :: "nat" shows "upt_raw n m = (if n \<le> m then Done else Yield m (Suc m))"
assms: shows "(\<forall>x<2 ^ n. g x = 0) \<or> (\<forall>x<2 ^ n. g x = 1)"
vec_of_basis_enum_uminus: fixes b2 :: "'a" shows "vec_of_basis_enum (- b2) = - vec_of_basis_enum b2"
opposite_ocircline_clmat_def: shows "opposite_ocircline_clmat \<equiv> map_fun Rep_circline_mat Abs_circline_mat opposite_ocircline_cmat"
pHom_rHom: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and f :: "'a \<Rightarrow> 'e" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y" shows "f \<in> rHom R A"
simps: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b + 'd"   and a2 :: "'c + 'e" shows "rel_sum R1 R2 a1 a2 = ((\<exists>a c. a1 = Inl a \<and> a2 = Inl c \<and> R1 a c) \<or> (\<exists>b d. a1 = Inr b \<and> a2 = Inr d \<and> R2 b d))"
lemma_SReal_lub: fixes x :: "real star" assumes "x \<in> \<real>" shows "isLub \<real> {s \<in> \<real>. s < x} x"
same_orbit_char: fixes m :: "'c"   and n :: "'c" assumes "m \<in> M"   and "n \<in> M" shows "((m, n) \<in> same_orbit_rel) = (\<exists>g\<in>carrier G. \<phi> g n = m)"
spr_repSim_def: shows "spr_repSim \<equiv> \<lambda>s. (sprFst s, sprLst s)"
less_minus_iff: fixes a :: "'a"   and b :: "'a" shows "(a < - b) = (b < - a)"
lt_of_comp_of_ords: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "lt_of_comp (comp_of_ords le lt) = lt"
coboundedI1: fixes c :: "'k"   and a :: "'k"   and b :: "'k" assumes "c \<le> a" shows "c \<le> max a b"
fold_A_chamber_system_image_fixespointwise: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "F \<in> ChamberComplex.chamber_system A" shows "fixespointwise (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) ((ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) ` F)"
one_mod_card_int: shows "1 mod int CARD('a) = 1"
R_2: shows "X = w +o \<langle>R\<rangle>"
mset_factors_exist: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.ufd times one plus zero minus uminus"   and "x \<noteq> zero"   and "\<not> dvd.dvd times x one" shows "\<exists>F. comm_semiring_1.mset_factors times one zero F x"
subcategory_axioms: shows "subcategory (\<cdot>) (right \<omega>)"