hcju/isabelleps huggingface dataset:
iso_bij_betw_individual_blocks: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and bl :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "bl \<in># \<B>" shows "bij_betw bij_map bl (bij_map ` bl)"
eqvtI: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "perm \<bullet> P \<leadsto><Rel'> perm \<bullet> Q"
ccompare_finite_1_def: shows "ccompare \<equiv> Some compare"
detB_rw: fixes \<phi> :: "nat \<Rightarrow> nat" assumes "\<phi> permutes {0..<n}" shows "(\<Sum>\<pi> | \<pi> permutes {0..<n}. signof (\<pi> \<circ> inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> inv \<phi>) i))) = (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
ht_phl_export1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<cdot> y \<le> y \<cdot> x \<cdot> y"   and "\<lbrace>x \<cdot> y\<rbrace>z\<lbrace>w\<rbrace>" shows "\<lbrace>x\<rbrace>y \<cdot> z\<lbrace>w\<rbrace>"
intros_1: shows "local.S (return_spmf (None, (), Void)) (return_spmf (None, Void))"
ennreal_eq_zero_iff: fixes x :: "real" assumes "0 \<le> x" shows "(ennreal x = 0) = (x = 0)"
ecl_ord_def: shows "ecl_ord = {(x, y). (mset_ecl x, mset_ecl y) \<in> mult (mult trm_ord)}"
Vs_reduce_to_themselves: fixes v :: "trm"   and T :: "ty" assumes "v \<in> V T" shows "v \<Down> v"
conc_fun_R_mono: fixes R :: "('a \<times> 'b) set"   and R' :: "('a \<times> 'b) set"   and M :: "'b nres" assumes "R \<subseteq> R'" shows "\<Down> R M \<le> \<Down> R' M"
prime_factorization_exists_nat: fixes n :: "nat" assumes "0 < n" shows "\<exists>M. (\<forall>p\<in>#M. prime p) \<and> n = (\<Prod>i\<in>#M. i)"
gen_ramanujan_sum_1_n: fixes f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex"   and n :: "nat" shows "s f g 1 n = f 1 * g 1"
rel_interior_subset: fixes S :: "'b set" shows "rel_interior S \<subseteq> S"
WT_restD_rfunc_adv: fixes \<I>_adv :: "('p, 'q) \<I>"   and \<I>_usr :: "('r, 's) \<I>"   and I :: "'t \<Rightarrow> bool"   and rest :: "('t, 'u, 'p, 'r, 'q, 's, 't) rest_scheme"   and y :: "'q"   and es :: "'u list"   and s' :: "'t"   and s :: "'t"   and x :: "'p" assumes "WT_rest \<I>_adv \<I>_usr I rest"   and "((y, es), s') \<in> set_spmf (rfunc_adv rest s x)"   and "x \<in> outs_\<I> \<I>_adv"   and "I s" shows "y \<in> responses_\<I> \<I>_adv x \<and> I s'"
approximating_semantics_imp_fun: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "approximating_bigstep_fun \<gamma> p rs s = t"
conjunctive_def: shows "Comp.conjunctive = {x. \<forall>y z. x \<circ> y \<sqinter> z = (x \<circ> y) \<sqinter> (x \<circ> z)}"
cong3_diff2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and C :: "'a"   and A :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "B \<noteq> C"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "B' \<noteq> C'"
to_tree_order_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set nodes" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes"
l: shows "l ` AA \<subseteq> {x. (0::'a) \<le> x}"
coset_mult_inv1: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Group.group G"   and "M #>\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y = M"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M #>\<^bsub>G\<^esub> x = M #>\<^bsub>G\<^esub> y"
current_plus_web: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current \<Gamma> g" shows "current \<Gamma> (f \<frown>\<^bsub>\<Gamma>\<^esub> g)"
1: shows "{w. llength w \<le> enat n} = (\<Union>k\<le>n. {w. llength w = enat k})"
simps_4: fixes v :: "Ferrack.num" shows "not (Lt v) = NOT (Lt v)"
good_item_simps_52: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and ys :: "'varSort"   and xa :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "good X"   and "xsa \<noteq> ys \<or> xa \<notin> {y, y1}" shows "Abs xsa xa X $[y1 // y]_ys = Abs xsa xa (X #[y1 // y]_ys)"
set_divide_def: fixes A :: "('a \<Rightarrow> 'b) set"   and B :: "('a \<Rightarrow> 'b) set" shows "set_divide A B = {\<lambda>x. f x / g x |f g. f \<in> A \<and> g \<in> B}"
VSLucky: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSLucky \<phi>) xs = eval \<phi> xs"
productF_succ_fail_imp: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and w :: "'d \<times> 'e"   and p1 :: "'f"   and p2 :: "'g" assumes "productF A B FAIL AB"   and "FAIL \<in> succ AB w (p1, p2)"   and "well_formed A"   and "well_formed B" shows "p1 \<in> nodes A \<and> p2 \<in> nodes B \<and> fst w \<in> inputs A \<and> snd w \<in> outputs A \<union> outputs B \<and> succ AB w (p1, p2) = {FAIL} \<and> succ A w p1 = {} \<and> succ B w p2 \<noteq> {}"
inf_greatest: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" assumes "x \<subseteq># y"   and "x \<subseteq># z" shows "x \<subseteq># y \<inter># z"
X2_X_in_pac_ideal: fixes c :: "nat"   and A :: "int mpoly set" shows "(Var c)\<^sup>2 - Var c \<in> pac_ideal A"
assms_9: shows "r = 2 ^ (2 * k - l) * f\<^sup>2 - 1"
unique_idx_append: fixes gs :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> nat \<times> 'h) list"   and data :: "nat \<times> 'i"   and hs :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> nat \<times> 'h) list"   and data' :: "nat \<times> 'i"   and aux :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> 'h) list \<times> 'i" assumes "unique_idx gs data"   and "(hs, data') = add_indices aux data" shows "unique_idx (gs @ hs) data'"
simps: fixes \<delta> :: "('c \<times> 'd \<times> 'c) set"   and q\<^sub>0 :: "'c"   and \<alpha> :: "(('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set set) set"   and w :: "nat \<Rightarrow> 'd" shows "accept\<^sub>G\<^sub>R_LTS (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 (Fin, Inf) w)"
oppositeLiteralsHaveSameVariable: fixes literal :: "Literal" shows "var (opposite literal) = var literal"
ccInf_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
parallel_eoracle_def: fixes eoracle1 :: "'x \<Rightarrow> 'r \<Rightarrow> (('t \<times> 'v list) \<times> 'x) spmf"   and eoracle2 :: "'y \<Rightarrow> 's \<Rightarrow> (('u \<times> 'w list) \<times> 'y) spmf"   and state :: "'x \<times> 'y" shows "parallel_eoracle eoracle1 eoracle2 state \<equiv> map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) \<circ> (eoracle1 \<ddagger>\<^sub>O eoracle2) state"
beta_star_trans: fixes M1 :: "lam"   and M2 :: "lam"   and M3 :: "lam" assumes " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2"   and " M2 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3" shows " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
no_markers_if_all_snapshotted: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "trace c t c'"   and "\<forall>p. ps c p \<noteq> None"   and "Marker \<notin> set (msgs c i)" shows "Marker \<notin> set (msgs c' i)"
oreturn_rule: fixes P :: "'a \<Rightarrow> bool"   and r :: "'a" assumes "P r" shows "ospec (Some r) P"
prime_imp_prime_elem': 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "comm_semiring_1.prime_elem times one zero x"
Nil_tl: fixes xs :: "'c list" shows "([] = list.tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
VEnd_composite_right: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and S :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T"   and "VectorSpaceEnd smult V S" shows "VectorSpaceEnd smult V (T \<circ> S)"
perm_4: fixes pi :: "(name \<times> name) list"   and x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "pi \<bullet> x3<x1>.x2 = (pi \<bullet> x3)<(pi \<bullet> x1)>.(pi \<bullet> x2)"
dom_MkArr: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a"   and F :: "'b" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) (concrete_category.MkArr A B F)" shows "partial_magma.dom (concrete_category.COMP Obj Hom Comp) (concrete_category.MkArr A B F) = concrete_category.MkArr A A (Id A)"
every_component_integrable: shows "\<forall>b\<in>basis. (\<lambda>x. F c \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
preprm_ext_symmetric: fixes x :: "('b \<times> 'b) list"   and y :: "('b \<times> 'b) list" assumes "x =p y" shows "y =p x"
set_of_apprs_Nil: fixes xrs :: "real list" shows "(xrs \<in> set_of_appr []) = (xrs = [])"
hyps_6: fixes b :: "lam" assumes "t \<rightarrow>\<^sub>\<beta> b" shows "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s b"
assms_5: shows "{v. `P\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk>`} \<subseteq> {v. `Q\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk>`}"
borel_measurable_restrict_space_iff: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(f \<in> borel_measurable (restrict_space M \<Omega>)) = ((\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable M)"
seq_char: fixes g :: "'a"   and f :: "'a" shows "hom'.seq g f = (hom'.arr f \<and> hom'.arr g \<and> seq g f)"
cc_subs: shows "set_mset CC \<subseteq> N - Rf N"
fmeasurable_def: fixes M :: "'a measure" shows "fmeasurable M = {A \<in> sets M. emeasure M A < \<infinity>}"
cadjoint_bounded_clinear: fixes A :: "'a \<Rightarrow> 'b" assumes "bounded_clinear A" shows "bounded_clinear (A\<^sup>\<dagger>)"
liftn_lift: fixes n :: "nat"   and t :: "dB"   and k :: "nat" shows "liftn (Suc n) t k = lift (liftn n t k) k"
finite_Sup_less_iff: fixes X :: "'c set"   and a :: "'c" assumes "finite X"   and "X \<noteq> {}" shows "(Sup X < a) = (\<forall>x\<in>X. x < a)"
zero_le_divide_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
trm_rep_compatible_with_structure: fixes S :: "'a eclause set" shows "value_is_compatible_with_structure (\<lambda>x. trm_rep x S)"
poly_dvd_antisym: fixes p :: "'a poly"   and q :: "'a poly" assumes "lead_coeff p = lead_coeff q"   and "p dvd q"   and "q dvd p" shows "p = q"
restrict_indets_pp_def: fixes t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "restrict_indets_pp t = (\<Sum>x\<in>keys t. lookup t x \<cdot> restrict_indets_subst x)"
borel_einterval: fixes a :: "ereal"   and b :: "ereal" shows "einterval a b \<in> sets borel"
lim1: shows "LIM k sequentially. real (nth_prime (k - 1)) :> at_top"
p: shows "partial_density_operator (DS (M1 * \<rho> * adjoint M1))"
continuous_closed_quotient_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "closed_map X X' f" shows "quotient_map X X' f = (f ` topspace X = topspace X')"
g'_def: shows "g' \<equiv> snd (bezout_coefficients g h)"
Step1_4_5a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCReply crCh rmCh rst p \<and> $S5 rmhist p \<and> S6 rmhist p$ \<and> unchanged (e p, c p, m p) \<longrightarrow> unchanged (rtrner memCh!p, resbar rmhist!p)"
pr_int_agrk_card: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and s :: "rat"   and b :: "rat"   and a :: "rat"   and d :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "NonceS (S (Card n, n, run)) = Some s"   and "IntMapK (S (Card n, n, run)) = Some b"   and "ExtMapK (S (Card n, n, run)) = Some a"   and "IntAgrK (S (Card n, n, run)) = Some d" shows "d * (s + a * b) \<noteq> 0"
map_disc_iff: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b option" shows "(map_option f a = None) = (a = None)"
simps_5: fixes vb :: "typ"   and vc :: "Core.term"   and va :: "Core.term list" shows "strip_comb_imp (Abs vb vc, va) = (Abs vb vc, va)"
diff_less_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) c = less a (plus c b)"
Ball_Collect: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
upper_right_closed: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set"   and t :: "real" assumes "\<forall>c\<in>\<X>. real (k c) < u c"   and "u \<in> R"   and "R \<in> \<R>"   and "0 \<le> t" shows "u \<oplus> t \<in> R"
assms_8: assumes "a \<noteq> b"   and "qs \<in> pattern" shows "real (T\<^sub>p [a, b] qs (OPT2 qs [a, b])) = opt"
case_ifexi_def: fixes fti :: "bdd \<Rightarrow> 'a option"   and ffi :: "bdd \<Rightarrow> 'a option"   and fii :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bdd \<Rightarrow> 'a option"   and ni :: "nat"   and s :: "bdd" shows "mi.case_ifexi fti ffi fii ni s \<equiv> destrmi' ni s \<bind> (\<lambda>dest. case dest of TD \<Rightarrow> fti s | FD \<Rightarrow> ffi s | IFD v ti ei \<Rightarrow> fii v ti ei s)"
loc_reach_refl': fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>c, mds, mem\<rangle> \<in> loc_reach \<langle>c, mds, mem\<rangle>"
cs_in_initial_state_implies_not_snapshotted: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = NotStarted"   and "channel cid = Some (p, q)" shows "\<not> ps (distributed_system.s channel trans send recv init t i) q \<noteq> None"
div_poly_eq: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "eucl_rel_poly x y (q, r)" shows "x div y = q"
signs_1: fixes q :: "real poly"   and xa :: "real" assumes "q \<in> set ps" shows "sgn (poly q xa) = sgn (poly d xa) * sgn (poly (q div d) xa)"
nonempty_rule: fixes G :: "('b, 'c) labeled_graph" assumes "graph G" shows "maintained nonempty_rule G = (vertices G \<noteq> {})"
simps_2: fixes ws :: "'b stream list" shows "stl (stranspose ws) = stranspose (map stl ws)"
put_get: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
inner_left_distrib: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner (x + y) z = inner x z + inner y z"
add: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
le_N: fixes a :: "'a" assumes "a \<in> set (diag_mat A)"   and "norm a = 1" shows "length (filter ((=) a) (diag_mat A)) \<le> N"
maintainedI: fixes A :: "('c, 'a) labeled_graph"   and G :: "('c, 'b) labeled_graph"   and B :: "('c, 'a) labeled_graph" assumes "\<And>f. graph_homomorphism A G f \<Longrightarrow> extensible (A, B) G f" shows "maintained (A, B) G"
Q'_2: fixes i :: "nat" shows "Q' i \<in> {Q \<in> sets M. emeasure N Q \<noteq> \<infinity>}"
p_coprime_right_nat: fixes p :: "nat"   and a :: "nat" assumes "prime p" shows "coprime a p = (\<not> p dvd a)"
trimming: fixes \<Gamma> :: "('a, 'b) web_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> g"   and "wave \<Gamma> g"   and "g \<le> f"   and "\<And>x. \<lbrakk>x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f); x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> KIR g x"   and "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) - A \<Gamma> = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - A \<Gamma>" shows "trimming \<Gamma> f g"
bij_betw_trans: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'h set"   and g :: "'h \<Rightarrow> 'i"   and C :: "'i set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (g \<circ> f) A C"
projection_on_empty_trace: fixes X :: "'b set" shows "[] \<upharpoonleft> X = []"
isPath_head: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and e :: "nat \<times> nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isPath c u (e # p) v = (fst e = u \<and> e \<in> Graph.E c \<and> Graph.isPath c (snd e) p v)"
map_ide_simp: fixes a :: "'c" assumes "Cop.ide a" shows "\<Psi>'.map a = S.inv (\<Phi>' a)"
intros_12: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and g :: "'e \<Rightarrow> 'e set"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i \<in> g (\<sigma> i)" shows "((\<sigma>, {l}\<langle>g\<rangle> p), \<tau>, \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
A_dest'_2: fixes A :: "'a set"   and y :: "'a"   and z :: "'a" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "y \<in> A"   and "z \<in> A" shows "(z, y) \<in> r"
LU: shows "GEN_OP lu op_map_lookup (Rk \<rightarrow> \<langle>Rk, Rv\<rangle>Rm \<rightarrow> \<langle>Rv\<rangle>option_rel)"
dvd_PM_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(p dvd q) = (divides_ff (content_ff (map_poly to_fract p)) (content_ff (map_poly to_fract q)) \<and> map_poly to_fract p dvd map_poly to_fract q)"
exhaust: fixes y :: "'a seq_action" obtains x1 :: "'a" where "y = broadcast x1"   | x21 :: "nat set" and x22 :: "'a" where "y = groupcast x21 x22"   | x31 :: "nat" and x32 :: "'a" where "y = unicast x31 x32"   | x4 :: "nat" where "y = \<not>unicast x4"   | x5 :: "'a" where "y = send x5"   | x6 :: "nat" where "y = deliver x6"   | x7 :: "'a" where "y = receive x7"   | "y = \<tau>\<^sub>s"
showsp_real_def: fixes p :: "nat"   and x :: "real" shows "showsp_real p x = showsp_float10 p (lfloat10 x)"
distinct_lists_children: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "local.CD.a_distinct_lists h"   and "known_ptr ptr"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
uminus_Pair: fixes a :: "'a"   and b :: "'b" shows "- (a, b) = (- a, - b)"
star_if_lang: fixes w :: "'b list"   and A :: "'b list set" assumes "w \<in> A" shows "w \<in> star A"
set_set_contI: fixes f :: "'b set \<Rightarrow> 'c set" assumes "\<And>S. f (\<Union> S) = \<Union> (f ` S)" shows "cont f"
fminusD2: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset"   and P :: "bool" assumes "c |\<in>| A |-| B"   and "c |\<in>| B" shows "P"
Ghost: fixes A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and \<O>s :: "nat set list"   and i :: "nat"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> \<O>s ! j = {}" shows "\<O>s,i\<turnstile> (Ghost A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
not_eq_extremum: fixes a :: "'a" shows "(a \<noteq> top) = (a < top)"
eq: shows "density M (RN_deriv M N) = density M (\<lambda>x. ennreal (enn2real (RN_deriv M N x)))"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "Go\<tau>.B.arr f"   and "Go\<tau>.B.cod f = b" shows "B b f = f"
WPO_SN_order_pair: shows "SN_order_pair WPO_S WPO_NS"
affine_Int: fixes s :: "'c set"   and t :: "'c set" assumes "affine s"   and "affine t" shows "affine (s \<inter> t)"
is_pot_ordI: fixes cmp_term :: "(('a, 'b) pp \<times> nat) nat_term_order" assumes "\<And>u v. snd u < snd v \<Longrightarrow> lt_of_nat_term_order cmp_term u v" shows "ordered_term.is_pot_ord (\<lambda>x. x) (lt_of_nat_term_order cmp_term)"
fls_const_def: fixes c :: "'b" shows "fls_const c \<equiv> Abs_fls (\<lambda>n. if n = 0 then c else (0::'b))"
reduced_GB_unique_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]"   and "punit.is_reduced_GB G"   and "ideal G = ideal F" shows "punit.reduced_GB F = G"
seq_compact_imp_Heine_Borel: fixes S :: "'c set" assumes "seq_compact S" shows "compact S"