hcju/isabelleps huggingface dataset:
unique_paths: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w" shows "\<exists>!xs. v \<leadsto>xs\<leadsto> w"
dy_fake_chan_mono_ik: fixes T :: "msg set"   and T' :: "msg set"   and bad :: "agent set"   and C :: "chan set" assumes "T \<subseteq> T'" shows "dy_fake_chan bad T C \<subseteq> dy_fake_chan bad T' C"
lt_nb: fixes t :: "tm" assumes "tmbound0 t" shows "bound0 (lt t)"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "D.arr (F f)"
distinct_map_eq: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list"   and x :: "'b"   and y :: "'b" assumes "distinct (map f l)"   and "f x = f y"   and "x \<in> set l"   and "y \<in> set l" shows "x = y"
prod_prime_factors: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "multiplicative_function f"   and "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f (p ^ multiplicity p n))"
r_search_def: shows "r_search \<equiv> Cn 1 r_enumdom [r_s, Mn 1 r_search_enum]"
normalized_last_not_None: fixes xs :: "('c, 'd) bintree option list" shows "normalized xs = (xs = [] \<or> last xs \<noteq> None)"
fold_set_fold: fixes y :: "'a"   and xs :: "'a list" shows "Finite_Set.fold (\<sqinter>) y (set xs) = fold (\<sqinter>) xs y"
homeomorphic_maps_sym: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphic_maps X Y f g = homeomorphic_maps Y X g f"
small_powers_of_2: fixes x :: "nat" assumes "3 \<le> x" shows "x < 2 ^ (x - 1)"
leq_h: shows "At = LeqUni (a, b, c) \<longrightarrow> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
disjCI: fixes Q :: "bool"   and P :: "bool" assumes "\<not> Q \<Longrightarrow> P" shows "P \<or> Q"
rotate2: fixes A2 :: "fmla"   and A1 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A2 (insert A1 H) \<turnstile> B" shows "insert A1 (insert A2 H) \<turnstile> B"
less_imp_diff_less: fixes j :: "nat"   and k :: "nat"   and n :: "nat" assumes "j < k" shows "j - n < k"
elims: fixes x :: "bool"   and xa :: "(nat \<times> nat) convnet"   and y :: "nat" assumes "count_weights x xa = y" obtains shared :: "bool" and M :: "nat" where "x = shared__" and "xa = Input M" and "y = 0"   | shared :: "bool" and r0 :: "nat" and r1 :: "nat" and m :: "(nat \<times> nat) convnet" where "x = shareda__" and "xa = Conv (r0, r1) m" and "y = r0 * r1 + count_weights shareda__ m"   | shared :: "bool" and m1 :: "(nat \<times> nat) convnet" and m2 :: "(nat \<times> nat) convnet" where "x = shared" and "xa = Pool m1 m2" and "y = (if shared then max (count_weights shared m1) (count_weights shared m2) else count_weights shared m1 + count_weights shared m2)"
Dynkin_system_axioms_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "Dynkin_system_axioms \<Omega> M \<equiv> \<Omega> \<in> M \<and> (\<forall>A. A \<in> M \<longrightarrow> \<Omega> - A \<in> M) \<and> (\<forall>A. disjoint_family A \<longrightarrow> range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M)"
init_imp_inv_implications_nonneg: fixes c :: "('b, 'c, 'd) Propagate.configuration_scheme" assumes "local.init_config c" shows "inv_implications_nonneg c"
inf_map_in_Inf: fixes \<iota> :: "'f inference" assumes "\<iota> \<in> Inf_F"   and "\<G>_I \<iota> \<noteq> None" shows "the (\<G>_I \<iota>) \<subseteq> Inf_G"
l_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))" shows "\<exists>y\<in>carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)). y \<otimes>\<^bsub>DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)\<^esub> x = \<one>\<^bsub>DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)\<^esub>"
eq: shows "opp p\<^sub>3 = ell_field_class.add a p\<^sub>1 p\<^sub>2"
iteratei_correct: fixes S :: "'s" assumes "invar S" shows "map_iterator (iteratei S) (\<alpha> S)"
bind_pmf_aux_code_aux: fixes A :: "'c set"   and p :: "'c pmf"   and f :: "'c \<Rightarrow> 'd pmf" assumes "finite A" shows "bind_pmf_aux p f A = fold_combine_plus (\<lambda>x. Mapping.map_values (\<lambda>_. (*) (pmf p x)) (mapping_of_pmf (f x))) A"
cases: fixes r :: "'b SecurityInvariant_configured" obtains c_sinvar :: "'b graph \<Rightarrow> bool" and c_offending_flows :: "'b graph \<Rightarrow> ('b \<times> 'b) set set" and c_isIFS :: "bool" where "r = \<lparr>c_sinvar = c_sinvar, c_offending_flows = c_offending_flows, c_isIFS = c_isIFS\<rparr>"
simps_2: fixes p :: "'b list \<Rightarrow> bool"   and a :: "'b"   and S :: "'b list" shows "\<box> p (a # S) = (p (a # S) \<and> \<box> p S)"
assms_3: shows "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)"
inverse_transformation_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" shows "inverse_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_isomorphism A B F G \<tau>"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (\<sigma> f) = G (A.cod f)"
open_dist: fixes S :: "'a set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
finite_char_closed: fixes C :: "'b set set" assumes "finite_char C" shows "subset_closed C"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (\<alpha> \<prec> P') A\<^sub>P \<Psi>\<^sub>P"   and "distinct (bn \<alpha>)"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<alpha> \<prec> P' \<parallel> Q) (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
swap_simp: fixes x :: "'c"   and y :: "'a" shows "prod.swap (x, y) = (y, x)"
antisym_classes1: fixes c :: "classes1"   and d :: "classes1" assumes "c \<le> d"   and "d \<le> c" shows "c = d"
snd_hyp_edge_at: fixes c :: "'form"   and "is" :: "nat list"   and n :: "nat"   and s :: "'subst" shows "snd (hyp_edge_at c is n s) = hyp_edge_to c is"
sing_in_sings: fixes x :: "'e"   and xd :: "'e set" shows "([x] \<in> (\<lambda>x. [x]) ` xd) = (x \<in> xd)"
trace_rest'_simps_3: fixes x1ca :: "('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme"   and x2ca :: "'f spmf"   and xa :: "'d"   and x :: "'e \<times> 'c list"   and xaa :: "('a \<times> 'b \<times> 'c list + 'd \<times> 'e \<times> 'c list) list" shows "trace_rest' x1ca x2ca (Inr (xa, x) # xaa) = trace_rest' x1ca (cond_spmf_fst (x2ca \<bind> (\<lambda>s. rfunc_usr x1ca s xa)) x) xaa"
not_outside_2: fixes x :: "'b"   and y :: "'b" assumes "R j x y" shows "y \<in> carrier"
non_Limit_ord_of_nat: fixes m :: "nat" shows "\<not> Limit (ord_of_nat m)"
C2: shows "C2 \<in> UNIV // K2.communicating"
2: shows "{} \<turnstile> (SyntaxN.Ex j (KRP \<guillemotleft>Var i\<guillemotright> (Var i) (Var j) AND \<alpha>(i::=Var j)))(i::=\<guillemotleft>\<beta>\<guillemotright>) IFF SyntaxN.Ex j (Var j EQ \<guillemotleft>\<beta>(i::=\<guillemotleft>\<beta>\<guillemotright>)\<guillemotright> AND \<alpha>(i::=Var j))"
necessaryPhi_def: fixes g :: "'g"   and v :: "'var"   and z :: "'node" shows "necessaryPhi g v z \<equiv> \<exists>n ns m ms. old.pathsConverge g n ns m ms z \<and> v \<in> oldDefs g n \<and> v \<in> oldDefs g m"
tree_rank_estimate: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) BinomialTree list" assumes "tree_invar (Node e a r ts)" shows "size (tree_to_multiset (Node e a r ts)) = 2 ^ r"
strong_impl_weak_reduction_simulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal" shows "weak_reduction_simulation Rel Cal"
select_convs_2: fixes variables_of :: "'a list"   and operators_of :: "('a, 'b) sas_plus_operator list"   and initial_of :: "'a \<Rightarrow> 'b option"   and goal_of :: "'a \<Rightarrow> 'b option"   and range_of :: "'a \<Rightarrow> 'b list option"   and more :: "'c" shows "\<lparr>sas_plus_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, range_of = range_of, \<dots> = more\<rparr>\<^sub>\<O>\<^sub>+ = operators_of"
dep3_event_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>" shows "MinkowskiPrimitive.dep3_event \<P> Q R S x \<equiv> Q \<noteq> R \<and> Q \<noteq> S \<and> R \<noteq> S \<and> Q \<in> MinkowskiPrimitive.SPRAY \<P> x \<and> R \<in> MinkowskiPrimitive.SPRAY \<P> x \<and> S \<in> MinkowskiPrimitive.SPRAY \<P> x \<and> (\<exists>T\<in>\<P>. T \<notin> MinkowskiPrimitive.SPRAY \<P> x \<and> (\<exists>y\<in>Q. y \<in> T) \<and> (\<exists>y\<in>R. y \<in> T) \<and> (\<exists>y\<in>S. y \<in> T))"
assms_2: shows "proj2_abs (i *\<^sub>R proj2_rep a + proj2_rep r) = proj2_abs (j *\<^sub>R proj2_rep a + proj2_rep r)"
fbox_segerberg: fixes y :: "'a"   and x :: "'a" shows "d y \<cdot>  |x\<^sup>\<star>] ( |x] y + ad y) =  |x\<^sup>\<star>] y"
inf_commute: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "inf_prod x y = inf_prod y x"
GatewaySystem_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and vc :: "nat \<Rightarrow> vcType list" shows "GatewaySystem req dt stop lose d ack vc \<equiv> \<exists>a i. Gateway req dt a stop lose d ack i vc \<and> ServiceCenter i a"
A4: shows "\<forall>q a b c. \<exists>x. B q a x \<and> a x \<equiv> b c"
c_tr_sources: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and step :: "'h \<Rightarrow> 'f \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'f \<Rightarrow> 'g"   and s :: "'h"   and xs :: "'f list" shows "c_sources I (c_dom D) u (c_tr step out s xs) = c_sources I D u xs"
rel_mset_Zero: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_mset R {#} {#}"
mod_less_imp_diff_div_conv: fixes n :: "nat"   and m :: "nat"   and r :: "nat" assumes "n mod m < r"   and "r \<le> m + n mod m" shows "(n - r) div m = n div m - Suc 0"
emeasure_pair_measure_alt: fixes X :: "('c \<times> 'a) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M M)" shows "emeasure (N \<Otimes>\<^sub>M M) X = \<integral>\<^sup>+ x. emeasure M (Pair x -` X) \<partial>N"
digraph_mapI: fixes M :: "'b pre_map" assumes "\<And>a. a \<notin> arcs G \<Longrightarrow> edge_rev M a = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> edge_rev M a \<noteq> a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> edge_rev M (edge_rev M a) = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> tail G (edge_rev M a) = head G a"   and "edge_succ M permutes arcs G"   and "\<And>v. \<lbrakk>v \<in> verts G; out_arcs G v \<noteq> {}\<rbrakk> \<Longrightarrow> cyclic_on (edge_succ M) (out_arcs G v)" shows "digraph_map G M"
sup_ge1: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
2: fixes kv :: "'a \<times> 'b" shows "set (update_by_pair kv xs) \<subseteq> insert kv (insert (k', v') (set xs))"
eq_iff_13: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Roll term = Roll terma) = (term = terma)"
H_exp1: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and q :: "'a" assumes "H (\<tt>\<tt> p \<cdot> \<tt>\<tt> r) x q" shows "H p (\<tt>\<tt> r \<cdot> x) q"
proj_onto_def: fixes a :: "'a"   and S :: "'a set" shows "proj_onto a S = sum (proj a) S"
fpxs_const_altdef: fixes x :: "'b" shows "fpxs_const x = fpxs_of_fls (fls_const x)"
arith_extra_simps_19: fixes a :: "'a" shows "a \<cdot> (0::'a) = (0::'a)"
goodInp_imp_qGoodInp_pickInp: fixes inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodInp inp" shows "qGoodInp (pickInp inp)"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "comm_group G"
simps_3: fixes e1 :: "expr"   and e2 :: "expr" shows "assignsV (e1.[e2]) = assignsE e1 \<union> assignsE e2"
assms_2: fixes m :: "nat" assumes "m < n"   and "0 < m" shows "(f ^^ m) s \<noteq> s"
Proj_on_own_range': fixes P :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "P o\<^sub>C\<^sub>L P = P"   and "P = P*" shows "Proj (P *\<^sub>S \<top>) = P"
assms_2: shows "a' \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
71989: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> y \<sqinter> (z \<squnion> top)) = - (x \<squnion> y)"
o_eq_elim: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'d \<Rightarrow> 'b"   and d :: "'a \<Rightarrow> 'd" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
vbrelation: fixes r :: "V" assumes "vbrelation r" shows "vpairs r = r"
id_aux1: fixes s :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "DenyAllFromTo (first_srcNet s) (first_destNet s) \<oplus> DenyAllFromTo (first_destNet s) (first_srcNet s) \<oplus> s \<in> set (insertDenies p)" shows "s \<in> set p"
permute: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "p permutes S" shows "prod g S = prod (g \<circ> p) S"
s_def: shows "s = (if 0 \<le> p then 1 else - 1)"
div_write_new_val_privilege: fixes s' :: "'b sparc_state"   and i :: "sparc_operation \<times> inst_operand list"   and result :: "word32"   and temp_V :: "word1"   and s :: "'b sparc_state" assumes "s' = snd (fst (div_write_new_val i result temp_V s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
mantissa_pos_iff: fixes x :: "float" shows "(0 < mantissa x) = (0 < x)"
SUP_commute: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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 bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and B :: "'e set"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>i. Sup (f i ` B)) ` A) = Sup ((\<lambda>j. Sup ((\<lambda>i. f i j) ` A)) ` B)"
x_sg_y_sg_mpmf_right: fixes x :: "'a pmf"   and y :: "'a pmf"   and b :: "real" assumes "x \<succeq>[\<R>] y \<and> (y, x) \<notin> \<R>"   and "b \<in> {0<..1}" shows "x \<succeq>[\<R>] mix_pmf b y x \<and> (mix_pmf b y x, x) \<notin> \<R>"
closedin_closed_eq: fixes S :: "'b set"   and T :: "'b set" assumes "closed S" shows "closedin (top_of_set S) T = (closed T \<and> T \<subseteq> S)"
ACIcl_set_eq: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r ~~ s" shows "set_rexp r = set_rexp s"
set_awalk_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "set (pawalk_verts u p) = {u} \<union> set (map fst p) \<union> set (map snd p)"
simps_5: fixes v :: "vname gexp"   and va :: "vname gexp" shows "enumerate_gexp_inputs (Nor v va) = enumerate_gexp_inputs v \<union> enumerate_gexp_inputs va"
simps_1: fixes name :: "name" shows "unconst (Const name) = Some name"
g_evol_def: fixes \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'd set" shows "EVOL \<phi> G U = (\<lambda>s. g_orbit (\<lambda>t. \<phi> t s) G (U s))"
distinct_33: fixes x7 :: "fm" shows "F \<noteq> NOT x7"
op_map_pick_remove_def: fixes m :: "'a \<Rightarrow> 'b option" shows "op_map_pick_remove m \<equiv> SPEC (\<lambda>((k, v), m'). m k = Some v \<and> m' = m |` (- {k}))"
a: shows "(l dvd k) = (Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l})"
le_cases: fixes x :: "'a option \<Rightarrow>\<^sub>0 nat"   and y :: "'a option \<Rightarrow>\<^sub>0 nat" obtains "extended_ord x y"   | "extended_ord y x"
Idempotent_ex: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "Idempotent (uex x)"
simps_1: fixes s :: "'a stream" shows "sdrop 0 s = s"
f_class_inv: shows "CT C = Some CDef \<and> cSuper CDef = D \<and> cFields CDef = Cf"
monomial_power_map_scale: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'c"   and n :: "nat" shows "monomial c t ^ n = monomial (c ^ n) (n \<cdot> t)"
LPHIL0_refines_LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m: shows "LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m 0 \<sqsubseteq> LPHIL0"
not_thesis: shows "\<exists>Q\<in>\<P>. a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
supt_size: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "s \<rhd> t" shows "size t < size s"
char_measurable: fixes a :: "'b set"   and S :: "'b set set"   and x :: "real set set" assumes "a \<in> S"   and "sigma_algebra S" shows "\<chi> a \<in> measurable S x"
rec: fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'c"   and x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool" shows "rec_char f (Char x1 x2 x3 x4 x5 x6 x7 x8) = f x1 x2 x3 x4 x5 x6 x7 x8"
subst_fm_subst_fm: fixes x :: "name"   and y :: "name"   and u :: "tm"   and A :: "fm"   and t :: "tm" assumes "x \<noteq> y"   and "atom x \<sharp> u" shows "(A(x::=t))(y::=u) = (A(y::=u))(x::=subst y u t)"
norm_15: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<box>\<box>P = \<box>P"
wt_elim_cases_12: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Acc va)\<Colon>T" obtains T :: "ty" where "Tb__ = Inl T" and "E,dt\<Turnstile>va\<Colon>=T"
fmdom_notD: fixes x :: "'c"   and m :: "'c \<rightharpoonup>\<^sub>f 'd" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
mat_of_cblinfun_scaleR: fixes a :: "real"   and F :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun (a *\<^sub>R F) = complex_of_real a \<cdot>\<^sub>m mat_of_cblinfun F"
are_local_morphisms: fixes x :: "'b"   and V :: "'a set set" assumes "x \<in> X"   and "is_zariski_open V"   and "f x \<in> V" shows "ind_mor_btw_stalks.is_local X is_open \<O>\<^sub>X \<rho> add_str mult_str zero_str one_str is_zariski_open sheaf_spec sheaf_spec_morphisms add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec f x V \<phi>\<^bsub>X is_open \<O>\<^sub>X \<rho> is_zariski_open sheaf_spec sheaf_spec_morphisms f \<phi>\<^sub>f x\<^esub>"
unitl_ex: fixes x :: "'a" shows "\<exists>e\<in>{1::'a}. D e x \<and> e \<oplus> x = x"
simps: fixes n :: "nat" shows "hensel_fpxs_aux n = (if n = 0 then (g, h) else let U = F $ n - (\<Sum>(i, j)\<in>{(i, j). i < n \<and> j < n \<and> i + j = n}. fst (hensel_fpxs_aux i) * snd (hensel_fpxs_aux j)) in (U * g' + g * (U * h' div h), U * h' mod h))"
CLF_set_def: shows "CLF_set = (SIGMA cl:CompleteLattice. {f \<in> pset cl \<rightarrow> pset cl. Tarski.monotone f (pset cl) (order cl)})"
simps_3: fixes C_info :: "(name, dt_def) fmap"   and css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "special_constants.vwelldefined' C_info (Vrecabs css name \<Gamma>) = (pred_fmap id (fmmap (special_constants.vwelldefined' C_info) \<Gamma>) \<and> pred_fmap (\<lambda>cs. list_all (\<lambda>(pat, t). consts t |\<subseteq>| fmdom \<Gamma> |\<union>| (constructors.C C_info |\<union>| fmdom css)) cs \<and> fdisjnt (constructors.C C_info) (fmdom \<Gamma>)) css \<and> name |\<in>| fmdom css \<and> fdisjnt (constructors.C C_info) (fmdom css))"
not_ev_iff: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "(\<not> ev P \<omega>) = alw (\<lambda>xs. \<not> P xs) \<omega>"
bind_cong: fixes x :: "'b option"   and y :: "'b option"   and f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "x \<bind> f = y \<bind> g"
CD_on_consistency: fixes ds :: "'d set" shows "consistency (CD_on ds)"
qintersect: fixes Quorum :: "'a set set"   and Q :: "'a set"   and Q' :: "'a set" assumes "quorum Quorum"   and "Q \<in> Quorum"   and "Q' \<in> Quorum" shows "Q \<inter> Q' \<noteq> {}"
case_3: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and x31 :: "'g"   and x32 :: "'f" shows "(case Yield x31 x32 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
euler_product_LIMSEQ': fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))"   and "completely_multiplicative_function f" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse ((1::'a) - f p) else (1::'a)) \<longlonglongrightarrow> (\<Sum>n. f n)"
bit_of_bool_iff: fixes b :: "bool"   and n :: "nat" shows "bit (of_bool b) n = (b \<and> n = 0)"
absorb1: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "a \<sqinter> b = a"
insertion_sum_var: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a mpoly"   and n :: "nat" shows "insertion f (sum g {0..<n}) = (\<Sum>i = 0..<n. insertion f (g i))"
healthy_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "healthy (wp (Embed t))"
init_def: fixes h :: "'a"   and w :: "nat" shows "init h w = sample_uniform (order \<G>) \<bind> (\<lambda>r. return_spmf (r, \<^bold>g [^] r))"
equality: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme" assumes "getF r = getF r'"   and "getM r = getM r'"   and "more r = more r'" shows "r = r'"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetMap less_eq less less_eq' less' P f"   and "OrderingSetIso_axioms less_eq less less_eq' less' P f" shows "OrderingSetIso less_eq less less_eq' less' P f"
assms_2: fixes F' :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "F' \<subseteq> P[X]"   and "ideal F \<subset> ideal F'" shows "ideal F' = UNIV"
f2: shows "prob0 3 (Matrix.mat 8 (Suc 0) (\<lambda>(i, j). [[0, 0, 0, 0, \<phi> $$ (0, 0) / complex_of_real (sqrt 2), - (\<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2)), - (\<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2)), \<phi> $$ (0, 0) / complex_of_real (sqrt 2)]] ! j ! i)) (Suc 0) = 1 / 2"