hcju/isabelleps huggingface dataset:
mgu\<^sub>l\<^sub>s_def: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "mgu\<^sub>l\<^sub>s \<sigma> L = (unifier\<^sub>l\<^sub>s \<sigma> L \<and> (\<forall>u. unifier\<^sub>l\<^sub>s u L \<longrightarrow> (\<exists>i. u = \<sigma> \<cdot> i)))"
D1H1: fixes Aa :: "int poly"   and Ba :: "int poly" shows "Aa * D1 + Ba * H1 =m Aa * D + Ba * H"
dconvfactor_pos: fixes c :: "'c \<Rightarrow>\<^sub>U 'd"   and d :: "(int, sdim) dimvec" shows "0 < dconvfactor c d"
I_prime: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "primeideal I R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes>\<^bsub>R\<^esub> b \<in> I" shows "a \<in> I \<or> b \<in> I"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'d list"   and x :: "'d" assumes "i < length xs" shows "xs[i := x] ! i = x"
bounded_matrix_for_marginals_ennreal: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ennreal"   and B :: "'d set"   and g :: "'d \<Rightarrow> ennreal"   and R :: "('c \<times> 'd) set" assumes "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) g"   and "integral\<^sup>N (count_space B) g \<noteq> \<top>"   and "\<And>X. X \<subseteq> A \<Longrightarrow> integral\<^sup>N (count_space X) f \<le> integral\<^sup>N (count_space (R `` X)) g"   and "countable A"   and "countable B"   and "R \<subseteq> A \<times> B" obtains h :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" where "\<And>x y. 0 < h x y \<Longrightarrow> (x, y) \<in> R" and "\<And>x y. h x y \<noteq> \<top>" and "\<And>x. x \<in> A \<Longrightarrow> integral\<^sup>N (count_space B) (h x) = f x" and "\<And>y. y \<in> B \<Longrightarrow> (\<Sum>\<^sup>+ x\<in>A. h x y) = g y"
hyperbola_method_semiring_sqrt: fixes x :: "real"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "0 \<le> x" shows "sum_upto (dirichlet_prod f g) x + sum_upto f (sqrt x) * sum_upto g (sqrt x) = sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) (sqrt x)"
GroupByPresentationInducedFun_S_P_id: shows "GroupByPresentationInducedFun S (map (charpair S) ` R) id"
Rats_cases': fixes x :: "real" assumes "x \<in> \<rat>" obtains p :: "int" and q :: "int" where "0 < q" and "x = real_of_int p / real_of_int q"
vproduct_cong: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and g :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f i = g i" shows "vproduct I f = vproduct I g"
zero_normh: fixes n :: "nat" shows "isnpolyh 0\<^sub>p n"
LN: fixes vs :: "e list"   and lholed :: "Lholed"   and n :: "nat"   and es' :: "e list"   and l :: "Lholed"   and es'' :: "e list"   and k :: "nat"   and es :: "e list"   and lfilledk :: "e list" assumes "const_list vs"   and "lholed = LRec vs n es' l es''"   and "Lfilled_exact k l es lfilledk" shows "Lfilled_exact (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')"
int_simps_4: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (\<not> \<not> P) = P"
build_stutter: fixes w :: "nat \<Rightarrow> 'a" shows "w 0 ## w \<approx> w"
fls_X_intpow_times_conv_shift_1: fixes i :: "int"   and f :: "'b fls" shows "fls_X_intpow i * f = fls_shift (- i) f"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end5_exit x (l, r) = (0 < x \<and> l = [] \<and> r = Bk # Oc \<up> x @ Bk # Oc \<up> x)"
arr_iff_in_hom: fixes f :: "'a" shows "L.arr f = L.in_hom f (L.dom f) (L.cod f)"
lexordp_simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "ord.lexordp less xs [] = False"
i_take_drop_eq_map: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Up> m \<Down> n = map f [m..<m + n]"
powr_closureE: fixes g :: "'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real" assumes "g \<in> powr_closure f" obtains p :: "real" where "g = (\<lambda>x. f x powr p)"
max_diff_distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.max less_eq x y) z = ord.max less_eq (minus x z) (minus y z)"
nat_mult_div_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "k * m div (k * n) = m div n"
ivls_of_aforms_def: fixes prec :: "nat" shows "ivls_of_aforms prec = map (\<lambda>a. Interval' (float_of (Inf_aform' prec a)) (float_of (Sup_aform' prec a)))"
order_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "prefix x y"   and "prefix y z" shows "prefix x z"
strip_lpfpc: fixes f :: "'b option acom \<Rightarrow> 'b option acom"   and c :: "com"   and c' :: "'b option acom" assumes "\<And>c. strip (f c) = strip c"   and "lpfp\<^sub>c f c = Some c'" shows "strip c' = c"
C3: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel"   and "\<Psi>' \<simeq> \<Psi>''" shows "(\<Psi>'', S, T) \<in> Rel"
equiv_def: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
inv_right: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "bijection f" shows "f (inv f a) = a"
permutation_finite_support: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "finite {x. p x \<noteq> x}"
Fubini': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f x y \<partial>lborel \<partial>lborel = \<integral>\<^sup>+ x. integral\<^sup>N lborel (f x) \<partial>lborel"
len_stable: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1--v2 \<longrightarrow> len v1 ts c = len v2 ts c"
m_defsAll_6: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'a"   and t :: "'b"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC_part c d s t I0 P F \<equiv> {} \<notin> P - {I0} \<and> {} \<notin> F ` (P - {I0}) \<and> I0 \<in> P \<and> part {..<brn c} P \<and> part {..<brn d} (F ` P)"
sumf'_leq_sumf: shows "\<forall>\<^sub>F x in finite_subsets_at_top B. sum f' x \<le> sum f x"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans lexext_rev"
outstanding_non_volatile_writes_unshared_axioms: shows "outstanding_non_volatile_writes_unshared \<S> ts"
homeomorphic_maps_Euclidean_space_euclidean_gen: fixes B :: "'a set"   and n :: "nat" assumes "independent B"   and "pairwise orthogonal B"   and "card B = n"   and "\<And>u. u \<in> B \<Longrightarrow> norm u = 1" obtains f :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" and g :: "'a \<Rightarrow> nat \<Rightarrow> real" where "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g" and "\<And>x. x \<in> topspace (Euclidean_space n) \<Longrightarrow> (norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)"
field_differentiable_within_Arccos: fixes z :: "complex"   and s :: "complex set" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "Arccos field_differentiable at z within s"
nofail_SPEC_iff: fixes m :: "'b nres" shows "nofail m = (m \<le> SPEC (\<lambda>_. True))"
dom_res_singleton: fixes m :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'c" assumes "m $$ k = Some v" shows "{k} \<lhd> m = {k $$:= v}"
slist_eq_TT_snil_2: fixes xs :: "[:'a:]" shows "(eq\<cdot>[::]\<cdot>xs = TT) = (xs = [::])"
A_ne: assumes "i \<in> set xs" shows "{k. k < length xs \<and> xs ! k = i} \<noteq> {}"
om_coinduct: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x \<cdot> y" shows "y \<le> x\<^sup>\<omega>"
generate_one_irrel: fixes A :: "'a set" shows "generate G A = generate G (A \<union> {\<one>})"
trans: fixes M :: "'b dfa"   and N :: "'b dfa"   and h :: "hf \<Rightarrow> hf"   and N' :: "'b dfa"   and h' :: "hf \<Rightarrow> hf" assumes "dfa_isomorphism M N h"   and "dfa_isomorphism N N' h'" shows "dfa_isomorphism M N' (h' \<circ> h)"
degree_filter_pdevs_le: fixes I :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b pdevs" shows "degree (filter_pdevs I x) \<le> degree x"
Compl_set_less_aux_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and s1 :: "'d"   and s2 :: "'e" shows "Compl_set_less_aux_fusion g1 g2 ao s1 s2 = Compl_set_less_aux ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
space_point_measure: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ennreal" shows "space (point_measure A f) = A"
intros: fixes \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and x :: "'a"   and m :: "'b"   and ta :: "('c, 'd, 'a, 'b, 'e, 'f) thread_action"   and x' :: "'a"   and m' :: "'b" assumes "\<tau>move (x, m) ta (x', m')" shows "\<tau>multithreaded.init_fin_\<tau>move \<tau>move ((Running, x), m) (convert_TA_initial (convert_obs_initial ta)) ((Running, x'), m')"
sgn_real_alg_def: fixes x :: "real_alg" shows "sgn x = of_rat_real_alg (sgn_real_alg_rat x)"
circ_decompose_10: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> = (x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> * x\<^sup>\<circ> * y\<^sup>\<circ>"
conf_f_def2: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and ST :: "ty list"   and LT :: "ty err list"   and "is" :: "'addr instr list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat" shows "conf_f P h (ST, LT) is (stk, loc, C, M, pc) \<equiv> P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is"
assms_5: shows "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d"
pH: shows "snd (config\<^sub>p (rTS []) (Lxy init {x, y}) (Lxy qs {x, y})) = rev (Lxy qs {x, y})"
basis_null_space_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "basis_null_space A = {row i (P_Gauss_Jordan (Finite_Cartesian_Product.transpose A)) |i. rank A \<le> mod_type_class.to_nat i}"
H: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (H i)"
Ngt0_def: shows "0<\<^sub>N = (\<lambda>(a, b). 0 < a)"
equiv'_def: shows "B_A.equiv' \<equiv> \<lambda>a b. b \<sim> a \<and> PB a \<and> PA b"
is_weakly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "is_weakly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<union> added_pointers \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
absorb1: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "a \<squnion> b = a"
quat_inner_k_right: fixes x :: "quat" shows "x \<bullet> \<k> = Im3 x"
\<Delta>\<^sub>d_nonneg: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "0 \<le> prob_space_with_indicators.\<Delta>\<^sub>d M I A"
elims: fixes x :: "order"   and y :: "order" assumes "invert_order x = y" obtains "x = Lt" and "y = Gt"   | "x = Gt" and "y = Lt"   | "x = Eq" and "y = Eq"
homeomorphic_space: fixes X :: "'a topology"   and Y :: "'b topology" shows "(X homeomorphic_space Y) = (\<exists>f. homeomorphic_map X Y f)"
lossless_exec_gpv: fixes gpv :: "('b, 'a, bool list) gpv"   and s :: "'a \<Rightarrow> bool list option" assumes "lossless_gpv \<I>_full gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>_full; finite (dom s)\<rbrakk> \<Longrightarrow> lossless_spmf (PRF.random_oracle s out)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "finite (dom s)" shows "lossless_spmf (exec_gpv PRF.random_oracle gpv s)"
Fubini_integral: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "integrable (restrict_space (measure_pmf A) (set_pmf A) \<Otimes>\<^sub>M restrict_space (measure_pmf B) (set_pmf B)) (\<lambda>(x, y). f x y)" shows "B.expectation (\<lambda>y. A.expectation (\<lambda>x. f x y)) = A.expectation (\<lambda>x. B.expectation (f x))"
that: fixes u :: "'a \<Rightarrow> 'b" assumes "u \<in> [M]\<^bsub>v,n\<^esub>"   and "u c1 = r" shows "thesis"
between_exchange4: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C"   and "Bet A C D" shows "Bet A B D"
filter_mset_False: fixes M :: "'b multiset" shows "{#y \<in># M. False#} = {#}"
d: shows "v \<equiv> Tensor.mat_of_cols_list 4 (map (map complex_of_real) [[1 / sqrt 2, 0, 1 / sqrt 2, 0], [0, 1 / sqrt 2, 0, 1 / sqrt 2], [1 / sqrt 2, 0, - 1 / sqrt 2, 0], [0, 1 / sqrt 2, 0, - 1 / sqrt 2]])"
BRParAux: fixes R :: "ccs"   and T :: "ccs"   and Pa :: "ccs"   and Qa :: "ccs" assumes "R \<approx> T"   and "(Pa, Qa) \<in> weakBisimulation O bangRel weakBisimulation O bisim" shows "(R \<parallel> Pa, T \<parallel> Qa) \<in> weakBisimulation O bangRel weakBisimulation O bisim"
length_rev: fixes xs :: "'b list" shows "\<^bold>|rev xs\<^bold>| = \<^bold>|xs\<^bold>|"
last_in_set: fixes l :: "'c list" assumes "l \<noteq> []" shows "last l \<in> set l"
update_convs_7: fixes getReason' :: "(Literal \<Rightarrow> nat option) \<Rightarrow> Literal \<Rightarrow> nat option"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getReason_update getReason' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason' getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
distr_id2: fixes M :: "'a measure"   and N :: "'a measure" assumes "sets M = sets N" shows "distr N M (\<lambda>x. x) = N"
even_plus_one_iff: fixes a :: "'a" shows "even (a + (1::'a)) = odd a"
normalize_gcd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "normalize (gcd a b) = gcd a b"
matrix_vector_mult_hom: fixes A :: "(('a, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and v :: "('a, 'e) Finite_Cartesian_Product.vec" shows "HMA_Connect.map_matrix hom A *v map_vector hom v = map_vector hom (A *v v)"
more_temp_simps1_2: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>\<diamond>F \<equiv> \<diamond>F"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>L 'c" shows "norm x = onorm (blinfun_apply x)"
fls_base_factor_to_fps_base: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "fls_base_factor_to_fps f $ 0 \<noteq> (0::'a)"
semiring_1_cancel_ow_axioms: shows "semiring_1_cancel_ow U (+\<^sub>o\<^sub>w) (-\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w 1\<^sub>o\<^sub>w (*\<^sub>o\<^sub>w)"
OrdS: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>O!,x\<rparr> in v] = (\<exists>o\<^sub>1 y. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y)"
f_def: shows "f \<equiv> \<lambda>x. - 1 + C1 * sqrt (x - a)"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>x xs y ys. \<lbrakk>x \<le> y \<Longrightarrow> P xs (y # ys); \<not> x \<le> y \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>xs. P xs []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
Union_rel_br: fixes l :: "'a \<Rightarrow> 'c set"   and lI :: "'a \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'b set"   and sI :: "'c \<Rightarrow> bool" shows "\<langle>br l lI, br s sI\<rangle>Union_rel = br (\<lambda>a. \<Union> (s ` l a)) (\<lambda>a. lI a \<and> (\<forall>c\<in>l a. sI c))"
cod_eqI: fixes b :: "'d"   and f :: "'d" assumes "D.ide b"   and "D.seq b f" shows "D.cod f = b"
foldli_conj: fixes list :: "'a list"   and P :: "'a \<Rightarrow> bool"   and b :: "bool" shows "foldli list id (\<lambda>kv \<sigma>. P kv) b = (b \<and> (\<forall>x\<in>set list. P x))"
sum_constant: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and y :: "'b"   and A :: "'c set" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum plus zero (\<lambda>x. y) A = times (semiring_1.of_nat one plus zero (card A)) y"
union_single_eq_member: fixes x :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" assumes "add_mset x M = N" shows "x \<in># N"
inverse_unique: fixes f :: "'s"   and g :: "'s" assumes "inverse_arrows f g" shows "local.inv f = g"
Hilbert_fun_Polys: fixes z :: "nat" assumes "X \<noteq> {}" shows "Hilbert_fun P[X] z = z + (card X - 1) choose (card X - 1)"
preserves_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "local.dom (HoHV f) = HoHV (VVV.dom f)"
simps_2: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a"   and y :: "'a"   and ys :: "'a llist" shows "lmember\<cdot>eq\<cdot>x\<cdot>(y :@ ys) = (lmember\<cdot>eq\<cdot>x\<cdot>ys orelse eq\<cdot>y\<cdot>x)"
tensor_mat_carrier: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "tensor_mat m1 m2 \<in> carrier_mat d d"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "inf_class.inf a b \<le> a"
assms: shows "list_all (finite \<circ> DCA.nodes) AA"
summable_comparison_test': fixes g :: "nat \<Rightarrow> real"   and N :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "summable g"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n" shows "summable f"
less_eqE: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'e" where "b = plus a c"
left_idem: fixes a :: "'b"   and b :: "'b" shows "sup a (sup a b) = sup a b"
simps_7: fixes c :: "int"   and e :: "MIR.num" shows "iszlfm (Gt (CN 0 c e)) = (\<lambda>bs. 0 < c \<and> numbound0 e \<and> isint e bs)"
integral_mult_right: fixes S :: "'b set"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a" shows "integral S (\<lambda>x. c * f x) = c * integral S f"
three_composable_identity_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data" shows "three_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<equiv> (three_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<and> identity_arrow_of_spans C \<mu> \<and> identity_arrow_of_spans C \<nu>) \<and> identity_arrow_of_spans C \<pi> \<and> two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<and> two_composable_identity_arrows_of_spans C prj0 prj1 \<nu> \<pi>"
subseteq_qle_trans: fixes A :: "'a list list set"   and B :: "'a list list set"   and C :: "'a list list set" assumes "A \<subseteq>\<^sub>\<simeq> B"   and "B \<subseteq>\<^sub>\<simeq> C" shows "A \<subseteq>\<^sub>\<simeq> C"
integral_less_AE: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "integrable M X"   and "integrable M Y"   and "emeasure M A \<noteq> 0"   and "A \<in> sets M"   and "AE x\<in>A in M. X x \<noteq> Y x"   and "AE x in M. X x \<le> Y x" shows "integral\<^sup>L M X < integral\<^sup>L M Y"
acute_obtuse__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Acute A B C"   and "Obtuse D E F" shows "A B C LtA D E F"
factor_unit: shows "Ker = \<one> \<cdot>| Ker"
cases: fixes a :: "ty" assumes "is_refT a" obtains "a = NT"   | C :: "String.literal" where "a = Class C"   | A :: "ty" where "a = A\<lfloor>\<rceil>"
straighten_def: fixes x :: "wall"   and y :: "wall" shows "straighten x y \<equiv> straighten_topdown x y \<or> straighten_downtop x y"
assms_4: shows "rel_set T (\<Union> K) xr"
ChamberComplex_def: fixes X :: "'b set set" shows "ChamberComplex X \<equiv> SimplicialComplex X \<and> ChamberComplex_axioms X"
continuous_map_if: fixes P :: "bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "P \<Longrightarrow> continuous_map X Y f"   and "\<not> P \<Longrightarrow> continuous_map X Y g" shows "continuous_map X Y (\<lambda>x. if P then f x else g x)"
segment_not_inc: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec> b" shows "b \<notin> segment D a"
normalize_while: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (while gs b c) = guards gs (While b (sequence Seq (flatten (Language.normalize c) @ flatten (guards gs Skip))))"
first_intersection_crossing_derivatives: fixes tg :: "real" assumes "t0 < tg"   and "tg \<le> t1"   and "w tg \<le> v tg" obtains x0 :: "real" where "t0 < x0" and "x0 \<le> tg" and "w' x0 \<le> v' x0" and "v x0 = w x0" and "\<And>t. \<lbrakk>t0 < t; t < x0\<rbrakk> \<Longrightarrow> v t < w t"
A: shows "Crypt (priSK CA) (Hash (pubAK g)) \<in> parts (A \<union> spies evsFR4) \<longrightarrow> (\<exists>n. g = priAK n)"
similar_mat_smult: fixes A :: "'b mat"   and B :: "'b mat"   and k :: "'b" assumes "similar_mat A B" shows "similar_mat (k \<cdot>\<^sub>m A) (k \<cdot>\<^sub>m B)"