hcju/isabelleps huggingface dataset:
sorted_list_of_set_3: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
assms_4: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r result \<rightarrow>\<^sub>h h'"
26339: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (- x \<sqinter> y) = - x"
atLeastLessThan_iff: fixes i :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'e"   and u :: "'e" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
runit_char_3: fixes f :: "'a" assumes "S.ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : R f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> R \<mu> = (f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a]"
nearly_healthy_monoD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "nearly_healthy t"   and "P \<tturnstile> Q"   and "unitary P"   and "unitary Q" shows "t P \<tturnstile> t Q"
process_rule_1: fixes P :: "'c process" shows "([], {}) \<in> failures P"
dghm_ArrMap_vsv: shows "vsv (\<FF>\<lparr>ArrMap\<rparr>)"
induct: fixes x1 :: "comb"   and x2 :: "comb"   and P :: "comb \<Rightarrow> comb \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^sup>1 x2"   and "\<And>x y. P (K \<bullet> x \<bullet> y) x"   and "\<And>x y z. P (S \<bullet> x \<bullet> y \<bullet> z) (x \<bullet> z \<bullet> (y \<bullet> z))"   and "\<And>x y z. \<lbrakk>x \<rightarrow>\<^sup>1 y; P x y\<rbrakk> \<Longrightarrow> P (x \<bullet> z) (y \<bullet> z)"   and "\<And>x y z. \<lbrakk>x \<rightarrow>\<^sup>1 y; P x y\<rbrakk> \<Longrightarrow> P (z \<bullet> x) (z \<bullet> y)" shows "P x1 x2"
rec_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
eq_Min_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(m = Min A) = (m \<in> A \<and> (\<forall>a\<in>A. m \<le> a))"
minus_add_eq_ereal: fixes a :: "ereal"   and b :: "ereal" assumes "\<not> (a = \<infinity> \<and> b = - \<infinity> \<or> a = - \<infinity> \<and> b = \<infinity>)" shows "- (a + b) = - a - b"
discharge_is_Leaf: fixes a :: "'c tree" shows "(discharge a = Leaf) = (a = Leaf)"
part_extensionality: fixes x :: "i"   and y :: "i" assumes "M3" shows "(x = y) = (\<forall>z. P z x = P z y)"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.dom (g \<cdot>\<^sub>C f) = C.dom f"
star_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
chain_empty: shows "chain_on P {} {x \<in> A. \<not> Q x}"
abP: shows "(p\<^sup>2 + N * q\<^sup>2) ^ 3 = (p * (p\<^sup>2 - 3 * N * q\<^sup>2))\<^sup>2 + N * (q * (3 * p\<^sup>2 - N * q\<^sup>2))\<^sup>2"
adds_lcs: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus s (lcs s t)"
known_ptr_not_shadow_root_ptr: fixes ptr :: "(_) object_ptr" assumes "ShadowRootClass.known_ptr ptr"   and "\<not> is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" shows "DocumentClass.known_ptr ptr"
abs_eq: fixes xa :: "'e \<Rightarrow> 'c"   and x :: "'e \<Longrightarrow> 'd" shows "mk\<^sub>e xa \<oplus>\<^sub>p x = mk\<^sub>e (\<lambda>b. xa (get\<^bsub>x\<^esub> b))"
that_6: fixes \<Gamma>' :: "char list \<Rightarrow> secLevel option"   and T :: "secLevel"   and c1 :: "com"   and c2 :: "com" assumes "x = \<Gamma>'"   and "xa = T"   and "xb = c1;; c2"   and "\<Gamma>',T \<turnstile> c1"   and "\<Gamma>',T \<turnstile> c2" shows "thesis"
STEP: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "I s" shows "if bval b s then wp \<pi> c (\<lambda>s'. I s' \<and> (s', s) \<in> R) s else Q s"
wide_subdg_Obj: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
ExtCl_def: shows "ExtCl = \<lparr>access = Public, cfields = [(vee, \<lparr>access = Public, static = False, type = PrimT Integer\<rparr>)], methods = [Ext_foo], init = Skip, super = Base, superIfs = []\<rparr>"
space_closed: shows "tail_events A \<subseteq> Pow (space M)"
types_preserved_block: fixes vs :: "e list"   and tn :: "t list"   and tm :: "t list"   and es :: "b_e list"   and m :: "nat"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and n :: "nat" assumes "\<lparr>vs @ [$Block (tn _> tm) es]\<rparr> \<leadsto> \<lparr>[Label m [] (vs @ ($* es))]\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> vs @ [$Block (tn _> tm) es] : ts _> ts'"   and "const_list vs"   and "length vs = n"   and "length tn = n"   and "length tm = m" shows "\<S>\<bullet>\<C> \<turnstile> [Label m [] (vs @ ($* es))] : ts _> ts'"
subelem_set_of_low: fixes x :: "ref"   and t :: "dag"   and low :: "ref \<Rightarrow> ref"   and p :: "ref"   and high :: "ref \<Rightarrow> ref" assumes "x \<in> set_of t"   and "x \<noteq> Null"   and "low x \<noteq> Null"   and "Dag p low high t" shows "low x \<in> set_of t"
Suc_lessI: fixes m :: "nat"   and n :: "nat" assumes "m < n"   and "\<not> Suc m = n" shows "Suc m < n"
frees_const: fixes name :: "name" shows "frees (const name) = {||}"
hidden_unmod: fixes Vs :: "char list list"   and i :: "nat"   and e :: "char list exp" assumes "hidden Vs i" shows "unmod (compE\<^sub>1 Vs e) i"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "LCons (xs $ n) (ldropn (Suc n) xs) = ldropn n xs"
min_add_distrib_right: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x + subset_mset.min y z = subset_mset.min (x + y) (x + z)"
independent_length_le: fixes Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K Us"   and "local.independent K Vs"   and "set Us \<subseteq> Span K Vs" shows "length Us \<le> length Vs"
sum_eq_0_iff: fixes F :: "'d set"   and f :: "'d \<Rightarrow> 'e multiset" assumes "finite F" shows "(comm_monoid_add.sum (+) {#} f F = {#}) = (\<forall>a\<in>F. f a = {#})"
integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "a \<le> c"   and "c \<le> b"   and "f integrable_on {a..b}" shows "integral {a..c} f + integral {c..b} f = integral {a..b} f"
hyps_7: shows "e\<^sub>2 \<Down> e\<^sub>2'"
col_conjugate: fixes i :: "nat"   and m :: "'a mat" assumes "i < dim_col m" shows "col (conjugate m) i = conjugate (col m i)"
norm_blinfun_euclidean_le: fixes a :: "'d \<Rightarrow>\<^sub>L 'e" shows "norm a \<le> (\<Sum>x\<in>Basis. norm (blinfun_apply a x))"
reachable_induct: fixes m :: "'a"   and y :: "'b"   and s :: "'a process_name \<Rightarrow> ('c, 'a, 'd, 'b) Model.local_state"   and P :: "'b \<Rightarrow> bool" assumes "mut_m.reachable m y s"   and "\<And>x. x \<in> roots (s (mutator m)) \<Longrightarrow> P x"   and "\<And>x. x \<in> ghost_honorary_root (s (mutator m)) \<Longrightarrow> P x"   and "\<And>x. x \<in> mut_m.tso_store_refs m s \<Longrightarrow> P x"   and "\<And>x y. \<lbrakk>mut_m.reachable m x s; (x points_to y) s; P x\<rbrakk> \<Longrightarrow> P y" shows "P y"
strong_cases: fixes a1 :: "trm"   and a2 :: "trm"   and x :: "vrs"   and X :: "tyvrs" assumes "a1 \<longmapsto> a2" obtains v\<^sub>2 :: "trm" and T\<^sub>11 :: "ty" and t\<^sub>12 :: "trm" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = (\<lambda>x:T\<^sub>11a__. t\<^sub>12__) \<cdot> v\<^sub>2 \<and> a2 = t\<^sub>12__[x \<mapsto> v\<^sub>2] \<and> x \<sharp> v\<^sub>2 \<and> val v\<^sub>2"   | t :: "trm" and t' :: "trm" and u :: "trm" where "a1 = ta__ \<cdot> u" and "a2 = t'__ \<cdot> u" and "ta__ \<longmapsto> t'__"   | v :: "trm" and t :: "trm" and t' :: "trm" where "a1 = v \<cdot> tb__" and "a2 = v \<cdot> t'a__" and "val v" and "tb__ \<longmapsto> t'a__"   | T\<^sub>11 :: "ty" and T\<^sub>2 :: "ty" and t\<^sub>12 :: "trm" where "\<lbrakk>X \<sharp> a1; X \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = (\<lambda>X<:T\<^sub>11. t\<^sub>12) \<cdot>\<^sub>\<tau> T\<^sub>2 \<and> a2 = t\<^sub>12[X \<mapsto>\<^sub>\<tau> T\<^sub>2] \<and> X \<sharp> (T\<^sub>11, T\<^sub>2)"   | t :: "trm" and t' :: "trm" and T :: "ty" where "a1 = t \<cdot>\<^sub>\<tau> T" and "a2 = t' \<cdot>\<^sub>\<tau> T" and "t \<longmapsto> t'"
soundness_mat_leqb_eqc2: fixes A :: "rat mat"   and c :: "rat vec"   and b :: "rat vec"   and X :: "(nat, rat) mapping" assumes "dim_row A\<^sup>T = dim_vec c"   and "dim_col A\<^sup>T = dim_vec b"   and "simplex (mat_leqb_eqc A b c) = Inr X" shows "\<exists>y. [y\<^sub>v*A]=c"
inf_sup_aci_1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
Dag_Null: fixes l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" shows "Dag Null l r t = (t = Tip)"
subgroup_int_pow_closed: fixes H :: "'c set"   and h :: "'c"   and k :: "int" assumes "subgroup H (H\<lparr>carrier := generate H {b}\<rparr>)"   and "h \<in> H" shows "h [^]\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub> k \<in> H"
swap_def: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'e" shows "Fun.swap a b f = f(a := f b, b := f a)"
f_inv: shows "\<forall>x\<in>carrier K - {\<zero>}. \<exists>x'\<in>carrier K. x' \<cdot>\<^sub>r x = 1\<^sub>r"
Allowed_totalize: fixes F :: "'c program" shows "Allowed (totalize F) = Allowed F"
has_finite_limits: fixes J :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "category J"   and "finite (Collect (partial_magma.arr J))" shows "has_limits_of_shape J"
ntr_is_tr_s: fixes c :: "'a list multiset"   and w :: "('b, 'c) label list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "(c, w, c') \<in> ntr fg" shows "(c, w, c') \<in> trcl (tr fg)"
simps_13: fixes sh :: "sheap"   and V :: "'a"   and T :: "ty"   and e :: "'a exp" shows "iconf sh {V:T; e} = iconf sh e"
tvs_def_3: shows "\<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "rep_list f \<noteq> 0"   and "rep_list p \<noteq> 0"   and "punit.lt (rep_list f) adds punit.lt (rep_list p)"   and "sing_reg (punit.lt (rep_list p) \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)" shows "thesis"
le_sup_cong: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a \<squnion> c \<le> b \<squnion> c"
interchange_columns_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and m :: "'e" shows "interchange_columns A n m = (\<chi>i j. if j = n then A $ i $ m else if j = m then A $ i $ n else A $ i $ j)"
Eapp: fixes e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "val list" shows "E (EApp e1 e2) \<rho> = apply_fun (E e1 \<rho>) (E e2 \<rho>)"
BVG_ODE_other: fixes y :: "variable"   and x :: "char"   and \<theta> :: "trm" assumes "y \<noteq> RVar x"   and "y \<noteq> DVar x" shows "y \<notin> BVG (ODE x \<theta>)"
F1_def: shows "F1 = (\<lambda>x. c / (a - x) ^ (n - 1) / real (n - 1))"
insert_ops_rem_last: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" assumes "insert_ops (xs @ [x])" shows "insert_ops xs"
below_monic_pair_split: fixes f :: "'a \<rightarrow> 'b"   and g :: "'c \<rightarrow> 'd" assumes "below_monic_cfun f"   and "below_monic_cfun g" shows "below_monic_cfun (\<Lambda> x. (f\<cdot>(fst x), g\<cdot>(snd x)))"
step: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "pfinite_gpv \<I>' (callee s x)"
path_split: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a"   and pth1 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list"   and e :: "('a \<times> 'b) \<times> 'a \<times> 'c"   and pth2 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" shows "Pre_Port_Graph.path edges v v' (pth1 @ [e] @ pth2) = (Pre_Port_Graph.path edges v (Pre_Port_Graph.edge_end e) (pth1 @ [e]) \<and> Pre_Port_Graph.path edges (Pre_Port_Graph.edge_end e) v' pth2)"
cs_implies_cbelow_on: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "relation_order r ord A"   and "relation.dw_closed r A"   and "relation.cs r a b"   and "a \<in> A"   and "b \<in> A"   and "ord a c"   and "ord b c" shows "cbelow_on A ord c (\<lambda>a b. r a b \<or> r b a) a b"
ln_upper_15_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_15 x"
Legendre_def: fixes a :: "int"   and p :: "int" shows "Legendre a p = (if [a = 0] (mod p) then 0 else if QuadRes p a then 1 else - 1)"
measurable_measure_Diff: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> fmeasurable M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
step: fixes xs :: "'a llist" assumes "X xs"   and "\<not> lnull xs" shows "lhd xs \<notin> lset (ltl xs) \<and> (X (ltl xs) \<or> ldistinct (ltl xs))"
vector_restrict_comp_conv: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> top = x" shows "x \<sqinter> y \<le> id x \<sqinter> y"
ev_wait_one_respects_policy: fixes tid :: "thread_id_t"   and u :: "partition_id_t"   and s :: "state_t"   and ipt :: "int_point_t"   and ev_wait_stage :: "ev_wait_stage_t" assumes "\<not> Policy.ifp (Step.partition tid) u"   and "atomic_step_invariant s"   and "atomic_step_precondition s tid ipt"   and "ipt = SK_EV_WAIT ev_wait_stage EV_CONSUME_ONE" shows "vpeq u s (atomic_step_ev_wait_one tid s)"
Borsukian_continuous_logarithm_circle: fixes S :: "'a set" shows "Borsukian S = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere 0 1 \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (g x))))"
deg': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "deg' x = deg_pp (rep_nat_pp x)"
in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" obtains "Cop.arr f" and "Cop.dom f = a" and "Cop.cod f = b"
dvd_gcdD2: 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"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (gcd b c)" shows "dvd.dvd times a c"
symp_combine_per_symp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "symp R" shows "symp (combine_per P R)"
gt'_irrefl: fixes z :: "'a" shows "\<not> gt' z z"
of_real_inverse: fixes x :: "real" shows "of_real (inverse x) = inverse (of_real x)"
injective: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set" assumes "injective_map \<alpha> S T" shows "inj_on \<alpha> S"
matrix_mul_lid: fixes A :: "(('a, 'b) vec, 'c) vec" shows "mat (1::'a) ** A = A"
divide_pos_pos: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less zero y" shows "less zero (divide x y)"
fmdrop_fset_null: fixes m :: "('a, 'b) fmap" shows "fmdrop_fset {||} m = m"
gga_from_list_def: fixes e :: "unit \<Rightarrow> 'a"   and a :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and u :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "'b list \<times> ('b \<times> 'c \<times> 'b) list" shows "gga_from_list e a u l \<equiv> let (nl, el) = l; g1 = foldl (\<lambda>g v. a v g) (e ()) nl in foldl (\<lambda>g (v, e, v'). u v e v' g) g1 el"
funion_commute: fixes A :: "'m fset"   and B :: "'m fset" shows "A |\<union>| B = B |\<union>| A"
A: shows "\<forall>zps'. length zps' < length zps \<longrightarrow> (\<forall>Z'. (zps', Z') \<in> futures (c_process step out s\<^sub>0) xps \<longrightarrow> (yp # ipurge_tr I (c_dom D) (c_dom D yp) zps', ipurge_ref I (c_dom D) (c_dom D yp) zps' Z') \<in> futures (c_process step out s\<^sub>0) xps)"
assms_1: shows "subterm_inj_on \<theta> (subst_domain \<theta>)"
domI: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b" assumes "m a = \<lfloor>b\<rfloor>" shows "a \<in> dom m"
assms_1: shows "(u \<longlongrightarrow> ennreal l) F"
ind_sheaf_def: fixes V :: "'b set" shows "local.ind_sheaf V \<equiv> \<O>\<^sub>X (X \<inter> V)"
step_3: shows "S' -T\<triangleright>TA\<rightarrow> S''"
F_def: shows "F = (\<lambda>i x. f ((T ^^ i) x))"
pw_leof_iff: fixes m :: "'e nres"   and m' :: "'e nres" shows "(m \<le>\<^sub>n m') = (nofail m \<longrightarrow> (\<forall>x. inres m x \<longrightarrow> inres m' x))"
intro: fixes k :: "nat set"   and c :: "nat \<times> nat \<Rightarrow> 'a" assumes "k \<subseteq> Graph.V c" shows "Cut c k"
OK_None_bot: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a option err" shows "OK None \<sqsubseteq>\<^bsub>Err.le (Opt.le r)\<^esub> x"
related: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b set"   and h :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "comm_monoid_set f z"   and "R z z"   and "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (f x1 y1) (f x2 y2)"   and "finite S"   and "\<forall>x\<in>S. R (h x) (g x)" shows "R (comm_monoid_set.F f z h S) (comm_monoid_set.F f z g S)"
maintain_upper_bound: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and D :: "'b \<Rightarrow> enat"   and S :: "'b set"   and u :: "'b"   and v :: "'b" assumes "Dijkstra_Invar w s D S" shows "WGraph.\<delta> w s u \<le> Dijkstra.relax_outgoing w v D u"
collinear_midpoint: fixes a :: "'b"   and b :: "'b" shows "collinear {a, midpoint a b, b}"
ipurge_ref_eq: fixes D :: "'b \<Rightarrow> 'a"   and x :: "'b"   and I :: "('a \<times> 'a) set"   and u :: "'a"   and xs :: "'b list"   and X :: "'b set" assumes "D x \<in> sinks I D u (xs @ [x])" shows "ipurge_ref I D u (xs @ [x]) X = ipurge_ref I D u xs {x' \<in> X. (D x, D x') \<notin> I}"
recv_stutter: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and m :: "'c" assumes "\<forall>j. U (\<sigma> j) (\<sigma>' j)"   and "\<sigma>' i = \<sigma> i" shows "P (\<sigma>, receive m, \<sigma>')"
distinct_15: fixes x2 :: "'b"   and x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PVar x2 \<noteq> PFn x41 x42 x43"
bounded: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "\<exists>K. \<forall>x. norm_class.norm (f x) \<le> norm_class.norm x * K"
card_of_Sigma_ordLeq_infinite_Field: fixes r :: "('c \<times> 'c) set"   and I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "infinite (Field r)"   and "Card_order r"   and "|I| \<le>o r"   and "\<forall>i\<in>I. |A i| \<le>o r" shows "|Sigma I A| \<le>o r"
qtable_empty_iff: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool" shows "qtable n A P Q empty_table = (\<forall>x. wf_tuple n A x \<longrightarrow> P x \<longrightarrow> Q x \<longrightarrow> False)"
quasi_inverse_hcomp': fixes f :: "'a"   and f' :: "'a"   and g :: "'a"   and g' :: "'a" assumes "equivalence_map f"   and "equivalence_map f'"   and "equivalence_map (f \<star> f')"   and "quasi_inverses f g"   and "quasi_inverses f' g'" shows "quasi_inverses (f \<star> f') (g' \<star> g)"
finite_keys: fixes f :: "'b \<Rightarrow>\<^sub>0 'c" shows "finite (keys f)"
advantage_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.advantage \<G> \<A> = \<bar>spmf (ddh.ddh_0 \<G> \<A>) True - spmf (ddh.ddh_1 \<G> \<A>) True\<bar>"
select_convs_4: fixes g_V :: "'d set"   and g_E :: "('d \<times> 'd) set"   and g_V0 :: "'d set"   and more :: "'b" shows "graph_rec.more \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = more"
dgrad_set_le_Un: fixes d :: "'b \<Rightarrow> nat"   and S :: "'b set"   and T :: "'b set"   and U :: "'b set" shows "dgrad_set_le d (S \<union> T) U = (dgrad_set_le d S U \<and> dgrad_set_le d T U)"
wfvarsoccs\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfvarsoccs\<^sub>s\<^sub>t\<^sub>p S))"
prems: shows "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None))"
p_pp_comp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> (\<ominus> \<ominus> x \<odot> y) = \<ominus> (x \<odot> y)"
set_bts_dfs_rev: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and ts :: "('d, 'e) bintree list" shows "set (bts_dfs f (rev ts)) = set (bts_dfs f ts)"
interval_upperbound_def: fixes s :: "'b set" shows "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x\<in>s. x \<bullet> i) *\<^sub>R i)"
filter_merge: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "Sorting_Algorithms.sorted cmp xs"   and "Sorting_Algorithms.sorted cmp ys" shows "filter P (local.merge xs ys) = local.merge (filter P xs) (filter P ys)"
all_relative_to: fixes P :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and Q :: "'c set \<Rightarrow> bool" shows "(\<forall>S. (P relative_to U) S \<longrightarrow> Q S) = (\<forall>S. P S \<longrightarrow> Q (U \<inter> S))"
from_nat_neq_cols: fixes i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "i < ncols A"   and "k < ncols A"   and "i \<noteq> k" shows "from_nat i \<noteq> from_nat k"
order_root: fixes p :: "'a poly"   and a :: "'a" shows "(poly p a = (0::'a)) = (p = 0 \<or> Polynomial.order a p \<noteq> 0)"
sup_state_opt_any_Some: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and s :: "ty list \<times> ty err list"   and X :: "(ty list \<times> ty err list) option" shows "P \<turnstile> \<lfloor>s\<rfloor> \<le>' X = (\<exists>s'. X = \<lfloor>s'\<rfloor> \<and> P \<turnstile> s \<le>\<^sub>i s')"
mwb_lens_weak: fixes x :: "'b \<Longrightarrow> 'c" assumes "mwb_lens x" shows "weak_lens x"