hcju/isabelleps huggingface dataset:
is_median_cong: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x = y"   and "mset xs = mset ys" shows "is_median x xs = is_median y ys"
fst_netgmap_none: fixes i :: "nat"   and s :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e" assumes "i \<notin> net_ips s" shows "fst (netgmap sr s) i = None"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ S')"   and "fv t' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> (V \<union> fv u)" shows "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
integral_add: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x + g x = integral\<^sup>L M f + integral\<^sup>L M g"
assms: shows "range w \<subseteq> Pow (atoms_ltlc \<phi>\<^sub>c)"
ArrVal_def: shows "ArrVal = []\<^sub>\<circ>"
C_bound: fixes C :: "real"   and hb :: "real" assumes "\<And>b. b \<in> set bs \<Longrightarrow> C < b"   and "0 < hb" shows "\<forall>\<^sub>F x in at_top. \<forall>b\<in>set bs. C * x \<le> b * x - hb * x / ln x powr (1 + e)"
hensel_1: shows "F' = hensel_fpxs1 * hensel_fpxs2"
simps_4: fixes p :: "real \<times> real \<times> real" shows "getPoly (LessUni p) = p"
pcr_cr_eq: shows "pcr_pfun (=) (=) = cr_pfun"
fresh_3: fixes a :: "name"   and x1 :: "name"   and x2 :: "lam" shows "a \<sharp> (Lam [x1].x2) = a \<sharp> [x1].x2"
remove_removes_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> Shadow_DOM.remove node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"
last_plus: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "m \<le> n" shows "prod g {m..n} = g n * prod g {m..<n}"
pt_set_prm_eq: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and Xs :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and " pi1 \<triangleq> pi2 " shows "pi1 \<bullet> Xs = pi2 \<bullet> Xs"
pr_gr_def: shows "pr_gr = PrimRecOp1 0 (\<lambda>a b. g_step b (c_fst a))"
clinear_continuous_within: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set" assumes "bounded_clinear f" shows "continuous (at x within s) f"
measure_preserving_total_measure: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "emeasure M (space M) = emeasure N (space N)"
assms_1: fixes x :: "'a" assumes "x \<in> box a b" shows "g x = f x"
complete: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A"   and "A \<in> null_sets M" shows "B \<in> sets M"
lm019: shows "converse ` injectionsUniverse = injectionsUniverse"
6567: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - - (x \<sqinter> y) = bot"
gen_entries_simps_3: fixes kvts :: "(('c \<times> 'd) \<times> ('c, 'd) rbt) list"   and c :: "color"   and l :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) rbt" shows "RBT_Impl.gen_entries kvts (Branch c l k v r) = RBT_Impl.gen_entries (((k, v), r) # kvts) l"
rel_pmf_conj_1: fixes P :: "bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a pmf"   and y :: "'b pmf" shows "rel_pmf (\<lambda>x y. P \<and> Q x y) x y = (P \<and> rel_pmf Q x y)"
fSum_def: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" shows "\<Sigma>\<^sub>f A f n m = (if n \<le> m then \<Sigma>\<^sub>e A cmp f (slide n) (m - n) else \<zero>\<^bsub>A\<^esub>)"
normalize_def: fixes p :: "int \<times> int" shows "Rat.normalize p = (if 0 < snd p then let a = gcd (fst p) (snd p) in (fst p div a, snd p div a) else if snd p = 0 then (0, 1) else let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a))"
Ioc_subset_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<or> less_eq c a \<and> less_eq b d)"
row_all1_matrix: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat" assumes "i < nr" shows "row (all1_matrix nr nc) i = all1_vec nc"
pause_state_def: fixes pause :: "'f \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> 'e"   and out :: "'f"   and c :: "'g \<Rightarrow> ('d, 'e) stateT" shows "pause_state pause out c = StateT (\<lambda>s. pause out (\<lambda>i. run_state (c i) s))"
INF_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Inf :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
round_down_ge1: fixes x :: "real"   and p :: "int" assumes "1 \<le> x"   and "- log 2 x \<le> real_of_int p" shows "1 \<le> round_down p x"
cases: fixes a :: "intv" assumes "isConst a" obtains uu_ :: "nat" where "a = Const uu_"
is_zero_row_eq_row_zero: fixes a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec" shows "is_zero_row a A = (row a A = 0)"
f_abs_plus_minus: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" shows "\<bar>f x\<bar> = pp f x + np f x"
simps_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vwellformed (Vrecabs css name \<Gamma>) = (pred_fmap id (fmmap vwellformed \<Gamma>) \<and> pred_fmap wellformed_clauses css \<and> name |\<in>| fmdom css \<and> True)"
Fun_gt_params: fixes f :: "'a"   and X :: "('a, 'b) Term.term list" shows "Fun f X \<notin> subterms\<^sub>s\<^sub>e\<^sub>t (set X)"
nof: fixes z :: "complex" assumes "A \<le> cmod z" shows "cmod (f z) \<le> B * cmod z ^ n"
assms_2: shows "R1 \<in> rcosets stabilizer G \<phi> x"
that_1: fixes j :: "'a" assumes "i = succ j"   and "local.aboveS j \<noteq> {}" shows "thesis"
inject_1: fixes x1 :: "agent"   and y1 :: "agent" shows "(msg.Agent x1 = msg.Agent y1) = (x1 = y1)"
SK_disjS_disjK: fixes xs :: "nat discr llist"   and ys :: "nat discr llist" shows "SK\<cdot>(disjS\<cdot>xs\<cdot>ys) = disjK\<cdot>(SK\<cdot>xs)\<cdot>(SK\<cdot>ys)"
impl_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a l\<rightarrow> b \<equiv> Abs_N ((Rep_N b - Rep_N a) \<sqinter> (0::'b))"
SUP_le_iff: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and u :: "'a" shows "(\<Squnion> (f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
def_scheme: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" shows "f X Y = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> \<and> (\<upsilon> Y) \<tau> = true \<tau> then g (X \<tau>) (Y \<tau>) else invalid \<tau>)"
is_sublist_eq: fixes vs :: "'a list"   and c :: "'a"   and y :: "'a"   and x :: "'a" assumes "distinct vs"   and "c \<noteq> y" shows "(nextElem vs c x = y) = is_sublist [x, y] vs"
proj_image: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "proj r ` A = A // r"
gcg_sd: shows "gen_contr_graph (with_proj (subdivide G' (u, v) w)) V = gen_contr_graph (with_proj G') V"
pref_cancel: fixes z :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "z \<cdot> u \<le>p z \<cdot> v" shows "u \<le>p v"
det_multrow: fixes k :: "nat"   and n :: "nat"   and A :: "'a mat"   and a :: "'a" assumes "k < n"   and "A \<in> carrier_mat n n" shows "det (multrow k a A) = a * det A"
Abs: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "phi X" shows "phiAbs (qAbs xs x X)"
le_dag_def: fixes s :: "dag"   and t :: "dag" shows "(s \<le> t) = (s = t \<or> s < t)"
sup_bot_left: fixes x :: "'b upred" shows "(false \<or> x) = x"
ord_spmf_None: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a spmf" shows "ord_spmf ord (return_pmf None) x"
q: shows "q = A * (MPoly_Type.monom (monomial (Suc 0) x') 1 + p) + b"
nt_abbrev_def: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and G :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" shows "NT f : F \<Longrightarrow> G \<equiv> NatTrans f \<and> NTDom f = F \<and> NTCod f = G"
Proj_is_Proj: fixes M :: "'a ccsubspace" shows "is_Proj (Proj M)"
wf_digraph_axioms: shows "wf_digraph (G \<restriction> verts H)"
GreenThm_typeI_typeII_divisible_region'_1: fixes one_chain_typeI :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain_typeII :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "only_vertical_division one_chain_typeI two_chain_typeI"   and "boundary_chain one_chain_typeI"   and "only_horizontal_division one_chain_typeII two_chain_typeII"   and "boundary_chain one_chain_typeII"   and "common_sudiv_exists one_chain_typeI one_chain_typeII" shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. F x \<bullet> j) i x - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
fin: fixes w :: "complex" assumes "w \<in> ball z r" shows "f w \<in> ball y B0"
minus_mult_minus: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) (uminus b) = times a b"
extremum_unique: fixes a :: "'d list" shows "prefix a [] = (a = [])"
res2: shows "wmin w1 w2 \<equiv>\<^sub>E real_of_int (sint (wmin w1 w2))"
equalityE: fixes A :: "'d set"   and B :: "'d set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
sel_rews_2: fixes cfun :: "ValD \<rightarrow> ValD" shows "appF\<cdot>(ValF\<cdot>cfun) = cfun"
bounded_bilinear: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_sesquilinear prod" shows "bounded_bilinear prod"
sum_upto_mangoldt_over_id_minus_phi_bounded: shows "(\<lambda>x. sum_upto (\<lambda>d. mangoldt d / real d) x - \<MM> x) \<in> O(\<lambda>_. 1)"
fold_empty: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e" shows "Finite_Set.fold f z {} = z"
monom_zero: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "monom (UP R) \<zero>\<^bsub>R\<^esub> n = \<zero>\<^bsub>UP R\<^esub>"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (\<tau>\<^sub>0 a) = F.map\<^sub>0 a"
h1: shows "\<forall>x\<in>set xs. h' \<turnstile> ok P x"
minus_extended_def: fixes x :: "'b extended"   and y :: "'b extended" shows "x - y = x + - y"
lexext_rev_total: fixes B :: "'b set"   and A :: "'b set"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "\<forall>y\<in>B. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x"   and "ys \<in> lists B"   and "xs \<in> lists A" shows "lexext_rev gt ys xs \<or> lexext_rev gt xs ys \<or> ys = xs"
ArityAnalysisLetSafe_axioms: shows "ArityAnalysisLetSafe Aexp Aheap"
cong_mod_right: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "[b = c mod a] (mod a) = [b = c] (mod a)"
APP'_def: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "f $' a \<equiv> f a"
dg_singleton_ArrI: fixes F :: "V"   and j :: "V"   and a :: "V"   and \<CC> :: "V" assumes "F = ZFC_in_HOL.set {\<langle>j, a\<rangle>}"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "F \<in>\<^sub>\<circ> (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)\<lparr>Arr\<rparr>"
powsum_mpoly_altdef: fixes X :: "nat set"   and k :: "nat" shows "powsum_mpoly X k = (\<Sum>x\<in>X. MPoly_Type.monom (Poly_Mapping.single x k) (1::'b))"
that_3: fixes \<Gamma>' :: "char list \<Rightarrow> secLevel option"   and Vn :: "char list"   and lev :: "secLevel" assumes "x = \<Gamma>'"   and "xa = Var Vn"   and "xb = lev"   and "\<Gamma>' Vn = Some lev" shows "thesis"
D1: shows "D1 \<in> carrier_mat (m - 1) n"
mono_hiding: fixes P :: "'b process"   and Q :: "'b process"   and A :: "'b set" assumes "P \<sqsubseteq> Q" shows "P \ A \<sqsubseteq> Q \ A"
mult_rel_carrier: fixes X :: "('c set \<times> 'd) set"   and Y :: "('c set \<times> 'd) set" assumes "X \<in> dlY.carrier_direct_lim"   and "Y \<in> dlY.carrier_direct_lim" shows "dlY.mult_rel X Y \<in> dlY.carrier_direct_lim"
not_enat_eq: fixes x :: "enat" shows "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
open_Arg2pi2pi_gt: fixes t :: "real" shows "open {z. t < Arg2pi z}"
order_linear_power: fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "order a ([:b, 1::'b:] ^ n) = (if b = - a then n else 0)"
deriv_sound: fixes R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and D :: "(nat \<times> ('sf, 'sc, 'sz) step) list" assumes "deriv_ok R D"   and "sound R" shows "sound (deriv_result R D)"
hereditarily_fs_eqvt: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (p \<bullet> t\<^sub>\<alpha>)"
alt_option_def: fixes alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m1 :: "('d, 'e) optionT"   and m2 :: "('d, 'e) optionT" shows "alt_option alt m1 m2 = OptionT (alt (run_option m1) (run_option m2))"
simps_13: fixes f :: "'h \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h llist" shows "lmap f (LCons x21 x22) = LCons (f x21) (lmap f x22)"
elims: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and y :: "nat" assumes "SkewBinomialHeapStruc.height_queue x = y" obtains "x = []" and "y = 0"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # ts" and "y = max (Suc (SkewBinomialHeapStruc.height_tree t)) (SkewBinomialHeapStruc.height_queue ts)"
Scheduler_def: fixes c :: "Config"   and activation :: "nat \<Rightarrow> nat list" shows "Scheduler c activation \<equiv> \<forall>t. let s = t mod cycleLength c in if s mem schedule c then activation t = [s] else activation t = []"
min_zero_index: fixes x :: "nat"   and k :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "x \<in> {1..k}"   and "T x = 0" shows "min\<^sub>k T k \<le> x"
add_to_coefficient_def: shows "add_to_coefficient = (\<lambda>(a, n) b. {#(a', uu_) \<in># b. a' \<noteq> a#} + (if n + \<Sum>\<^sub># (snd `# {#(a', uu_) \<in># b. a' = a#}) = 0 then {#} else {#(a, n + \<Sum>\<^sub># (snd `# {#(a', uu_) \<in># b. a' = a#}))#}))"
SpanD_lincomb: fixes ms :: "'m list" shows "Span ms = {rs \<bullet>\<cdot> ms |rs. set rs \<subseteq> UNIV \<and> length rs = length ms}"
quorum_of_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and Q :: "'b set" assumes "stellar slices W" shows "stellar.quorum_of slices W p Q \<equiv> stellar.quorum slices W Q \<and> (p \<notin> W \<or> (\<exists>Sl\<in>slices p. Sl \<subseteq> Q))"
prob_conj_imp1: fixes Q :: "'key \<times> 'message list \<Rightarrow> bool"   and Pr :: "'key \<times> 'message list \<Rightarrow> bool" assumes "prob ({x. Q x} \<inter> msgs) = 0" shows "prob ({x. Pr x \<and> Q x} \<inter> msgs) = 0"
SUP_UN_eq: fixes r :: "'c \<Rightarrow> 'b set"   and S :: "'c set" shows "(SUP i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> \<Union> (r ` S))"
face_distance_eq_chamber_distance_compare_other_chamber: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set"   and E :: "'b set" assumes "ThinChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "z \<lhd> C"   and "z \<lhd> D"   and "C \<noteq> D"   and "order.greater_eq (ChamberComplex.chamber_distance X D E) (ChamberComplex.chamber_distance X C E)" shows "ChamberComplex.face_distance X z E = ChamberComplex.chamber_distance X C E"
split_def: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list" shows "split split \<equiv> (\<forall>xs p ls rs. split xs p = (ls, rs) \<longrightarrow> xs = ls @ rs) \<and> (\<forall>xs p ls sub sep rs. split xs p = (ls @ [(sub, sep)], rs) \<longrightarrow> sorted_less (separators xs) \<longrightarrow> sep < p) \<and> (\<forall>xs p ls sub sep rs. split xs p = (ls, (sub, sep) # rs) \<longrightarrow> sorted_less (separators xs) \<longrightarrow> p \<le> sep)"
meta_eq_to_obj_eq: fixes A :: "'d"   and B :: "'d" assumes "A \<equiv> B" shows "A = B"
basis_enum_of_vec_unit_vec: fixes i :: "nat" assumes "i < length canonical_basis" shows "basis_enum_of_vec (unit_vec (length canonical_basis) i) = canonical_basis ! i"
CSP1_meet: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is CSP1 healthy"   and "y is CSP1 healthy" shows "x \<and> y is CSP1 healthy"
cInf_superset_mono: 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 A :: "'a set"   and B :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq B"   and "A \<subseteq> B" shows "less_eq (Inf B) (Inf A)"
hfprod_def: fixes RR :: "('a \<Rightarrow> 'c \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'c \<Rightarrow> assn)"   and SS :: "('b \<Rightarrow> 'd \<Rightarrow> assn) \<times> ('b \<Rightarrow> 'd \<Rightarrow> assn)" shows "RR *\<^sub>a SS \<equiv> (fst RR \<times>\<^sub>a fst SS, snd RR \<times>\<^sub>a snd SS)"
l_def: fixes "is" :: "int list" shows "l is = sum_list (concat (map (\<lambda>i. if i < 0 then [i] else []) is))"
not_coprime_iff_common_factor: fixes p :: "'a"   and q :: "'a" shows "(\<not> coprime p q) = (\<exists>r. r dvd p \<and> r dvd q \<and> \<not> r dvd (1::'a))"
BisT_MtransC: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.BisT tval aval indis"   and "PL.mustT tval aval c s"   and "PL.mustT tval aval d t"   and "(s, t) \<in> indis"   and "PL.MtransC tval aval (c, s) (c', s')" obtains d' :: "('b, 'c) com" and t' :: "'a" where "PL.MtransC tval aval (d, t) (d', t')" and "(s', t') \<in> indis" and "(c', d') \<in> PL_Indis.BisT tval aval indis"
poly_monom: fixes a :: "'a"   and n :: "nat"   and x :: "'a" shows "poly (Polynomial.monom a n) x = a * x ^ n"
eq_1: shows "H \<turnstile> x1 EQ x1'"
N: shows "\<forall>s\<in>N. \<Delta> s \<le> 0"
rnd_def: fixes \<eta> :: "nat" shows "rnd \<eta> \<equiv> nlists UNIV \<eta>"
Quotient_Bonk_Schramm_extension: shows "Quotient (\<lambda>x y. x \<in> extended_distance_set \<and> y \<in> extended_distance_set \<and> extended_distance x y = 0) abs_Bonk_Schramm_extension rep_Bonk_Schramm_extension cr_Bonk_Schramm_extension"
IntD2: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
fmrel_on_fset_refl_strong: fixes S :: "'a fset"   and m :: "('a, 'b) fmap"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x |\<in>| S; fmlookup m x = Some y\<rbrakk> \<Longrightarrow> P y y" shows "fmrel_on_fset S P m m"
obtuse_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Obtuse A B C" shows "A \<noteq> B \<and> C \<noteq> B \<and> A \<noteq> C"
assms_2: shows "\<theta> \<doteq> \<sigma> \<lozenge> \<eta>"
nhds_def: fixes a :: "'b" shows "nhds a = Inf (principal ` {S. open S \<and> a \<in> S})"
axioms_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_isomorphism A B F G \<tau>" shows "natural_transformation A B F G \<tau>"
subadditive: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "norm (x + y) \<le> norm x + norm y"
hyp_2: shows "mat (row_length (a # M)) (length (a # M)) (a # M)"
H2: fixes speca :: "int mpoly"   and x1b :: "nat set"   and x2a :: "('a, int mpoly) fmap"   and b :: "int mpoly multiset" assumes "spec - speca \<in> More_Modules.ideal polynomial_bool"   and "vars spec \<subseteq> x1b"   and "\<Union> (vars ` set_mset (ran_m A)) \<subseteq> x1b"   and "\<Union> (vars ` set_mset (ran_m x2a)) \<subseteq> x1b"   and "speca \<in> pac_ideal (set_mset (ran_m x2a))"   and "restricted_ideal_to\<^sub>I x1b b \<subseteq> restricted_ideal_to\<^sub>I x1b (ran_m x2a)" shows "spec \<in> pac_ideal (set_mset (ran_m x2a))"
simps: fixes n :: "nat"   and xs :: "'b list" shows "bal_tm n xs =1 (if n = 0 then return (\<langle>\<rangle>, xs) else let m = n div 2 in bal_tm m xs \<bind> (\<lambda>(l, ys). bal_tm (n - 1 - m) (tl ys) \<bind> (\<lambda>(r, zs). return (\<langle>l, hd ys, r\<rangle>, zs))))"
less_numeral_extra_1: fixes one :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
servTicket_form: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "servK \<notin> range shrK \<and> servK \<in> symKeys \<and> (\<exists>A. servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>)"
state_to_strips_state_dom_is: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and s :: "'a \<Rightarrow> 'b option" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<phi>\<^sub>S \<Psi> s) = (\<Union>v\<in>{v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None}. {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
cod_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "cod (local.dom f) = local.dom f"
lubI: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "L \<in> pset cl"   and "\<forall>x\<in>S. (x, L) \<in> order cl"   and "\<And>z. \<lbrakk>z \<in> pset cl; \<forall>y\<in>S. (y, z) \<in> order cl\<rbrakk> \<Longrightarrow> (L, z) \<in> order cl" shows "L = lub S cl"
comp_converter_assoc: fixes conv1 :: "('y, 'z, 'ae, 'af) converter"   and conv2 :: "('ae, 'af, 'ac, 'ad) converter"   and conv3 :: "('ac, 'ad, 'aa, 'ab) converter" shows "(conv1 \<odot> conv2) \<odot> conv3 = conv1 \<odot> conv2 \<odot> conv3"
zsgn_def: fixes i :: "Int.int" shows "sgn i = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
intros: fixes c :: "'a"   and T :: "'a list"   and c' :: "'a" assumes "trace c T c'"   and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'"   and "stack c' = stack c" shows "bal c T c'"
oper_type_def: fixes op :: "String.literal \<times> 'a OCL_Types.type \<times> (String.literal \<times> 'a OCL_Types.type \<times> param_dir) list \<times> 'a OCL_Types.type \<times> bool \<times> 'b option" shows "oper_type op \<equiv> let params = oper_out_params op in if length params = 0 then oper_result op else Tuple (fmap_of_list (map (\<lambda>p. (param_name p, param_type p)) (params @ [(STR ''result'', oper_result op, Out)])))"