hcju/isabelleps huggingface dataset:
gcd'_zero: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and gcd' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.GCD_ring times one plus zero minus uminus"   and "GCD_ring.is_gcd times gcd'" shows "(gcd' x y = zero) = (x = zero \<and> y = zero)"
m_weakly_prefers_def: fixes m :: "M"   and \<mu> :: "(M \<times> W) set" shows "Non_Strict.m_weakly_prefers m \<mu> = {w' \<in> Field (Pm m). \<forall>w\<in>{w. (m, w) \<in> \<mu>}. (w, w') \<in> Pm m}"
ltl_const_equiv_def: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C \<psi> \<equiv> \<phi> = \<psi> \<or> eval \<phi> = eval \<psi> \<and> eval \<psi> \<noteq> Maybe"
streamsp_streams_eq: fixes A :: "'d set" shows "streamsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> streams A)"
nat_zero_less_power_iff: fixes x :: "nat"   and n :: "nat" shows "(0 < x ^ n) = (0 < x \<or> n = 0)"
col124__nos: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and B :: "'a"   and A :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet P Q B" shows "\<not> Tarski_neutral_dimensionless.OS Bet P Q A B"
ESem_subst_same: fixes \<rho> :: "Vars.var \<Rightarrow> 'a"   and x :: "Vars.var"   and y :: "Vars.var"   and Fn :: "('a \<rightarrow> 'a) \<rightarrow> 'a"   and Fn_project :: "'a \<rightarrow> 'a \<rightarrow> 'a"   and B :: "bool discr \<rightarrow> 'a"   and B_project :: "'a \<rightarrow> 'a \<rightarrow> 'a \<rightarrow> 'a"   and tick :: "'a \<rightarrow> 'a"   and e :: "exp" assumes "\<rho> x = \<rho> y" shows "semantic_domain.ESem Fn Fn_project B B_project tick e\<cdot>\<rho> = semantic_domain.ESem Fn Fn_project B B_project tick e[x::=y]\<cdot>\<rho>"
PUTListGD_def: fixes PUT :: "'c \<Rightarrow> 'd decision option"   and x :: "'c"   and p :: "('c \<Rightarrow> 'd decision option) list" shows "PUTListGD PUT x p = ((x \<notin> gatherDomain p \<and> x \<in> dom (last p) \<longrightarrow> PUT x = last p x) \<and> PUTList PUT x (butlast p))"
weak_inf_empty: shows "\<^bold>\<Sqinter>{} .=\<^bsub>thy_order\<^esub> \<^bold>\<top>"
map_spmf_conv_bind_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
2: shows "AE x in lborel. incseq (\<lambda>n. indicat_real {l n..u n} x *\<^sub>R f x)"
HFail_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2a s"   and "HFail s s' p" shows "Inv2a s'"
tuple_in_hom: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "\<guillemotleft>\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> : C.dom h \<rightarrow>\<^sub>C f \<down>\<down> g\<guillemotright>"
inv_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
I_def: fixes insert_message :: "('a, 'b) insert_message" shows "I insert_message = (case insert_message of InsertMessage x1 x2 x3 x4 \<Rightarrow> x2)"
onl_invariant_sterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('i \<times> ('f, 'g, 'e, 'h) seqp, 'j) automaton"   and I :: "'j \<Rightarrow> bool"   and P :: "'i \<times> 'h \<Rightarrow> bool"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and p' :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<xi>, l)"
external_events_neq: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> \<E>"   and "b \<notin> Q" shows "a \<noteq> b"
evalOP_inv_1: fixes P :: "db"   and Q :: "db"   and v :: "db" assumes "DBApp P Q \<Down> v" obtains M :: "db" where "P \<Down> DBAbsN M__" and "M__<Q/0> \<Down> v"   | M :: "db" and q :: "db" where "P \<Down> DBAbsV M" and "Q \<Down> q" and "M<q/0> \<Down> v"
d_def: shows "d = nat \<bar>int_of_fbint x - int_of_fbint y\<bar> + 1"
union_less_iff: fixes i :: "V"   and j :: "V"   and k :: "V" assumes "Ord i"   and "Ord j" shows "(i \<squnion> j < k) = (i < k \<and> j < k)"
gfp_in_Fix: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "gfp f \<in> Fix f"
D: shows "LeftDerivation [\<SS>] D ((terminals p @ [X]) @ \<omega>)"
simps_2: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and p :: "'a process" shows "CONT sys (PROC p) = PROC (contOf p)"
fmsubset_divides: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and a :: "'c"   and b :: "'c" assumes "comm_monoid_cancel G"   and "fmset G as \<subseteq># fmset G bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a divides\<^bsub>G\<^esub> b"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) ((\<otimes>) \<I>) local.map \<ll>"
sd_D_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sd_D x y \<equiv> D (fst x) (fst y) \<and> Dla (fst x) (snd y) \<and> D (snd x) (\<alpha> (fst x) (snd y))"
hyps_2: fixes u' :: "Syntax.term" assumes "\<pi>\<^sub>2 = []"   and "\<lless>[], e\<^sub>1\<ggreater> I\<rightarrow>i \<lless>[], u'\<ggreater>" shows "e\<^sub>2 = u'"
P_H2H2: shows "(P_H2, H2) = reduce_column div_op H"
1: shows "F \<a>\<^sub>C[f, g, h] = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
uminus_nonneg_Reals_iff: fixes x :: "'a" shows "(- x \<in> \<real>\<^sub>\<ge>\<^sub>0) = (x \<in> \<real>\<^sub>\<le>\<^sub>0)"
strict_trans2: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry"   and c :: "'b DBMEntry" assumes "b \<prec> a"   and "c \<preceq> b" shows "c \<prec> a"
supp_of_atom_list: fixes as :: "atom list" shows "supp as = set as"
vec_eq_iff: fixes x :: "('e, 'f) vec"   and y :: "('e, 'f) vec" shows "(x = y) = (\<forall>i. x $ i = y $ i)"
dagger_of_ket_is_bra: fixes v :: "complex Matrix.vec" shows "|v\<rangle>\<^sup>\<dagger> = \<langle>|v\<rangle>|"
o_inv_o_cancel: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "inj f" shows "g \<circ> inv f \<circ> f = g"
times_bit1_def: fixes x :: "'b bit1"   and y :: "'b bit1" shows "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
subStepB: shows "(!p \<cdot> l + y + p \<cdot> r + !p \<cdot> r)\<^sup>\<infinity> = (!p \<cdot> l + y + r \<cdot> p + r \<cdot> !p)\<^sup>\<infinity>"
nodes_of_graph_from_list: fixes l :: "('a \<times> 'a) list" shows "nodes (graph_from_list l) = fst ` set l \<union> snd ` set l"
comp_char: fixes g :: "'a"   and f :: "'a" shows "Left.comp g f = (if Left.arr f \<and> Left.arr g \<and> seq g f then g \<cdot> f else null)"
nodes_of_phis'_correct: fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_phis (uninst_code.ssa.phiNodes_of (p_g g s1_phis) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)"   and "uninst_code.ssa.trivial (u_g g s1_uses) (p_g g s1_phis) g (snd next)" shows "uninst_code.phi_equiv_mapping (p_g g s1_phis) g (uninst_code.nodes_of_phis' g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_phis) (uninst_code.ssa.phiNodes_of (\<lambda>gb. uninst_code.phis'_codem (p_g g s1_phis) g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_phis) g)"
lookup_poly_eval_focus: fixes a :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'c \<Rightarrow>\<^sub>0 nat" shows "lookup (poly_eval (\<lambda>x. monomial (a x) 0) (focus X p)) t = poly_eval a (lookup (focus (- X) p) t)"
dd1: fixes l :: "'a" assumes "a + l \<in> affine hull S"   and "l \<noteq> (0::'a)" shows "0 < dd l \<and> a + dd l *\<^sub>R l \<in> rel_frontier S"
inner2: shows "(\<chi>i. (if i \<in> ODE_vars I (OUrename x y ODE) then c else a) $ i, \<chi>i. (if i \<in> ODE_vars I (OUrename x y ODE) then ODE_sem I (OUrename x y ODE) c else b) $ i) = (\<chi>i. (if swap x y i \<in> ODE_vars I ODE then c else a) $ i, \<chi>i. (if swap x y i \<in> ODE_vars I ODE then ODE_sem I (OUrename x y ODE) c else b) $ i)"
LTL_nxt_2_means_vend: fixes i :: "(String.literal \<times> value list) stream" shows "alw (\<lambda>xs. nxt (state_eq (Some 2)) xs \<longrightarrow> state_eq (Some 1) xs) (watch drinks i)"
Cfun_app_eqvt: fixes \<pi> :: "perm"   and f :: "'e \<rightarrow> 'd"   and x :: "'e" shows "\<pi> \<bullet> f\<cdot>x = (\<pi> \<bullet> f)\<cdot>(\<pi> \<bullet> x)"
fset_of_list_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "fset_of_list (map f xs) = f |`| fset_of_list xs"
Send: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S"   and "ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
resr_unit: fixes x :: "'a" shows "(1::'a) \<rightarrow> x = x"
finiteV: fixes G :: "'c graph" assumes "wf_graph G" shows "finite (nodes G)"
iprev_neq_imp_greater: fixes n :: "nat"   and I :: "nat set" assumes "n \<noteq> iprev n I" shows "iprev n I < n"
subset_refl: fixes A :: "'f set" shows "A \<subseteq> A"
worklist_tree_aux_unfold: fixes succs :: "'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and ws :: "'c list"   and s :: "'d" shows "worklist_tree_aux succs f (ws, s) = (case ws of [] \<Rightarrow> Some ([], s) | x # ws' \<Rightarrow> worklist_tree_aux succs f (succs x @ ws', f x s))"
separated_tmps_nth_update: fixes i :: "nat"   and t :: "nat"   and s :: "stmt"   and sb :: "(stmt \<times> nat) memref list"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "valid_sops_stmt t s"   and "\<forall>(s', t')\<in>prog_configs sb. valid_sops_stmt t' s'"   and "\<forall>i\<in>load_tmps is. i < t"   and "\<forall>i\<in>read_tmps sb. i < t"   and "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"   and "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"   and "dom \<theta> \<union> load_tmps is = {i. i < t}" shows "separated_tmps (ts[i := ((s, t), is, \<theta>, sb, \<D>, \<O>)])"
1: shows "0 \<le> (x * y + sqrt ((x\<^sup>2 - 1) * (y\<^sup>2 - 1)))\<^sup>2 - 1"
mapS_defined: fixes a :: "('b, 'c) Stream"   and f :: "'b \<rightarrow> 'd" assumes "a \<noteq> \<bottom>" shows "mapS\<cdot>f\<cdot>a \<noteq> \<bottom>"
pp_arc_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.pp_arc f" shows "\<exists>e. f e \<noteq> bot \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)"
closed_scaleR_image_iff: fixes r :: "real"   and f :: "'c \<Rightarrow> 'b"   and R :: "'c set" shows "closed ((\<lambda>x. r *\<^sub>R f x) ` R) = (r = 0 \<or> closed (f ` R))"
resPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and x :: "name" assumes "P \<leadsto>[Rel] Q"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (\<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'" shows "\<lparr>\<nu>x\<rparr>P \<leadsto>[Rel'] \<lparr>\<nu>x\<rparr>Q"
intros_4: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<in> i_set_ind"
isidx_def: fixes xs :: "nat list" shows "isidx xs = to_nat (Some xs)"
Suc: shows "Abs_freelist (rev (alternating_list m s t) @ alternating_list m s t) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
lm062: fixes P :: "('d \<times> 'e) set" shows "inj_on fst P = inj_on snd (P\<inverse>)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C"   and "category_with_finite_products_axioms C" shows "category_with_finite_products C"
perm_empty: fixes xs :: "'a list" shows "([] <~~> xs) = (xs = [])"
unit_disc_fix_preserve_poincare_collinear: fixes M :: "moebius"   and A :: "complex_homo set" assumes "unit_circle_fix M"   and "poincare_collinear A" shows "poincare_collinear (moebius_pt M ` A)"
finite_char_subset: fixes C :: "'a set set" assumes "subset_closed C" shows "C \<subseteq> mk_finite_char C"
prime_elem_mult_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and p :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "comm_semiring_1.prime_elem times one zero (times a p) = comm_semiring_1.prime_elem times one zero p"
natural_isomorphism_\<ll>: shows "natural_isomorphism Left_a.comp Left_a.comp Left_a.L Left_a.map Left_a.\<ll>"
push_down_configuration_token_run: fixes x :: "nat"   and q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and y :: "nat" assumes "x \<in> \<UU>.configuration q n"   and "y \<in> \<UU>.configuration q n" shows "x \<le> n \<and> y \<le> n \<and> \<UU>.token_run x n = \<UU>.token_run y n"
\<phi>_def: shows "\<phi> \<equiv> \<phi>J \<psi>"
steps'_rests: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list"   and r1 :: "'c list"   and r2 :: "(nat \<times> nat list) list" assumes "length qs = length as"   and "n \<le> length as" shows "steps' s qs as n = steps' s (qs @ r1) (as @ r2) n"
simps_5: fixes P :: "pol"   and i :: "nat"   and Q :: "pol"   and c :: "int" shows "PX P i Q \<langle>*\<rangle> Pc c = (if c = 0 then Pc 0 else mkPX (P \<langle>*\<rangle> Pc c) i (Q \<langle>*\<rangle> Pc c))"
marked_child_ll: shows "\<forall>n\<in>set_of (Node lt p rt). if mark n = mark p then n \<in> set (ll ! var n) \<and> (\<forall>nt pa. Dag n low high nt \<and> pa \<in> set_of nt \<longrightarrow> mark pa = mark p) else n \<notin> set (concat ll)"
sorted_snoc: fixes xs :: "'b list"   and y :: "'b" assumes "sorted (xs @ [y])" shows "sorted xs"
PreValid_def: fixes A :: "'b \<Rightarrow> bool" shows "|~ A \<equiv> \<forall>w. A w"
Adef: shows "A = four_block_mat B (0\<^sub>m n m) (0\<^sub>m m n) D"
t_contains_open: fixes x :: "'a" assumes "x \<in> t" shows "\<exists>t'. t' open \<and> x \<in> t' \<and> t' \<subseteq> t"
push_bit_minus_one_eq_not_mask: 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 n :: "nat" 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 "push_bit n (uminus one) = not (mask n)"
prems: fixes x :: "'b" assumes "x \<in> insert a A" shows "\<exists>a. P x a"
assms_2: shows "(G ===> (=)) \<alpha>e \<alpha>e2"
3: shows "case vo of None \<Rightarrow> pending s `` {u} = {} | Some v \<Rightarrow> v \<in> pending s `` {u}"
cycle_decomp: fixes u :: "'d list" assumes "u \<noteq> []" shows "cycle u = u @- cycle u"
cfun_eqI: fixes f :: "'e \<rightarrow> 'f"   and g :: "'e \<rightarrow> 'f" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
a': shows "( \<Gamma> \<oplus> Compound F Fs \<Rightarrow>* \<Delta>, n) \<in> derivable R*"
FGModHom_composite_left: 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 W :: "'d set"   and smult'' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'d \<Rightarrow> 'e" assumes "FGModuleHom G smult V smult' T"   and "FGModuleHom G smult' W smult'' S"   and "T ` V \<subseteq> W" shows "FGModuleHom G smult V smult'' (S \<circ> T)"
tyenv_eq_sym: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "tyenv_wellformed mds \<Gamma> \<S> P"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" shows "mem\<^sub>2 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>1"
nx: shows "(0::'a) < (n / x + x) / (2::'a)"
succ_is_subset_of_verts: shows "\<forall>v\<in>set (g_V G). succs v \<subseteq> set (g_V G)"
strict_order_iff: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subset># b) = (a = a \<inter># b \<and> a \<noteq> b)"
closed_glob_consistent_def: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>) \<in> \<R> \<longrightarrow> (\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v\<^sub>1)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v\<^sub>2)\<rangle>) \<in> \<R>)) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v)\<rangle>) \<in> \<R>))))"
span_Diff_zero: fixes B :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "phull (B - {0}) = phull B"
A_updates_vars: shows "\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x"
LSC_Resolution: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n" shows "\<Union> (cnf ` nnf ` set_mset \<Gamma>) \<turnstile> \<box>"
S_def: shows "S = {k. [[a\<^sub>1 f k b]] \<and> k < card Y}"
bconfs_def: fixes P :: "'a prog"   and sh :: "sheap"   and es :: "'b exp list"   and b :: "bool" shows "P,sh \<turnstile>\<^sub>b (es,b) \<surd> \<equiv> b \<longrightarrow> (\<exists>C. icheck P C (the (ss_exps es)) \<and> curr_inits P es = \<lfloor>C\<rfloor> \<and> initPD sh C)"
smallest_prime_beyond_def: fixes n :: "nat" shows "smallest_prime_beyond n = (LEAST p. prime p \<and> n \<le> p)"
8: fixes x :: "'a"   and y :: "'a" shows "\<not> x \<le> y \<or> x \<squnion> y = y"
Iff_MP_left: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A IFF B"   and "insert A H \<turnstile> C" shows "insert B H \<turnstile> C"
domain_and_range_permutation_def: fixes A :: "'c set"   and B :: "'d set" shows "domain_and_range_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p\<^sub>A p\<^sub>B. p\<^sub>A permutes A \<and> p\<^sub>B permutes B \<and> (\<forall>x\<in>A. f x = p\<^sub>B (f' (p\<^sub>A x))))}"
map2_elemE2: fixes xs :: "'a list"   and ys :: "'b list"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "length xs = length ys"   and "y \<in> set ys" obtains x :: "'a" where "x \<in> set xs" and "f x y \<in> set (Term_Utils.map2 f xs ys)"
inv_imps_work_sum_zcount: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.inv_imps_work_sum summary results_in c = dataflow_topology.inv_imps_work_sum_zcount summary results_in c"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "\<And>v. P (NOT (Le v))"   and "\<And>v. P (NOT (Lt v))"   and "\<And>v. P (NOT (Eq v))"   and "\<And>v. P (NOT (NEq v))"   and "\<And>v va. P (NOT (And v va))"   and "\<And>v va. P (NOT (Or v va))"   and "\<And>v va. P (NOT (Imp v va))"   and "\<And>v va. P (NOT (Iff v va))"   and "\<And>v. P (NOT (E v))"   and "\<And>v. P (NOT (A v))"   and "P T"   and "P F"   and "\<And>t. P (Lt t)"   and "\<And>t. P (Le t)"   and "\<And>t. P (Eq t)"   and "\<And>t. P (NEq t)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Or v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
list_eq2_refl: fixes xs :: "'b list" shows "xs \<approx>2 xs"
typ_ok_Tv: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and S :: "String.literal set"   and a :: "variable" assumes "wf_sort (subclass (osig \<Sigma>)) S" shows "wf_type \<Sigma> (Tv a S)"
strict_prefix_minus_not_empty: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys" shows "ys - xs \<noteq> []"
pR: shows "set (coeffs p) \<subseteq> \<real>"
part_equivpI: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<exists>x. R x x"   and "symp R"   and "transp R" shows "part_equivp R"
connected_root_aux: fixes x :: "'a"   and r :: "'a" shows "x\<^sup>T\<^sup>+ ; r \<le> x\<^sup>T ; 1"
bigo_const_mult1: fixes c :: "'d"   and f :: "'c \<Rightarrow> 'd" shows "(\<lambda>x. c * f x) \<in> O(f)"
r_distr_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a"   and r'' :: "'a"   and s'' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel"   and "(r'', s'') \<in> carrier rel" shows "((r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s')) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'')"
ex_witness_list: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and q :: "('b, 'c) fm"   and used :: "'c set" assumes "p \<in>. ps"   and "proper_dia p = Some q" shows "\<exists>i. {\<^bold>@ i q, \<^bold>\<diamond> Nom i} \<subseteq> set (witness_list ps used)"
k_adju: fixes y :: "'d" shows "\<exists>F. \<forall>x. F \<stileturn> (\<lambda>k. k y)"
vector_smult_lid: fixes x :: "('d, 'e) vec" shows "(1::'d) *s x = x"
monoid_axioms: shows "monoid (inverse ` H) (\<cdot>) \<one>"
vars_term_subset_subst_eq: fixes t :: "('b, 'a) Term.term"   and s :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "fv t \<subseteq> fv s"   and "s \<cdot> \<sigma> = s \<cdot> \<tau>" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
orbits: shows "pmf (sds R28) b = pmf (sds R28) c"