hcju/isabelleps huggingface dataset:
intros_7: fixes c :: "('d, 'e, 'f) com"   and f :: "'f"   and g :: "'d set" shows "com_relp c (Guard f g c)"
size_fmap_overloaded_def: shows "size = size_fmap (\<lambda>_. 0) (\<lambda>_. 0)"
simps_1: fixes p :: "'a" shows "Dioid_Models.last (p, []) = p"
Node_2: fixes i :: "int"   and j :: "int" assumes "inorder r = [i..j]" shows "wpl i j r = Wpl i r"
hyps: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "set vs \<subseteq> set (fst (sym_preproc_addnew gs vs fs v))"
to_mset_def: fixes t :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.to_mset t \<equiv> BsSkewBinomialHeapStruc.bs_to_mset (Rep_SkewBinomialHeap t)"
simps_1: shows "invc \<langle>\<rangle> = True"
fin: fixes C :: "'a set set" assumes "C \<subseteq> \<B>"   and "U \<subseteq> \<Union> C" shows "\<exists>C'. finite C' \<and> C' \<subseteq> C \<and> U \<subseteq> \<Union> C'"
exec_append': fixes xs :: "('d, 'c) instr list"   and ys :: "('d, 'c) instr list"   and stack :: "'c list"   and env :: "'d \<Rightarrow> 'c" shows "exec (xs @ ys) stack env = exec ys (exec xs stack env) env"
chamber_vertices: fixes x :: "'a set" assumes "domain.chamber x" shows "order.greater_eq (\<Union> X) x"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and v :: "tm"   and i :: "tm"   and sl :: "name"   and sl' :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name"   and t :: "tm"   and u :: "tm" assumes "atom l \<sharp> (s, k, v, i, sl, sl', m, n, sm, sm', sn, sn')"   and "atom sl \<sharp> (s, v, i, sl', m, n, sm, sm', sn, sn')"   and "atom sl' \<sharp> (s, v, i, m, n, sm, sm', sn, sn')"   and "atom m \<sharp> (s, n, sm, sm', sn, sn')"   and "atom n \<sharp> (s, sm, sm', sn, sn')"   and "atom sm \<sharp> (s, sm', sn, sn')"   and "atom sm' \<sharp> (s, sn, sn')"   and "atom sn \<sharp> (s, sn')"   and "atom sn' \<sharp> s" shows "SeqStTermP v i t u s k = VarP v AND LstSeqP s k (HPair t u) AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND ((Var sl EQ v AND Var sl' EQ i OR (IndP (Var sl) OR Var sl NEQ v) AND Var sl' EQ Var sl) OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ Q_Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
val_ord_Zp: fixes p :: "int"   and a :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>" shows "padic_integers.val_Zp p a = eint (padic_integers.ord_Zp p a)"
assms_1: shows "\<forall>v\<in>A. card {n. R v n} = k"
map_spmf_eq_map_spmf_iff: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and g :: "'e \<Rightarrow> 'c"   and q :: "'e spmf" shows "(map_spmf f p = map_spmf g q) = rel_spmf (\<lambda>x y. f x = g y) p q"
integrable_pmf: fixes X :: "'b set"   and M :: "'b pmf" shows "integrable (count_space X) (pmf M)"
simps_8: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a acom"   and vf :: "'a" shows "(v ::= va {vb} \<le> IF vc THEN vd ELSE ve {vf}) = False"
inv_atD: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat"   and i :: "nat" assumes "local.inv_at p A m"   and "i < n" shows "p A i m"
is_functor: shows "functor \<F>C.CC.comp D.CC.comp FF.map"
lead_coeff_in_carrier: fixes K :: "'a list set"   and a :: "'a list"   and p :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K (a # p)" shows "a \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}"
InitConf_def: fixes A :: "('d, 'e, 'f) hierauto" shows "InitConf A \<equiv> (HAInitStates A \<times> HAInitStates A \<inter> ChiRel A)\<^sup>* `` {HAInitState A}"
assms_2: fixes b :: "'a set" assumes "b \<in># \<B>" shows "b \<noteq> {}"
scaleR_minus1_left: fixes x :: "'c" shows "- 1 *\<^sub>R x = - x"
prime_eq_prime_mult: fixes p :: "'a" assumes "p \<in> carrier R" shows "prime R p = prime (Ring_Divisibility.mult_of R) p"
override_on_apply_in: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "a \<in> A" shows "(f ++\<^bsub>A\<^esub> g) a = g a"
mset_factors_exist: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" 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"
prems_6: shows "is_zero_row_upt_k i k A"
f_expand_0: fixes xs :: "'b message_af list" shows "xs \<odot> 0 = []"
has_integral_substitution_general_': fixes s :: "real set"   and a :: "real"   and b :: "real"   and s' :: "real set"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real"   and f :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> real" assumes "finite s"   and "a \<le> b"   and "finite s'"   and "g ` {a..b} \<subseteq> {c..d}"   and "f integrable_on {c..d}"   and "continuous_on ({c..d} - s') f"   and "continuous_on {a..b} g"   and "\<forall>x\<in>s'. finite (g -` {x})"   and "surj_on s' g"   and "inj_on g ({a..b} \<union> (s \<union> g -` s'))"   and "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})" shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral integral {g a..g b} f - integral {g b..g a} f) {a..b}"
is_extensional: fixes f :: "'a" assumes "\<not> A.arr f" shows "\<sigma> f = \<sigma>.B.null"
assms: shows "\<C> \<turnstile> [] : ts _> ts'"
get_root_node_same_no_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r 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" shows "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r None"
f: fixes x :: "nat" assumes "x \<notin> range g" shows "f x = 0"
heap_axioms: shows "heap addr2thread_id thread_id2addr allocate typeof_addr heap_write P"
inject_9: fixes x101 :: "stmt"   and x102 :: "stmt"   and y101 :: "stmt"   and y102 :: "stmt" shows "(x101 Finally x102 = y101 Finally y102) = (x101 = y101 \<and> x102 = y102)"
x_zero_meet_closed: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<sqinter> y \<cdot> (0::'a) = (x \<sqinter> y) \<cdot> (0::'a)"
reachable_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<in> verts G"
hcomp_in_hhom': fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and \<nu> :: "'c"   and a :: "'c"   and c :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "src \<mu> = a"   and "trg \<nu> = c"   and "src \<nu> = trg \<mu>" shows "horizontal_homs.in_hhom V src trg (H \<nu> \<mu>) a c"
prime_power_inj'_2: 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 p :: "'b"   and q :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "power.power one times p m = power.power one times q n"   and "0 < m"   and "0 < n" shows "m = n"
eq: fixes m :: "nat" assumes "0 < m" shows "(\<Sum>x\<in>{0<..m}. 1 / of_nat x ^ n) = contour_integral (\<gamma> m) g * (complex_of_real (2 * pi) * \<i>) ^ n / (complex_of_real (4 * pi) * \<i>) - complex_of_real (bernoulli n / fact n) * (complex_of_real (2 * pi) * \<i>) ^ n / 2"
Equiv: fixes A :: "'t"   and e1 :: "('t, 'f) Expression"   and e2 :: "('t, 'f) Expression"   and B :: "'t" assumes "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk> \<rightarrow> IBool True" shows "\<exists>g. L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk> \<rightarrow> IMor g \<and> L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk> \<rightarrow> IMor g"
ortho: fixes w :: "'a vec"   and x :: "'a vec" assumes "w \<in> W"   and "x \<in> set Ls" shows "w \<bullet> x = (0::'a)"
pdiv_pmod: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p = q \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> UP.pdiv p q \<oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> UP.pmod p q"
next_events_def: fixes P :: "'a process"   and xs :: "'a list" shows "next_events P xs \<equiv> {x. xs @ [x] \<in> traces P}"
simps_13: fixes v :: "\<I>"   and va :: "Formula.formula" shows "remove_neg (formula.Next v va) = formula.Next v va"
lrev_work_final_def: shows "lrev_work_final \<equiv> fix\<cdot>lrev_body_final"
\<delta>0_def: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" shows "\<delta>0 qd1 qd2 \<equiv> let c1 = qdfst qd1; c2 = qdfst qd2; k1 = qdsnd qd1; k2 = qdsnd qd2 in if c1 < c2 \<and> k2 < k1 then (c2 - c1) / (k1 - k2) else 1"
h2: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A"   and "\<not> i < dim_row (take_rows A [0..<length (pivot_positions A)])" shows "0\<^sub>m (dim_row A - length (pivot_positions A)) (dim_col A) $$ (i - dim_row (take_rows A [0..<length (pivot_positions A)]), j) = A $$ (i, j)"
bigo_plus_absorb: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "f +o O(g) = O(g)"
verit_la_disequality: fixes a :: "'q"   and b :: "'q" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
dagger_of_ket_is_bra: fixes v :: "complex vec" shows "|v\<rangle>\<^sup>\<dagger> = \<langle>|v\<rangle>|"
RG_n0: shows "ActingGroup.group_ring \<noteq> 0"
sum_equal: fixes N :: "nat" assumes "0 < N" shows "(\<Sum>n<N. 1 / real N) = 1"
assms_7: shows "a \<noteq> 0\<^sub>h"
rotate_downpos_def: fixes x :: "wall"   and y :: "wall" shows "rotate_downpos x y \<equiv> x = basic [cup, vert] \<circ> basic [vert, over] \<and> y = basic [vert, cup] \<circ> basic [brick.under, vert]"
step: fixes s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "s \<in> reachable A I"   and "P s"   and "Q (s, a, s')"   and "I a" shows "P s'"
open_scaling: fixes c :: "real"   and s :: "'a set" assumes "c \<noteq> 0"   and "open s" shows "open ((*\<^sub>R) c ` s)"
1: shows "\<forall>i<length (annos C2). m\<^sub>o (annos C2 ! i) (vars (strip C2)) \<le> m\<^sub>o (annos C1 ! i) (vars (strip C2))"
assms_2: shows "ps (S t i) p \<noteq> None"
execute_parallel_operator_sas_plus_cons: fixes s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator"   and ops :: "('c, 'd) sas_plus_operator list" shows "execute_parallel_operator_sas_plus s (op # ops) = execute_parallel_operator_sas_plus (s ++ map_of (effect_of op)) ops"
dom_combination_dom_union: fixes \<tau> :: "'d \<Rightarrow> 'e option"   and \<tau>' :: "'d \<Rightarrow> 'e option" shows "dom (\<tau>;;\<tau>') = dom \<tau> \<union> dom \<tau>'"
lebesgue_integral_count_space_finite_support: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite {a \<in> A. f a \<noteq> (0::'c)}" shows "integral\<^sup>L (count_space A) f = (\<Sum>a | a \<in> A \<and> f a \<noteq> (0::'c). f a)"
env_restr_HSem: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "domA \<Gamma> \<inter> S = {}" shows "(\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>) f|` S = \<rho> f|` S"
identity\<^sub>3_def: shows "(\<^bold>=) \<equiv> \<lambda>F G. \<^bold>\<forall>x y. (\<^bold>\<lambda>z. \<lparr>F,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>)"
induct: fixes P :: "('c, 'd) strand_step list \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step list" assumes "P []"   and "\<And>t t' S. P S \<Longrightarrow> P (\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (receive\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>va vb S. P S \<Longrightarrow> P (\<langle>check: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va S. P S \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0"
mat_delete_submatrix_insert: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and I :: "nat set"   and k :: "nat"   and J :: "nat set"   and a :: "nat"   and b :: "nat"   and a' :: "nat"   and b' :: "nat" assumes "A \<in> carrier_mat n m"   and "card I = k"   and "card J = k"   and "I \<subseteq> {0..<n}"   and "J \<subseteq> {0..<m}"   and "a < n"   and "b < m"   and "k < min n m"   and "a \<notin> I"   and "b \<notin> J"   and "a' < Suc k"   and "b' < Suc k"   and "pick (insert a I) a' = a"   and "pick (insert b J) b' = b" shows "mat_delete (submatrix A (insert a I) (insert b J)) a' b' = submatrix A I J"
create_reservation_restrict_union: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic"   and v :: "view" shows "ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts' \<longrightarrow> restrict v (res ts') c = restrict v (res ts) c \<squnion> restrict v (clm ts) c"
strictly_generalizes_atm_def: fixes A :: "'a"   and B :: "'a" shows "strictly_generalizes_atm A B = (generalizes_atm A B \<and> \<not> generalizes_atm B A)"
otherassms_6: shows "(\<sigma>, pp) \<in> oreachable A S U"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) Term.term list \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Var x1 of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = f1 x1"
n: shows "0 < prob {\<omega> \<in> space M. X n \<omega> = s}"
signed_take_bit_signed_take_bit: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (min m n) a"
get_host_reads: fixes element_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads get_host_locs (get_host element_ptr) h h'"
assms: shows "D.is_rendered_commutative_by p0' p1'"
neq_Nil_conv: fixes xs :: "'k list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
a1: shows "s' = snd (fst (save_retore_sub1 result new_cwp rd s)) \<and> get_S (cpu_reg_val PSR s) = 0"
cod_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.cod C (C g f) = partial_magma.cod C g"
maybe_and_true: fixes x :: "trilean"   and y :: "trilean" shows "(x \<and>? y = true) = (x = true \<and> y = true)"
GroupWithGeneratorsRelators_S_R: shows "GroupWithGeneratorsRelators S R"
infinite_rangeIt: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "inj f"   and "\<forall>y. x \<noteq> f y" shows "infinite (rangeIt x f)"
ipurge_fail_aux_t_invariance_aux: fixes Z :: "('b, 'c) ipurge_rec"   and Y :: "('b, 'c) ipurge_rec" assumes "Z \<in> ipurge_fail_aux_t_set Y" shows "Pol Z = Pol Y \<and> Map Z = Map Y \<and> Doms Z = Doms Y"
weakly_precedes_sub: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c\<^sub>1 weakly precedes c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c\<^sub>1 weakly precedes c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
thin_Un: fixes H :: "fm set"   and A :: "fm"   and H' :: "fm set" assumes "H \<turnstile> A" shows "H \<union> H' \<turnstile> A"
ordered_term_axioms: shows "ordered_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
gtr_def: fixes gtrs :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" shows "gtr gtrs \<equiv> {(add_mset s c, e, add_mset s' c') |s c e s' c'. ((s, c), e, s', c') \<in> gtrs}"
plus_uexpr_def: fixes u :: "('c, 'd) uexpr"   and v :: "('c, 'd) uexpr" shows "u + v = bop (+) u v"
weakenL: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset"   and F :: "'a formula" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "F, \<Gamma> \<Rightarrow> \<Delta>"
cr_trm_def: shows "cr_trm \<equiv> \<lambda>x. (=) (abs_trm x)"
Un_mono: fixes A :: "'x set"   and C :: "'x set"   and B :: "'x set"   and D :: "'x set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
has_typ1_tsubst: fixes Ts :: "typ list"   and t :: "Core.term"   and T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "Ts \<turnstile>\<^sub>\<tau> t : T" shows "map (\<lambda>T. tsubstT T \<rho>) Ts \<turnstile>\<^sub>\<tau> tsubst t \<rho> : tsubstT T \<rho>"
h2: shows "h \<turnstile> (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent node_ptr) \<rightarrow>\<^sub>h h2"
simps: fixes a :: "exp list" shows "vals(a) = (a = [] \<or> (\<exists>vh vt. a = vh # vt \<and> val(vh) \<and> vals(vt)))"
rquot_DFS_axioms: shows "rquot_DFS A n"
univ_second_countable: obtains \<B> :: "'d set set" where "countable \<B>" and "\<And>C. C \<in> \<B> \<Longrightarrow> open C" and "\<And>S. open S \<Longrightarrow> \<exists>U\<subseteq>\<B>. S = \<Union> U"
Zp_converges_to_def: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.Zp_converges_to p s a = ((a \<in> carrier (padic_int p) \<and> s \<in> carrier padic_int p\<^bsup>\<omega>\<^esup>) \<and> (\<forall>n. \<exists>k. \<forall>m>k. eint n < padic_integers.val_Zp p (s m \<ominus>\<^bsub>padic_int p\<^esub> a)))"
IH_4: fixes as :: "accent list"   and bs :: "accent list" assumes "d # ds = as @ ([Acute] @ [Macron]) @ bs" shows "thesis"
STA_Dup: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and q :: "('c, 'b) fm"   and i :: "'b"   and xs :: "(nat \<times> nat) set" assumes "A, n \<turnstile> branch"   and "Dup q i branch xs" shows "A, n \<turnstile> omit_branch xs branch"
finishedL_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "finishedL cl \<equiv> \<forall>c\<in>set cl. finished c"
closure_halfspace_lt: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}"
r_prefixes_0: fixes j :: "nat" shows "eval r_prefixes [0, j] \<down>= singleton_encode j"
injw: shows "inj_on (f \<circ> inv_into A g) (g ` A)"
vars_plus: fixes p1 :: "linear_poly"   and p2 :: "linear_poly" shows "vars (p1 + p2) \<subseteq> vars p1 \<union> vars p2"
under_Under_trans: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "trans r"   and "a \<in> under r b"   and "b \<in> Under r C" shows "a \<in> Under r C"
\<gamma>_eq_trans: fixes x :: "real list"   and y :: "real list"   and z :: "real list" assumes "\<gamma>_eq (x, y)"   and "\<gamma>_eq (y, z)" shows "\<gamma>_eq (x, z)"
inf_sup_ord_3: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
out_def: fixes sp\<^sub>\<nu> :: "('b, 'c) sp\<^sub>\<nu>" shows "out sp\<^sub>\<nu> = (case sp\<^sub>\<nu> of In x \<Rightarrow> x)"
member_bind: fixes x :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "(x \<in> A \<bind> f) = (x \<in> \<Union> (f ` A))"
Lim_bounded2: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b"   and N :: "nat"   and C :: "'b" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>N. C \<le> f n" shows "C \<le> l"
length_vec_to_iarray: fixes xa :: "('c, 'd) vec" shows "IArray.length (vec_to_iarray xa) = CARD('d)"
a1: shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
transp_less: fixes ko :: "'c key_order" shows "transp (ko.lt ko)"
less_powr_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(x < b powr y) = (log b x < y)"
distinct_13: fixes x2 :: "'b"   and x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PVar x2 \<noteq> PApp x31 x32"
step: fixes x :: "name"   and xs :: "name list"   and P :: "pi" shows "resChain (x # xs) P = <\<nu>x>resChain xs P"
run_detE: fixes c :: "'a Heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'a"   and \<tau> :: "heap option"   and s :: "'a" assumes "run c \<sigma> \<sigma>' r"   and "run c \<sigma> \<tau> s"   and "\<not> is_exn \<sigma>" obtains "is_exn \<sigma>'" and "\<sigma>' = \<tau>"   | "\<not> is_exn \<sigma>'" and "\<sigma>' = \<tau>" and "r = s"
mins_subset: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "QI c_k c w" shows "Args_min_on (c_k i j) {i + 1..j} \<subseteq> {i + 1..j}"
subst_floatarith_Add: fixes s :: "nat \<Rightarrow> floatarith"   and fa :: "floatarith"   and x :: "floatarith"   and y :: "floatarith" assumes "subst_floatarith s fa = Add x y" obtains a :: "floatarith" and b :: "floatarith" where "fa = Add a b" and "x = subst_floatarith s a" and "y = subst_floatarith s b"   | n :: "nat" where "fa = Var n" and "Add x y = s n"
IInf_us_inv: shows "IInf \<circ> us = id"
coeff_inject: fixes x :: "'a poly"   and y :: "'a poly" shows "(coeff x = coeff y) = (x = y)"
antisym_conv3: fixes y :: "'t"   and x :: "'t" assumes "\<not> y \<prec>\<^sub>t x" shows "(\<not> x \<prec>\<^sub>t y) = (x = y)"
fBallE: fixes A :: "'e fset"   and P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"