hcju/isabelleps huggingface dataset:
P: shows "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q \<or> P = fv_r\<^sub>e\<^sub>q ac"
semigroup_add_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semigroup_add plus \<equiv> \<forall>a b c. plus (plus a b) c = plus a (plus b c)"
fr_Send3: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "nReturn k t \<noteq> []" shows "nSend k t \<noteq> []"
disjoint_insert_2: fixes A :: "'c set"   and b :: "'c"   and B :: "'c set" shows "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
sorted_sequences: fixes key :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "\<forall>x\<in>set (sequences key xs). sorted (map key x)"
simps: fixes a :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" shows "(a \<in> p_e R M N) = (\<exists>Ps c \<Gamma> \<Delta>. a = extendRule ( M \<cdot> \<Gamma> \<Rightarrow>* N \<cdot> \<Delta>) (Ps, c) \<and> (Ps, c) \<in> R \<and> R \<subseteq> modRules2)"
cfun_arg_cong: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<rightarrow> 'b" assumes "x = y" shows "f\<cdot>x = f\<cdot>y"
run_snth: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd"   and k :: "nat" assumes "grun A w V r p" shows "case gtarget (stake k r) p of (ka, p) \<Rightarrow> w !! ka \<in> alphabet A \<and> r !! k \<in> {Suc ka} \<times> transition A (w !! ka) p \<inter> V"
subst_clss_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b literal multiset set"   and \<sigma> :: "'c" shows "substitution_ops.subst_clss subst_atm AA \<sigma> = (\<lambda>A. substitution_ops.subst_cls subst_atm A \<sigma>) ` AA"
convert_neg: fixes var :: "nat"   and F :: "atom fm"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and F' :: "atomUni fmUni" assumes "convert_fm var F (xs' @ x # xs) = Some F'"   and "length xs' = var" shows "eval (Neg F) (xs' @ x # xs) = evalUni (negUni F') x"
i_take_nth: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "i < n" shows "f \<Down> n ! i = f i"
1: shows "prv (imp (PP \<langle>neg \<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>)))"
zminus_minus_pos: fixes n :: "int" assumes "n < 0" shows "0 < - n"
ccSUP_sup_distrib: 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 A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "sup (Sup (f ` A)) (Sup (g ` A)) = Sup ((\<lambda>a. sup (f a) (g a)) ` A)"
contour_integrable_joinI: fixes f :: "complex \<Rightarrow> complex"   and g1 :: "real \<Rightarrow> complex"   and g2 :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g1"   and "f contour_integrable_on g2"   and "valid_path g1"   and "valid_path g2" shows "f contour_integrable_on g1 +++ g2"
simps: fixes a1 :: "'a"   and a2 :: "'a"   and succs :: "'a \<Rightarrow> 'a list" shows "(a1 [succs]\<rightarrow>* a2) = ((\<exists>g. a1 = g \<and> a2 = g) \<or> (\<exists>g g' g''. a1 = g \<and> a2 = g'' \<and> g' \<in> set (succs g) \<and> g' [succs]\<rightarrow>* g''))"
simps: fixes S :: "'d \<Rightarrow> bool"   and a :: "'d tree" shows "treesp S a = (a = \<langle>\<rangle> \<or> (\<exists>l r v. a = \<langle>l, v, r\<rangle> \<and> treesp S l \<and> treesp S r \<and> S v))"
generate_sincl: fixes A :: "'c set" shows "A \<subseteq> generate H A"
ordinal_rec0_oSuc: fixes z :: "'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"   and x :: "ordinal" shows "ordinal_rec0 z s l (oSuc x) = s x (ordinal_rec0 z s l x)"
aprimedivisor_primepow_power: fixes n :: "'a"   and k :: "nat" assumes "primepow n"   and "0 < k" shows "aprimedivisor (n ^ k) = aprimedivisor n"
cut_greater_Int_left: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>> t = A \<down>> t \<inter> B"
conjugate_multl: fixes y :: "'a" shows "conjugate (\<lambda>x. x \<cdot> y) = (\<lambda>x. x \<lhd> y)"
convergent_prod_iff_nz_lim: fixes f :: "nat \<Rightarrow> 'e" assumes "\<And>i. f i \<noteq> (0::'e)" shows "convergent_prod f = (\<exists>L. (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> L \<and> L \<noteq> (0::'e))"
seqSubstZero: fixes \<sigma> :: "(name \<times> name) list" shows "\<zero>[<\<sigma>>] = \<zero>"
map_idI: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "list.map f xs = xs"
g: shows "rec_exec rec_conf [code tp, bl2wc (<lm>), stp] = trpl_code (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
zero_uint64_def: shows "0 \<equiv> Abs_uint64 0"
iTILL_add_neg1: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "[\<dots>n] \<oplus>- k = [\<dots>n - k]"
variant_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "variant \<pi> b c s \<equiv> THE n. count_it \<pi> b c s n"
sum: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "(\<Sum>x\<in>A. f x *s x) = (0::'b)"   and "x \<in> A" shows "f x = (0::'a)"
IH: fixes As :: "'a set set" assumes "disjoint As" shows "ipPartition (set ts) (partitioning ts As)"
distinct_4: fixes x3 :: "num" shows "num.Bit1 x3 \<noteq> num.One"
thetaSSeqD_Sbis: shows "thetaSSeqD \<subseteq> Sbis"
inputFreeTrans: fixes a :: "name"   and x :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "a<x>.P \<longmapsto> \<alpha> \<prec> P'" shows "False"
nodes_of_component: fixes g :: "'b ugraph"   and r :: "'b" shows "nodes (component_of g r) = reachable_nodes g r"
adjacent_V_2: fixes v :: "'v"   and v' :: "'v" assumes "adjacent v v'" shows "v' \<in> V"
abs_eq: fixes x :: "'a \<times> 'a" assumes "fractrel x x" shows "- abs_fract x = abs_fract (- fst x, snd x)"
assms: shows "negligible (S - T \<union> (T - S))"
carry_def: fixes a :: "bool"   and b :: "bool"   and c :: "bool" shows "carry a b c = (a \<and> (b \<or> c) \<or> b \<and> c)"
lmeasurable_division: fixes \<D> :: "'a set set"   and S :: "'a set" assumes "\<D> division_of S" shows "S \<in> lmeasurable"
nontrivial_def: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "nontrivial Q \<equiv> \<exists>t t'. Q t t'"
SIod_carr_Ssegment: fixes D :: "'c Order"   and a :: "'c" assumes "Order D" shows "carrier (SIod D (Ssegment D a)) = Ssegment D a"
finite_accessible_states: fixes s :: "('i, 'a) state" shows "finite {s'. (s, s') \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*}"
sublens_pres_mwb: fixes Y :: "'a \<Longrightarrow> 'b"   and X :: "'c \<Longrightarrow> 'b" assumes "mwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "mwb_lens X"
b: assumes "x \<noteq> y"   and "rel x y" shows "filter P (merge_wrt rel xs (y # ys)) = filter P (y # ys)"
diff: shows "\<gamma> C1_differentiable_on {0..1} - S"
poly_cond2D_1: fixes p :: "int poly" assumes "poly_cond2 p" shows "poly_cond p"
prob0_leq_one: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" assumes "state n v" shows "prob0 n v i \<le> 1"
Rep_Store_inverse: fixes x :: "Store" shows "Abs_Store (Rep_Store x) = x"
psubstEnv_vsubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[y1 // y]_ys &[rho'] = rho &[rho' [y \<leftarrow> Var ys y1 #[rho']]_ys]"
StuckProp: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> =n\<Rightarrow> Stuck"
exp_divide_power_eq: fixes n :: "nat"   and x :: "'a" assumes "0 < n" shows "exp (x / of_nat n) ^ n = exp x"
ideI: fixes a :: "'a" assumes "\<guillemotleft>a : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"   and "ide a" shows "hom.ide a"
subst_atm_id_subst: fixes A :: "'a" shows "A \<cdot>a id_subst = A"
stableI: fixes F :: "'d program"   and A :: "'d set" assumes "F \<in> A co A" shows "F \<in> stable A"
b: shows "supp x = supp x - set as \<union> supp x \<inter> set as"
intros_6: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "(WHILE b DO c, s) \<Rightarrow> s"
l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs \<equiv> l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs \<and> l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
Abs_freelist_Cons: fixes x :: "'a"   and xs :: "'a list" shows "Abs_freelist (x # xs) = Abs_freeletter x + Abs_freelist xs"
contractible_imp_holomorphic_arccos: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "contractible S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> - 1" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (g z)"
minus_gt: fixes a :: "'a"   and b :: "'a" shows "(a < b) = ((0::'a) < b - a)"
edomIff: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "(a \<in> edom m) = (m a \<noteq> \<bottom>)"
hensel_seq_comp_f: shows "res_lim ((\<bullet>) f \<circ> ns) = \<zero>"
IH: shows "vars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<theta>) \<subseteq> vars\<^sub>s\<^sub>t S"
cut_less_Max_all: fixes I :: "'b set"   and t :: "'b" assumes "finite I"   and "Max I < t" shows "I \<down>< t = I"
totient_less: fixes n :: "nat" assumes "1 < n" shows "totient n < n"
w: shows "ws (action_translation (J (Suc n)) a'') = action_translation (J (Suc n)) w"
polymul_eq0_iff: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
subcocycle_lim_def: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_lim u = (\<lambda>x. real_of_ereal (subcocycle_lim_ereal u x))"
assms_4: fixes e :: "real" assumes "0 < e" shows "\<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < e"
trimming_\<E>: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "wave \<Gamma> f"   and "trimming \<Gamma> f g" shows "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g)"
l_get_dom_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_elements_by_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
beta_reduction_left_varE: fixes x :: "'a"   and M' :: "'a trm" assumes "Var x \<rightarrow>\<beta> M'" shows "False"
simps: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and F :: "atom fm"   and z :: "nat" shows "quadratic_sub_fm_helper var a b c d F z = liftmap (\<lambda>x. quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d)) F z"
pw_bind_inres: fixes M :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres" shows "inres (M \<bind> f) = (\<lambda>x. nofail M \<longrightarrow> (\<exists>y. inres M y \<and> inres (f y) x))"
eq_swap_word: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and p :: "'state" assumes "w\<^sub>1 =\<^sub>S w\<^sub>2"   and "path w\<^sub>1 p" shows "path w\<^sub>2 p"
distinct_129: fixes x5 :: "Syntax.term"   and x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term" shows "Inj1 x5 \<noteq> Syntax.Let x81 x82 x83"
itrm_brackets_def: shows "itrm_brackets = foldr_option itrm_bracket"
topspace_prod_topology: fixes X :: "'b topology"   and Y :: "'c topology" shows "topspace (prod_topology X Y) = topspace X \<times> topspace Y"
odd_plus1: fixes x :: "nat" assumes "ya_even x" shows "ya_odd (x + 1)"
daMk: fixes k :: "nat" assumes "k < n" shows "adjoint (M k) \<in> carrier_mat d d"
coplanar_perm_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet A C B D"
aref_def: fixes dynamic_array :: "'a dynamic_array" shows "aref dynamic_array = (case dynamic_array of Dyn_Array x1 x2 \<Rightarrow> x2)"
nat_pow_mult_of: fixes R :: "('b, 'c) ring_scheme" shows "([^]\<^bsub>mult_of R\<^esub>) = ([^]\<^bsub>R\<^esub>)"
inv_hom: fixes t :: "'t" assumes "inv' t" shows "inv (hom t)"
qp3: shows "is_quantum_predicate (adjoint (exexH_k (n - 1)) * (adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)) * exexH_k (n - 1))"
ConcAssoc: fixes A :: "((int \<times> int) set set, int) Combinators"   and B :: "((int \<times> int) set set, int) Combinators"   and D :: "((int \<times> int) set set, int) Combinators" shows "C ((A \<oplus> B) \<oplus> D) = C (A \<oplus> B \<oplus> D)"
conv_radius_altdef: fixes f :: "nat \<Rightarrow> 'b" shows "conv_radius f = liminf (\<lambda>n. inverse (ereal (root n (norm (f n)))))"
has_derivative_abs: fixes a :: "real" assumes "a \<noteq> 0" shows "(abs has_derivative (*) (sgn a)) (at a)"
binomial_coeffs_induct: fixes n :: "nat"   and k :: "nat" shows "(1 + fps_X) ^ n $ k = of_nat (n choose k)"
proj_stoch_range_singleton: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and r :: "'a stream"   and n :: "nat" assumes "borel_adapt_stoch_proc F X"   and "r \<in> range (proj_stoch_proc X n)" shows "\<exists>A\<in>sets (stream_space borel). range (proj_stoch_proc X n) \<inter> A = {r}"
z'_def: shows "z' \<equiv> getFr [out, inp, z, zz] [] [\<chi>, \<sigma>, \<tau>]"
hom_char: fixes f :: "'d"   and b :: "'d"   and a :: "'d" shows "Dop.in_hom f b a = \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"
disconnected_nodes_h3: shows "h3 \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r remove1 node disconnected_nodes_h2"
runit_commutes_with_R: fixes a :: "'a" assumes "ide a" shows "\<r>[a \<otimes> \<I>] = \<r>[a] \<otimes> \<I>"
prems: shows "prefix (h' # as') (rem_condless_act s [h] (a # as))"
per_union_def: fixes R :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" shows "per_union R a b \<equiv> R \<union> {(x, y). (x, a) \<in> R \<and> (y, b) \<in> R} \<union> {(y, x). (x, a) \<in> R \<and> (y, b) \<in> R}"
decide_exis_lem_helper_R: fixes fml_struct :: "nat fml"   and polys :: "real poly list"   and fml :: "real poly fml" assumes "(fml_struct, polys) = convert fml" shows "(\<exists>x. lookup_sem fml_struct (map (\<lambda>p. poly p x) polys)) = decide_existential_R fml"
homeomorphic_map_derived_set_of: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "homeomorphic_map X Y f"   and "S \<subseteq> topspace X" shows "Y derived_set_of f ` S = f ` (X derived_set_of S)"
intros: fixes v :: "hf"   and x :: "hf"   and ax :: "hf"   and y :: "hf"   and sx :: "hf" assumes "AbstForm v 0 x ax"   and "SubstForm v y x sx"   and "Form x"   and "is_Var v"   and "Term y" shows "q_Imp sx (q_Ex ax) \<in> Special_ax"
subresultant'_zero_ge: fixes f :: "'b poly"   and J :: "nat"   and g :: "'b poly"   and k :: "nat" assumes "degree f - J + (degree g - J) \<noteq> 0"   and "degree f + (degree g - J) \<le> k" shows "det (subresultant'_mat J k f g) = (0::'b)"
IH: fixes xa :: "'a Bonk_Schramm_extension_unfolded" assumes "xa \<in> range u" shows "map_aux xa \<in> Field wo_aux"
intv_CL_lub: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "\<forall>S\<subseteq>interval r a b. \<exists>L. isLub S \<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> L"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes obj :: "(_) Object" shows "cast obj = (case RObject.more obj of Inl x \<Rightarrow> Map.empty x | Inr (Inl document) \<Rightarrow> Some (RObject.extend (RObject.truncate obj) document) | Inr (Inr x) \<Rightarrow> Map.empty x)"
map_nth_Cons_atLeastLessThan: fixes x :: "'b"   and xs :: "'b list"   and m :: "nat"   and n :: "nat" shows "map ((!) (x # xs)) [Suc m..<n] = map ((!) xs) [m..<n - 1]"
atom_eq_iff: fixes a :: "'b"   and b :: "'b" shows "(atom a = atom b) = (a = b)"
tts_AB_transfer: fixes f :: "'b \<Rightarrow> 'a"   and \<UU>A :: "'b set"   and \<UU>B :: "'a set"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'd \<Rightarrow> bool" assumes "f ` \<UU>A \<subseteq> \<UU>B"   and "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B" shows "\<exists>rcdt. rel_fun A B f rcdt"
simps_2: fixes NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m" shows "NTCod \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTCod"
inv_partition_approx: fixes S :: "'a set"   and P :: "'a set set" assumes "inv_partition S V P" shows "approximation_miv \<Delta> S"
has_bochner_integral_indicator: fixes A :: "'c set"   and M :: "'c measure"   and c :: "'d" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "has_bochner_integral M (\<lambda>x. indicat_real A x *\<^sub>R c) (Sigma_Algebra.measure M A *\<^sub>R c)"
degree_C: fixes a :: "'b" shows "degree (C a) = 0"
mono_f: shows "\<forall>x\<in>#M. \<forall>y\<in>#N. x < y \<longrightarrow> f x < f y"
shift2_invD_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "shift2_inv qs" shows "standard_decomp k qs"
get_root_node_is_l_get_root_node: shows "l_get_root_node Shadow_DOM.get_root_node Shadow_DOM.get_parent"
square_mod_8_eq_1_iff: fixes x :: "nat" shows "[x\<^sup>2 = 1] (mod 8) = odd x"
dyn_ref_eq_dyn_ref_red: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "dynamically_complete_calculus Bot Inf entails Red_I Red_F = dynamically_complete_calculus Bot Inf entails (calculus.Red_Red_I Inf Red_I Red_F) Red_F"
Bernstein_changes_01_def: fixes p :: "nat"   and P :: "real poly" shows "Bernstein_changes_01 p P = int (nat (changes (Bernstein_coeffs_01 p P)))"
obj_ty_obj_class: fixes G :: "prog"   and obj :: "obj"   and statC :: "qtname" shows "G\<turnstile>obj_ty obj\<preceq>Class statC = G\<turnstile>obj_class obj\<preceq>\<^sub>C statC"
pwd_spy_iii_def: fixes n :: "nat" shows "pwd_spy_iii n \<equiv> case pwd_owner_iii n of (s, S, A, B, C, x) \<Rightarrow> (insert (Spy, \<lbrace>Num 4, PubKey S\<rbrace>) s, S, A, B, C, S)"
elims_3: fixes x :: "const"   and xa :: "const" assumes "\<not> x \<le> xa" obtains xb :: "const" and y :: "const" where "x = xb" and "xa = y" and "\<not> (y = Any \<or> xb = y)"
rrep_prrep: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:4..PRrep-:6} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> the (flag (rt \<xi>) (dip \<xi>)) = val)"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f tree" shows "map_tree g (map_tree f v) = map_tree (g \<circ> f) v"
alg_dvd_0_left_iff: fixes x :: "'b" shows "((0::'b) alg_dvd x) = (x = (0::'b))"
\<mu>_small_row_def: fixes n :: "nat"   and i :: "nat"   and fs :: "int vec list"   and j :: "nat" shows "LLL.\<mu>_small_row n i fs j = (\<forall>j'\<ge>j. j' < i \<longrightarrow> \<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j'\<bar> \<le> inverse 2)"
real_integrable_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity> \<and> \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>)"
valid_without_offending_flows: fixes F :: "('a \<times> 'a) set"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "F \<in> set_offending_flows G nP" shows "sinvar (delete_edges G F) nP"
intros_1: fixes b :: "nat" shows "C [] \<in> hbase b"
Null_staticD: fixes a' :: "val"   and m :: "'a member_scheme"   and e :: "expr" assumes "a' = Null \<longrightarrow> is_static m" shows "invmode m e = IntVir \<longrightarrow> a' \<noteq> Null"
inf_apply: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(f \<sqinter> g) x = f x \<sqinter> g x"
est: shows "TS t = \<lfloor>x\<rfloor>"
minim_iff_In_Under: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "(a = minim B) = (a \<in> B \<and> a \<in> local.Under B)"
brickcount_zero_implies_brick_is_cup: fixes x :: "brick" assumes "domain x = 0" shows "brick_is_cup x"
i2: shows "i < dim_row A + dim_row (0\<^sub>m 0 (dim_col B))"