hcju/isabelleps huggingface dataset:
affine_hull_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "bounded_linear f" shows "f ` (affine hull s) = affine hull f ` s"
RELATESI: shows "RELATES (S \<inter> set li \<times> set l)"
fv_subst_obtain_var: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "x \<in> fv (t \<cdot> \<delta>)" shows "\<exists>y\<in>fv t. x \<in> fv (\<delta> y)"
bw_diff_sum_pow: fixes k :: "nat"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and I :: "'d set" shows "(bw_diff ^^ k) (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<lambda>x. \<Sum>i\<in>I. (bw_diff ^^ k) (f i) x)"
intros_1: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> synth H"
predicate2I: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
lhd_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lhd (iterates f x) = x"
A5_DAcc_level1: shows "DAcc level1 sA5 = {}"
scaling_mono: fixes u :: "'a"   and v :: "'a"   and r :: "'a"   and s :: "'a" assumes "u \<le> v"   and "(0::'a) \<le> r"   and "r \<le> s" shows "u + r * (v - u) / s \<le> v"
multiset_inter_def: fixes A :: "'c multiset"   and B :: "'c multiset" shows "A \<inter># B = A - (A - B)"
igWlsAbsDisj_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igWlsAbsDisj MOD \<equiv> \<forall>xs s xs' s' A. isInBar (xs, s) \<and> isInBar (xs', s') \<and> igWlsAbs MOD (xs, s) A \<and> igWlsAbs MOD (xs', s') A \<longrightarrow> xs = xs' \<and> s = s'"
p: shows "\<forall>s\<in>P. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' s Z) Call (p s) (Q' s Z),(A' s Z)"
dim_diag_block_mat_1: fixes As :: "'b mat list" shows "dim_row (diag_block_mat As) = sum_list (map dim_row As)"
above_AboveS_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "trans r"   and "antisym r"   and "a \<in> above r b"   and "b \<in> AboveS r C" shows "a \<in> AboveS r C"
ge: fixes x :: "'a" shows "pmf N x \<le> pmf M x"
lin_indpt_list_nonzero: fixes G :: "'a vec list" assumes "lin_indpt_list G" shows "0\<^sub>v n \<notin> set G"
set_conv_list: fixes g :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "prod g (set xs) = prod_list (map g (remdups xs))"
cod_closed: fixes f :: "'a" assumes "\<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>" shows "\<guillemotleft>cod f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
mono_inv_aval': fixes r1 :: "'a"   and r2 :: "'a"   and S1 :: "'a st option"   and S2 :: "'a st option"   and e :: "aexp" assumes "r1 \<le> r2"   and "S1 \<le> S2" shows "inv_aval' e r1 S1 \<le> inv_aval' e r2 S2"
map2_val_compat'_map_val_raw: fixes f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f" shows "map2_val_compat' (map_val_raw f)"
Rf_mono: fixes N :: "'b literal multiset set"   and N' :: "'b literal multiset set" assumes "N \<subseteq> N'" shows "standard_redundancy_criterion.Rf N \<subseteq> standard_redundancy_criterion.Rf N'"
lift_valid_node: fixes m :: "'b" assumes "valid_node m" shows "CFGExit.valid_node (Node m)"
set_vars_list: fixes lp :: "linear_poly" shows "set (Abstract_Linear_Poly.vars_list lp) = vars lp"
cobounded2: fixes b :: "'a word"   and a :: "'a word" shows "b \<le>s signed.max a b"
fresh_floatarith_max_Var: fixes ea :: "floatarith"   and i :: "nat" assumes "max_Var_floatarith ea \<le> i" shows "fresh_floatarith ea i"
fps_lr_inverse_zero_2: fixes y :: "'a" shows "fps_right_inverse 0 y = fps_const y"
Suc: shows "\<Z> k u \<subseteq> \<W> (\<P> k u) k"
in_degree_def: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" shows "in_degree G v \<equiv> card (in_arcs G v)"
OrdSum_second: fixes b :: "'c" shows "(1::'b, b) \<in> OrdSum"
hb_consistent_append_elim_ConsD: fixes y :: "'a"   and ys :: "'a list" assumes "hb_consistent (y # ys)" shows "hb_consistent ys"
map_sel_3: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b le_constraint" shows "lec_const (map_le_constraint f a) = f (lec_const a)"
hom_zero: fixes c :: "'b" shows "c * (0::'b) = (0::'b)"
sel_3: fixes x1 :: "('a, 'b) hd" shows "fun (Hd x1) = Hd x1"
Shuffle_def: fixes A :: "'c list set"   and B :: "'c list set" shows "A \<parallel> B = \<Union> {shuffles xs ys |xs ys. xs \<in> A \<and> ys \<in> B}"
eng_mult: fixes h1 :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and h2 :: "'a" assumes "h1 \<in> generate_field R H"   and "h2 \<in> generate_field R H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> generate_field R H"
WTrtFAss: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : Class C"   and "P \<turnstile> C has F:T in D"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T" shows "P,E,h \<turnstile> e\<^sub>1\<bullet>F{D} := e\<^sub>2 : Void"
rotate_inv2: fixes n :: "nat"   and ls :: "'c list"   and ms :: "'c list" assumes "rotate n ls = ms" shows "ls = rotate (|ls| - n mod |ls|) ms"
tObsC_tStep_eq_inv: fixes t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState" assumes "tObsC t' = tObsC (t \<leadsto> s)" shows "\<exists>t'' s'. t' = t'' \<leadsto> s'"
ide_compE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.ide (VoV.comp g f)" obtains "VoV.seq g f" and "VoV.seq f g" and "VoV.comp g f = VoV.dom f" and "VoV.comp g f = VoV.cod g"
if_polynomial_0_evaluate: fixes wd :: "nat \<Rightarrow> real"   and Z :: "nat"   and ws :: "nat \<Rightarrow> real" assumes "polynomial_f wd \<noteq> 0"   and "\<forall>inputs. input_sizes (deep_model_l rs) = map dim_vec inputs \<longrightarrow> evaluate_net (insert_weights shared_weights (deep_model_l rs) wd) inputs = evaluate_net (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws) inputs" shows "r ^ N_half \<le> Z"
find_augmenting_spec_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" shows "find_augmenting_spec f \<equiv> assert (NFlow c s t f) \<bind> (\<lambda>_. select p. NPreflow.isAugmentingPath c s t f p)"
alphaAbs_qAbs_iff_ex_distinct_qFresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
max_prob_Cons: fixes s :: "'b"   and o\<^sub>1 :: "'a"   and o :: "'a"   and os :: "'a list" shows "Max {emeasure (T (s, o\<^sub>1)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as (o # os) @- \<omega>'} |as. length as = length (o # os) \<and> set as \<subseteq> \<S>} = (MAX t\<in>\<S>. ennreal (pmf (\<O> t) o * pmf (\<K> s) t) * (MAX as\<in>{as. length as = length os \<and> set as \<subseteq> \<S>}. emeasure (T (t, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}))"
rep_eq: fixes x :: "'b list" shows "mapping_rbt.impl_of (bulkload x) = rbt_comp_bulkload ccomp (map (\<lambda>x. (x, ())) x)"
group_def: fixes G :: "('b, 'c) monoid_scheme" shows "group G \<equiv> monoid G \<and> group_axioms G"
raw_ports_invert: fixes ps :: "(16 word \<times> 16 word) list" shows "ports_to_set (raw_ports_invert ps) = - ports_to_set ps"
fg: fixes x :: "'v" assumes "v \<noteq> x" shows "f x = g x"
lower_higher_zeroI: fixes u :: "'a"   and v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "u \<preceq> v" shows "punit.lower (punit.higher p v) u = 0"
intros_1: fixes R :: "('b \<times> 'b) set" shows "Linorder_Relations.sorted_wrt R []"
fin_A: shows "finite {x. h x (f x) (g x) \<noteq> (0::'b)}"
close_app_comm: fixes A :: "'b list"   and B :: "'b list"   and x :: "'b" shows "close (A @ B) x = close A x @ close B x"
dual_spaceI: fixes S :: "'c set"   and E :: "'c \<Rightarrow> real" assumes "extensional0 S E"   and "linear_fun_on S E" shows "E \<in> dual_space S"
nres_more_simps_2: fixes X :: "'b set" shows "(RES X = SUCCEED) = (X = {})"
dg_prod_2_is_arrI: fixes a :: "V"   and c :: "V"   and g :: "V"   and b :: "V"   and d :: "V"   and f :: "V" assumes "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c"   and "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d" shows "[g, f]\<^sub>\<circ> : [a, b]\<^sub>\<circ> \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G \<BB>\<^esub> [c, d]\<^sub>\<circ>"
uniform_discrete_imp_discrete: fixes S :: "'c set" assumes "uniform_discrete S" shows "discrete S"
direct_decompI: fixes ss :: "'a set list"   and A :: "'a set" assumes "inj_on sum_list (listset ss)"   and "sum_list ` listset ss = A" shows "direct_decomp A ss"
simps_8: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Lt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f2"
Dlog: fixes t :: "'b tree" shows "real (rlh t) \<le> log 2 (real (size1 t))"
INF_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
InvariantReasonClausesAfterUnitPropagate: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and uClause :: "Literal list"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantReasonClauses F M"   and "formulaEntailsClause F uClause"   and "isUnitClause uClause uLiteral (elements M)"   and "M' = M @ [(uLiteral, False)]" shows "InvariantReasonClauses F M'"
RETURN_refine: fixes x :: "'c"   and x' :: "'d"   and R :: "('c \<times> 'd) set" assumes "(x, x') \<in> R" shows "RETURN x \<le> \<Down> R (RETURN x')"
le_ref1: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<le> Q" shows "D Q \<subseteq> D P"
limv: shows "((\<lambda>n. real_of_ereal (v n)) \<longlongrightarrow> m) F"
v0_edge_w0: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "v0 \<rightarrow>\<^bsub>G\<^esub> vmc_path_no_deadend.w0 P"
ideal_subset1: fixes R :: "('h, 'i) Ring_scheme"   and I :: "'h set" assumes "Ring R"   and "ideal R I" shows "I \<subseteq> carrier R"
assms_5: shows "are_operators_interfering op\<^sub>1 op\<^sub>2"
pred_map: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d pmf" shows "pred_pmf Q (map_pmf f x) = pred_pmf (Q \<circ> f) x"
PFUadmit_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c, 'd) formula"   and \<theta> :: "('b, 'c + 'a, 'd) formula"   and U :: "('d + 'd) set" shows "PFUadmit \<sigma> \<theta> U = True"
simps_2: fixes uv :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uw :: "'g \<Rightarrow> 'e"   and e :: "'f" shows "ap_sum uv (pure uw) (Inr e) = Inr e"
iso_tuple_update_accessor_eq_assist_idI: fixes v' :: "'b"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" assumes "v' = f v" shows "Record.iso_tuple_update_accessor_eq_assist id id v f v' v"
Borsukian_retraction_gen: fixes S :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and T :: "'d set"   and k :: "'d \<Rightarrow> 'c" assumes "Borsukian S"   and "continuous_on S h"   and "h ` S = T"   and "continuous_on T k"   and "k ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> h (k y) = y" shows "Borsukian T"
wlog_x_axis: fixes u :: "complex_homo"   and v :: "complex_homo"   and P :: "complex_homo \<Rightarrow> complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "\<And>M u v. \<lbrakk>unit_disc_fix M; u \<in> unit_disc; v \<in> unit_disc; P (moebius_pt M u) (moebius_pt M v)\<rbrakk> \<Longrightarrow> P u v"   and "\<And>x. \<lbrakk>is_real x; 0 \<le> Re x; Re x < 1\<rbrakk> \<Longrightarrow> P 0\<^sub>h (of_complex x)" shows "P u v"
eventually_nhds_metric_le: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "eventually P (nhds a) = (\<exists>d>0. \<forall>x. dist x a \<le> d \<longrightarrow> P x)"
that: fixes v :: "'a llist"   and a :: "'a" assumes "w = v $ <a>"   and "n \<le> llength v" shows "thesis"
transfer'_return: fixes P :: "heap \<Rightarrow> bool"   and Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and dp :: "'c \<Rightarrow> 'd" assumes "dp_heap P Q lookup lookup_st update update_st" shows "rel_fun R (dp_heap.crel_vs P Q dp lookup_st R) Wrap return"
p_def: shows "p \<equiv> \<lambda>x. 1 / real (card subU) * (\<Sum>t\<in>subU. pf t x)"
A: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"
smult_closed: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and m :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "m \<in> M" shows "smult r m \<in> M"
cr_\<Pi>\<^sub>1_def: shows "cr_\<Pi>\<^sub>1 \<equiv> \<lambda>x y. x = eval\<Pi>\<^sub>1 y"
fold_union_elemI: fixes x :: "'b"   and ys :: "'b set"   and xss :: "'b set list"   and xs :: "'b set" assumes "x \<in> ys"   and "ys \<in> set xss \<union> {xs}" shows "x \<in> fold (\<union>) xss xs"
Iff_trans: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> B IFF C" shows "H \<turnstile> A IFF C"
distinct_11: fixes x3 :: "'e" shows "Fault x3 \<noteq> Stuck"
split_vec_dims_2: fixes i :: "nat"   and j :: "nat"   and x :: "(nat, 'a) mapping"   and a :: "'a vec"   and b :: "'a vec" assumes "split_i_j_x i j x = (a, b)" shows "dim_vec b = j - i"
setops\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('a, 'b) stateful_strand_step list"   and a :: "('a, 'b) stateful_strand_step" shows "setops\<^sub>s\<^sub>s\<^sub>t A \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (a # A)"
isS: shows "statediff \<omega> \<omega>' \<subseteq> - FVT \<theta>"
bisim1BlockSome2: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and T :: "ty"   and v :: "'addr val"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,{V:T=\<lfloor>v\<rfloor>; e},h \<turnstile> ({V:T=\<lfloor>v\<rfloor>; e}, xs) \<leftrightarrow> ([v], xs, Suc 0, None)"
abs_eq: fixes xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "MPoly xa + MPoly x = MPoly (xa + x)"
plossless_converter_bot1: fixes \<I> :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" shows "plossless_converter \<bottom> \<I> conv"
left_neutral: fixes a :: "'a multiset" shows "{#} \<union># a = a"
full_list_subseqs: fixes xs :: "'c list" shows "{ys \<in> set (subseqs xs). length ys = length xs} = {xs}"
size_mset_SucE: fixes A :: "'b multiset"   and n :: "nat" assumes "size A = Suc n" obtains a :: "'b" and B :: "'b multiset" where "A = {#a#} + B" and "size B = n"
scale_sum_right: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "a * sum f A = (\<Sum>x\<in>A. a * f x)"
transfer_eq: fixes x :: "'c star"   and X :: "nat \<Rightarrow> 'c"   and y :: "'c star"   and Y :: "nat \<Rightarrow> 'c" assumes "x \<equiv> star_n X"   and "y \<equiv> star_n Y" shows "x = y \<equiv> \<forall>\<^sub>F n in \<U>. X n = Y n"
Sup_sup_pres: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_pres f" shows "sup_pres f"
cases: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and a1 :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and a2 :: "'form fset \<times> 'form" assumes "iwf fc a1 a2" obtains n :: "('form, 'rule) graph_node" and p :: "'form" and s :: "'subst" and i :: "nat" and \<Gamma> :: "'form fset" and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" and c :: "'form" where "a1 = INode n p i__ s__ ants" and "a2 = (\<Gamma>__ \<turnstile> c__)" and "n \<in> sset nodes" and "Reg p |\<in>| outPorts n" and "list_all2 (\<lambda>ip t. iwf fc t ((\<lambda>h. subst s__ (freshen i__ (labelsOut n h))) |`| hyps_for n ip |\<union>| \<Gamma>__ \<turnstile> subst s__ (freshen i__ (labelsIn n ip)))) (inPorts' n) ants" and "fc n i__ s__ (\<Gamma>__ \<turnstile> c__)" and "c__ = subst s__ (freshen i__ p)"   | c :: "'form" and \<Gamma> :: "'form fset" and s :: "'subst" and i :: "nat" where "a1 = HNode i s []" and "a2 = (\<Gamma> \<turnstile> c)" and "c |\<notin>| ass_forms" and "c |\<in>| \<Gamma>" and "c = subst s (freshen i anyP)"
wf_projectTable: fixes n :: "nat"   and X :: "nat set \<times> 'b option list set"   and I :: "nat set" assumes "wf_atable n X" shows "wf_atable n (New_max.projectTable I X) \<and> fst (New_max.projectTable I X) = fst X \<inter> I"
vdiff_vempty_right: fixes A :: "V" shows "A -\<^sub>\<circ> []\<^sub>\<circ> = A"
pointermap_sane_def: fixes m :: "('a, 'b) pointermap_scheme" shows "pointermap_sane m \<equiv> distinct (entries m) \<and> (\<forall>n\<in>{..<length (entries m)}. getentry m (entries m ! n) = Some n) \<and> (\<forall>p i. getentry m p = Some i \<longrightarrow> entries m ! i = p \<and> i < length (entries m))"
roots1: fixes p :: "'b poly" assumes "degree p = 1" shows "{x. poly p x = (0::'b)} = {roots1 p}"
linear_singular_image_hyperplane: fixes f :: "'c \<Rightarrow> 'c" assumes "linear f"   and "\<not> inj f" obtains a :: "'c" where "a \<noteq> (0::'c)" and "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
inf_sup_aci_3: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "inf_prod x (inf_prod y z) = inf_prod y (inf_prod x z)"
lborel_eqI: fixes M :: "'b measure" assumes "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = ennreal (prod ((\<bullet>) (u - l)) Basis)"   and "sets M = sets borel" shows "lborel = M"
assms_1: fixes ts :: "dB list"   and n :: "nat" assumes "\<forall>t\<in>set ts. P t" shows "P (Var n \<degree>\<degree> ts)"
remove1_remdups: fixes xs :: "'d list"   and x :: "'d" assumes "distinct xs" shows "remove1 x (remdups xs) = remdups (remove1 x xs)"
preserves_reflects_arr: fixes f :: "'b" shows "D.arr (src\<^sub>D f) = D.arr f"
prob_dom_def: fixes prob :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" shows "prob_dom prob \<equiv> \<Union>(s1, s2)\<in>prob. action_dom s1 s2"
imp_simps_2: fixes P :: "fm" shows "imp P T = T"
map_cong: fixes x :: "('a, 'b) fmap"   and ya :: "('a, 'b) fmap"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> fmran' ya \<Longrightarrow> f z = g z" shows "fmmap f x = fmmap g ya"
Fs_dim_2: shows "dim_col (mat_of_rows n (map gso [0..<length fs])) = n"
sel: fixes x1 :: "'e" shows "prot_atom.un_Atom (prot_atom.Atom x1) = x1"
select_convs_2: fixes g_V :: "'b set"   and g_E :: "('b \<times> 'b) set"   and g_V0 :: "'b set"   and bg_F :: "'b set"   and more :: "'a" shows "b_graph_rec.more \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, bg_F = bg_F, \<dots> = more\<rparr> = more"
is_meetI: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y"   and "\<And>a. \<lbrakk>a \<sqsubseteq> x; a \<sqsubseteq> y\<rbrakk> \<Longrightarrow> a \<sqsubseteq> z" shows "x \<sqinter> y = z"
lemma3: fixes a :: "letter"   and ws :: "letter list list"   and zs :: "letter list list" assumes "T a ws zs"   and "good ws" shows "good zs"
assms_2: shows "b \<in> Mapping.keys (phis g)"
coerce_eq_ID: shows "COERCE('c, 'c) = ID"
b_less_square_base_b: fixes b :: "proper_base" shows "get_base b < get_base (square_base b)"
nxt_alw: fixes P :: "'a stream \<Rightarrow> bool"   and s :: "'a stream" assumes "nxt (alw P) s" shows "alw (nxt P) s"
balL_def: fixes AB :: "('b \<times> nat) tree"   and b :: "'b"   and C :: "('b \<times> nat) tree" shows "balL AB b C = (if ht AB = ht C + m + 1 then case AB of \<langle>A, (a, x), B\<rangle> \<Rightarrow> if ht B \<le> ht A then node A a (node B b C) else case B of \<langle>B\<^sub>1, (ab, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) ab (node B\<^sub>2 b C) else node AB b C)"
one_extra_update_def: fixes t1 :: "'a transition_scheme"   and t2 :: "transition"   and s2 :: "nat"   and e2 :: "((nat \<times> nat) \<times> transition) fset" shows "one_extra_update t1 t2 s2 e2 = (Label t1 = Label t2 \<and> Arity t1 = Arity t2 \<and> set (Guards t1) \<subseteq> set (Guards t2) \<and> Outputs t1 = Outputs t2 \<and> Updates t1 \<noteq> [] \<and> tl (Updates t1) = Updates t2 \<and> (\<exists>r\<in>set (map fst (Updates t1)). fst (hd (Updates t1)) = r \<and> not_updated r t2 \<and> initially_undefined_context_check e2 r s2))"
step_fun_end: fixes F :: "'fenv_inca"   and g :: "'fun"   and gd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and \<Sigma>\<^sub>f :: "'dyn list"   and pc\<^sub>g :: "nat"   and frame\<^sub>g :: "('fun, 'dyn) frame"   and \<Sigma>\<^sub>g :: "'dyn list"   and frame\<^sub>f :: "('fun, 'dyn) frame"   and f :: "'fun"   and pc\<^sub>f :: "nat"   and frame\<^sub>f' :: "('fun, 'dyn) frame"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F g = Some gd"   and "arity gd \<le> length \<Sigma>\<^sub>f"   and "pc\<^sub>g = length (body gd)"   and "frame\<^sub>g = Frame g pc\<^sub>g \<Sigma>\<^sub>g"   and "frame\<^sub>f = Frame f pc\<^sub>f \<Sigma>\<^sub>f"   and "frame\<^sub>f' = Frame f (Suc pc\<^sub>f) (\<Sigma>\<^sub>g @ drop (arity gd) \<Sigma>\<^sub>f)" shows "Sinca.step (State F H (frame\<^sub>g # frame\<^sub>f # st)) (State F H (frame\<^sub>f' # st))"
final_no_step: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "final e" shows "((e, s, b), e', s', b') \<notin> red P"
mk_disjoint_insert: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
homeomorphic_map_subtopologies_alt: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and S :: "'a set" assumes "homeomorphic_map X Y f"   and "\<And>x. \<lbrakk>x \<in> topspace X; f x \<in> topspace Y\<rbrakk> \<Longrightarrow> (f x \<in> T) = (x \<in> S)" shows "homeomorphic_map (subtopology X S) (subtopology Y T) f"
healthy_intros_5: fixes t :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy (wp t)"   and "healthy (wp u)" shows "healthy (wp (t ;; u))"
EndPhase2_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "EndPhase2 s s' p = (IsMajority {d \<in> disksWritten s p. \<forall>q\<in>UNIV - {p}. hasRead s p d q} \<and> phase s p = 2 \<and> outpt s' = (outpt s)(p := inp (dblock s p)) \<and> dblock s' = dblock s \<and> phase s' = (phase s)(p := phase s p + 1) \<and> InitializePhase s s' p \<and> inpt s' = inpt s \<and> disk s' = disk s)"
ass: shows "\<exists>s'. s (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> s'"
induces_unit_counit_adjunction: shows "unit_counit_adjunction (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F G Adj.\<eta> Adj.\<epsilon>"