hcju/isabelleps huggingface dataset:
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "lattice L"   and "weak_partial_order_bottom L"   and "weak_partial_order_top L" shows "bounded_lattice L"
dec_interp_enc_Inr: fixes n :: "nat"   and FO :: "nat set"   and w :: "'a list"   and I :: "(nat + nat set) list"   and i :: "nat"   and P' :: "nat set"   and P :: "nat set" assumes "dec_interp n FO (enc (w, I)) ! i = Inr P'"   and "I ! i = Inr P"   and "i \<notin> FO"   and "i < n"   and "length I = n"   and "\<forall>p\<in>P. p < length w" shows "P = P'"
simps_3: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and v :: "nat" shows "C M2 M1 \<Omega> V m (Suc (Suc v)) = append_set (C M2 M1 \<Omega> V m (Suc v) - RM M2 M1 \<Omega> V m (Suc v)) (inputs M2) - TS M2 M1 \<Omega> V m (Suc v)"
Q: shows "peak ars (\<gamma>1'', \<gamma>2)"
rep_eq: fixes x :: "uint16"   and xa :: "uint16" shows "Rep_uint16 (x mod xa) = Rep_uint16 x mod Rep_uint16 xa"
sum_set_simps_4: fixes x :: "'c" shows "setr (Inr x) = {x}"
simps_4: fixes \<Gamma> :: "name list"   and x :: "name"   and t :: "nterm" shows "nterm_to_term \<Gamma> (\<Lambda>\<^sub>n x. t) = \<Lambda> nterm_to_term (x # \<Gamma>) t"
insert3_insert2: fixes n :: "nat"   and t :: "'a tree"   and x :: "'a" assumes "bal_i n (height t)" shows "insert3 x (t, n) = insert2 x (t, n)"
extend_map_affine_to_sphere_cofinite: fixes S :: "'a set"   and T :: "'a set"   and r :: "real"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and L :: "'a set" assumes "compact S"   and "affine T"   and "S \<subseteq> T"   and "aff_dim T \<le> int DIM('b)"   and "0 \<le> r"   and "continuous_on S f"   and "f ` S \<subseteq> sphere a r"   and "\<And>C. \<lbrakk>C \<in> components (T - S); bounded C\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}" obtains K :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite K" and "K \<subseteq> L" and "K \<subseteq> T" and "disjnt K S" and "continuous_on (T - K) g" and "g ` (T - K) \<subseteq> sphere a r" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
size_23: fixes x6 :: "Formula.formula" shows "size (formula.Neg x6) = size x6 + Suc 0"
Inr_Inl_False: fixes x :: "'d"   and y :: "'c" shows "(Inr x = Inl y) = False"
Lp_measurable: fixes f :: "'b \<Rightarrow> real"   and p :: "ennreal"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> p M)" shows "f \<in> borel_measurable M"
continuous_map_compose: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and X'' :: "'c topology"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_map X X' f"   and "continuous_map X' X'' g" shows "continuous_map X X'' (g \<circ> f)"
process_mru_is_None: fixes v_f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and a :: "'c" shows "(process_mru v_f a = None) = (vote_set v_f {a} = {})"
c_idem: shows "1\<^sub>\<pi> \<cdot> 1\<^sub>\<pi> = 1\<^sub>\<pi>"
union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "prod_mset (M + N) = prod_mset M * prod_mset N"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_has_all_failures_Un: fixes P :: "'b process" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F P"
lnth_0_conv_lhd: fixes xs :: "'a llist" assumes "\<not> lnull xs" shows "lnth xs 0 = lhd xs"
GFP_least: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> carrier L"   and "\<And>u. \<lbrakk>u \<in> carrier L; u \<sqsubseteq> f u\<rbrakk> \<Longrightarrow> u \<sqsubseteq> x" shows "GFP f \<sqsubseteq> x"
fps_of_poly_mult: fixes p :: "'c poly"   and q :: "'c poly" shows "fps_of_poly (p * q) = fps_of_poly p * fps_of_poly q"
h4a1: shows "isolate_variable_sparse p x (i + 1) \<noteq> 0 \<longrightarrow> MPoly_Type.degree (isolate_variable_sparse p x (i + 1) * (Var x ^ i * Const (real (i + 1)))) x = MPoly_Type.degree (isolate_variable_sparse p x (i + 1)) x + MPoly_Type.degree (Const (real (i + 1)) * Var x ^ i) x"
compat_ioas_def: fixes ioa1 :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and ioa2 :: "('d set \<times> 'd set \<times> 'd set) \<times> 'f set \<times> ('f \<times> 'd \<times> 'f) set" shows "compat_ioas ioa1 ioa2 \<equiv> compat_asigs (asig_of ioa1) (asig_of ioa2)"
cases: fixes a :: "'b set" assumes "finite a" obtains "a = {}"   | A :: "'b set" and aa :: "'b" where "a = insert aa A" and "finite A"
abs_square_le_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times x 2) one = less_eq (abs x) one"
eq_span_insert_eq: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "x - y \<in> More_Modules.ideal S" shows "More_Modules.ideal (insert x S) = More_Modules.ideal (insert y S)"
not_col_permutation_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col B C A"
contour_integrable_holomorphic: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and k :: "complex set"   and g :: "real \<Rightarrow> complex" assumes "continuous_on S f"   and "open S"   and "finite k"   and "valid_path g"   and "path_image g \<subseteq> S"   and "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x" shows "f contour_integrable_on g"
wset_init_fin_lift_state: fixes s :: "status"   and \<sigma> :: "('c, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'c \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'b wait_set_status option) \<times> 'a set" shows "wset (init_fin_lift_state s \<sigma>) = wset \<sigma>"
is_full_pmdlD: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_full_pmdl B"   and "component_of_term ` keys p \<subseteq> component_of_term ` Keys B" shows "p \<in> pmdl B"
hiding_def: fixes P :: "'b process"   and A :: "'b set" shows "P \ A \<equiv> Abs_process ({(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}, div_hide P A)"
Nats_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_1 one times plus zero"   and "a \<in> semiring_1.Nats one plus zero"   and "b \<in> semiring_1.Nats one plus zero" shows "plus a b \<in> semiring_1.Nats one plus zero"
diff_smult_distrib_vec: fixes x :: "'b"   and y :: "'b"   and v :: "'b vec" shows "(x - y) \<cdot>\<^sub>v v = x \<cdot>\<^sub>v v - y \<cdot>\<^sub>v v"
polyhedron_Int: fixes S :: "'b set"   and T :: "'b set" assumes "polyhedron S"   and "polyhedron T" shows "polyhedron (S \<inter> T)"
set_zmset_def: fixes M :: "'a zmultiset" shows "set_zmset M = {x. zcount M x \<noteq> 0}"
not_eq_order_implies_strict: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
tendsto_within_open_NO_MATCH: fixes a :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'c"   and l :: "'c" assumes "a \<in> S"   and "NO_MATCH UNIV S"   and "open S" shows "(f \<longlongrightarrow> l) (at a within S) = f \<midarrow>a\<rightarrow> l"
Chernoff_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "Chernoff_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. f B \<inter> C \<subseteq> f C)"
h2: shows "\<forall>i\<in>{p..n}. f i = 0"
pred_mono_strong: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b uprod"   and Pa :: "'b \<Rightarrow> bool" assumes "pred_uprod P x"   and "\<And>z. \<lbrakk>z \<in> set_uprod x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_uprod Pa x"
lsl_trans: shows "\<nu> \<circ> \<nu> \<le> \<nu>"
ereal_uminus_eq_reorder: fixes a :: "ereal"   and b :: "ereal" shows "(- a = b) = (a = - b)"
is_acc_def: fixes r :: "nat \<Rightarrow> 'v" shows "G'.is_acc r \<equiv> \<forall>A\<in>G'.F. \<exists>\<^sub>\<infinity>i. r i \<in> A"
s_prod_im: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<cdot> S = {(a, A). \<exists>B. (a, B) \<in> R \<and> (\<exists>f. (\<forall>b\<in>B. (b, f b) \<in> S) \<and> A = \<Union> (f ` B))}"
assms_1: shows "(x1, w1) \<in> Rel1"
range_eqI: fixes b :: "'h"   and f :: "'i \<Rightarrow> 'h"   and x :: "'i" assumes "b = f x" shows "b \<in> range f"
inv_one: shows "inv\<^bsub>G i\<^esub> \<one>\<^bsub>G i\<^esub> = \<one>\<^bsub>G i\<^esub>"
name_calc_1: fixes a :: "name"   and b :: "name"   and pi :: "(name \<times> name) list"   and x :: "name" shows "((a, b) # pi) \<bullet> x = swap (a, b) (pi \<bullet> x)"
lm024: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" assumes "f \<circ> g = id" shows "inj g"
zero_eq_power2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a 2 = zero) = (a = zero)"
X: shows "{x. \<not> token_succeeds x} = {x. \<not> token_succeeds x \<and> token_squats x} \<union> {x. \<not> token_succeeds x \<and> \<not> token_squats x}"
s_S: shows "S = {x \<in> I. \<nexists>n. I \<leftarrow> n = x}"
max_step_axioms: shows "max_step t (t0 + e) 1 B' (euler_C TYPE('a)) r"
components_are_iso: fixes a :: "'a" assumes "B.ide a" shows "B.iso (B.L a)"
identify_block_le': fixes A :: "'b mat"   and j :: "nat"   and i :: "nat" assumes "??.Jordan_Normal_Form_Existence.identify_block A j = i" shows "i \<le> j"
simps_1: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "asym.implInsec A B M = \<langle>InsecTag, Messages.Agent A, Messages.Agent B, M\<rangle>"
rotate_is_Nil_conv: fixes n :: "nat"   and xs :: "'b list" shows "(rotateLeft n xs = []) = (xs = [])"
simple_ruleset_append: fixes rs\<^sub>1 :: "'b rule list"   and rs\<^sub>2 :: "'b rule list" shows "simple_ruleset (rs\<^sub>1 @ rs\<^sub>2) = (simple_ruleset rs\<^sub>1 \<and> simple_ruleset rs\<^sub>2)"
fT_3: shows "\<Gamma> t' = Fun f (map \<Gamma> T)"
field_inverse_zero: shows "inverse (0::'c) = (0::'c)"
LI_preserves_subst_wf_4: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>* (S\<^sub>2, \<theta>\<^sub>2)"   and "fv\<^sub>s\<^sub>t S\<^sub>1 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>1 = {}"   and "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>\<^sub>1"   and "subst_domain \<theta>\<^sub>1 \<inter> vars\<^sub>s\<^sub>t S\<^sub>1 = {}"   and "range_vars \<theta>\<^sub>1 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>1 = {}" shows "range_vars \<theta>\<^sub>2 \<inter> bvars\<^sub>s\<^sub>t S\<^sub>2 = {}"
inf_Sup: fixes a :: "'a"   and B :: "'a set" shows "a \<sqinter> \<Squnion> B = \<Squnion> ((\<sqinter>) a ` B)"
table_eq_rel: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm"   and n :: "nat" assumes "is_simple_eq t1 t2" shows "table n (fv_trm t1 \<union> fv_trm t2) (eq_rel n t1 t2)"
lm078: fixes x :: "'c"   and y :: "'d" shows "{x} \<times> {y} = {(x, y)}"
subsumes_accessible_globally_def: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "\<A>\<^sub>G s = (\<forall>r\<^sub>1 r\<^sub>2. r\<^sub>1 \<in> dom s \<longrightarrow> r\<^sub>2 \<in> dom s \<longrightarrow> \<A> r\<^sub>1 r\<^sub>2 s)"
mat_of_cols_carrier_2: fixes n :: "nat"   and vs :: "'b vec list" shows "dim_row (mat_of_cols n vs) = n"
that: assumes "P,t \<turnstile> \<sigma> -\<lbrace>\<rbrace>-jvm\<rightarrow> \<sigma>'"   and "check P \<sigma>"   and "\<tau>Move2 P \<sigma>" shows "thesis"
\<tau>move2BlockSome1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and T :: "ty"   and v :: "'addr val"   and e :: "'addr expr1" shows "\<tau>move2 P h stk {V:T=\<lfloor>v\<rfloor>; e} 0 None"
shunting: fixes X :: "'b program set"   and Y :: "'b program set" shows "X guarantees Y = UNIV guarantees - X \<union> Y"
index_smult_mat_2: fixes a :: "'a"   and A :: "'a Matrix.mat" shows "dim_row (a \<cdot>\<^sub>m A) = dim_row A"
G_reflects_isomorphic: fixes f :: "'c"   and f' :: "'c"   and \<psi> :: "'d" assumes "C.ide f"   and "C.ide f'"   and "src\<^sub>C f = src\<^sub>C f'"   and "trg\<^sub>C f = trg\<^sub>C f'"   and "\<guillemotleft>\<psi> : G f \<cong>\<^sub>D G f'\<guillemotright>" shows "\<guillemotleft>\<Phi> \<psi> f f' : f \<cong>\<^sub>C f'\<guillemotright>"
ind_zero_str_def: fixes V :: "'a set" shows "ind_zero_str V \<equiv> \<zero>\<^bsub>U \<inter> V\<^esub>"
getRel_def: fixes l :: "'d"   and G :: "('d, 'c) labeled_graph" shows "getRel l G = {(x, y). (l, x, y) \<in> edges G}"
ts_conc: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c |\<in>| Abstract_Task.conc_forms conclusions" shows "snd (fst (tree.root (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c))) = c"
B: shows "(\<forall>za\<in>ball 0 e - {0}. cmod (f (z + complex_of_real r * za) / a) \<le> B) \<or> (\<forall>za\<in>ball 0 e - {0}. B \<le> cmod (f (z + complex_of_real r * za) / a))"
assms: shows "A \<in> set (subseqs (unlabel B))"
distinct_find_base_vectors: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "distinct (find_base_vectors A)"
span_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "subspace T" shows "span S \<subseteq> T"
equal_if_bisimilar: fixes K :: "'a list set"   and L :: "'a list set" assumes "bisimilar K L" shows "K = L"
sums_emeasure: fixes F :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "disjoint_family F"   and "\<And>i. F i \<in> sets M" shows "(\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union> (range F))"
Uniq_I: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
reduced_letter_set_subset: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero" shows "order.greater_eq A (\<Union> (set ` Collect (monoid_add.reduced_word_for plus zero A a)))"
atMost_iff: fixes i :: "'b"   and k :: "'b" shows "(i \<in> {..k}) = (i \<le> k)"
gcd_proj1_if_dvd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a b" shows "gcd a b = normalize a"
TID_sym: fixes x :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "turn_into_distance extended_predist x y = turn_into_distance extended_predist y x"
stake_invert_Nil: fixes n :: "nat"   and s :: "'b stream" shows "(stake n s = []) = (n = 0)"
sequence_result_Rval: fixes rs :: "'a list" shows "sequence_result (map Rval rs) = Rval rs"
sum_in_Ints: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<int>" shows "sum f A \<in> \<int>"
degree_aform_err_sqrt_aform_err: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "sqrt_aform_err p X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
N_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "N f \<equiv> adjoint f \<cdot> DRA_Models.bot \<sqinter> id"
aff_dim_convex_Int_open: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "open T"   and "S \<inter> T \<noteq> {}" shows "aff_dim (S \<inter> T) = aff_dim S"
eqpoll_sym: fixes A :: "'b set"   and B :: "'c set" assumes "A \<approx> B" shows "B \<approx> A"
simps_1: fixes i :: "'b"   and j :: "'b"   and cs :: "'b list" shows "cycle_of_list (i # j # cs) = Fun.swap i j id \<circ> cycle_of_list (j # cs)"
f2: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real" assumes "Table0 f1 f2 f1' f2' e c" shows "0 < f2"
empty_dom_iff_empty_subst: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "(subst_domain \<theta> = {}) = (\<theta> = Var)"
F_finite_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq" shows "d_OUT f x \<noteq> \<top>"
dist_triangle_lt: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and z :: "'c"   and y :: "'c"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z < e" shows "dist x y < e"
assert_s_def: fixes a :: "'i \<times> 'a atom"   and s :: "('i, 'a) state" shows "assert_s a s = (let s' = assert_bound a s in if \<U> s' then Inl (the (\<U>\<^sub>c s')) else Inr s')"
map_definedness_contra: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a"   and z :: "'b" assumes "f x = Some y"   and "z \<notin> dom f" shows "x \<noteq> z"
simps_2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r1.cond_action_ok s t Yield = True"
monomial_decomp_concat: fixes pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list" assumes "\<And>ps. ps \<in> set pss \<Longrightarrow> monomial_decomp ps" shows "monomial_decomp (concat pss)"
vcomp_VLambda_vid_on: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda A f \<circ>\<^sub>\<circ> vid_on A = VLambda A f"
l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> (known_ptr = ShadowRootClass.known_ptr \<and> type_wf = ShadowRootClass.type_wf) \<and> get_child_nodes = Shadow_DOM.get_child_nodes \<and> get_child_nodes_locs = Shadow_DOM.get_child_nodes_locs"
cindex_polyE_mod: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_polyE a b q p = cindex_polyE a b (q mod p) p"
some_gcd_dvd_gcd: fixes a :: "'a"   and b :: "'a" shows "some_gcd a b dvd gcd a b"
Y: shows "(yp # yps, Y) \<in> futures (c_process step out s\<^sub>0) xps"
assms_4: shows "mat_det (a, b, cnj b, cnj a) \<noteq> 0"
Step_DEAxiomSchema: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and ODE :: "('sf, 'sz) ODE"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, DEAxiomSchema ODE \<sigma>) = (closeI SG i, C)"
list_graph_correct_5: fixes G :: "'c list_graph"   and E :: "('c \<times> 'c) list" shows "FiniteGraph.delete_edges (list_graph_to_graph G) (set E) = list_graph_to_graph (FiniteListGraph.delete_edges G E)"
word_plus_one_nonzero: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> x + y"   and "y \<noteq> 0" shows "x + 1 \<noteq> 0"
index_mat_addrow_2: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_addrow_gen ad mul a i l A $$ (i, j) = ad (mul a (A $$ (l, j))) (A $$ (i, j))"
count_correct: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.count t = length (FingerTreeStruc.toList t)"
ldropWhile_LCons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b llist" shows "ldropWhile P (LCons x xs) = (if P x then ldropWhile P xs else LCons x xs)"
decomposable_function_subst: fixes f :: "'c ltl \<Rightarrow> 'c ltl"   and \<phi> :: "'c ltl" assumes "f true = true"   and "f false = false"   and "\<And>\<phi> \<psi>. f (\<phi> and \<psi>) = f \<phi> and f \<psi>"   and "\<And>\<phi> \<psi>. f (\<phi> or \<psi>) = f \<phi> or f \<psi>" shows "f \<phi> = subst \<phi> (\<lambda>\<chi>. Some (f \<chi>))"
translation: fixes G :: "('b, 'c) labeled_graph"   and x :: "'c"   and y :: "'c"   and e :: "'b allegorical_term" assumes "graph G" shows "((x, y) \<in> :G:\<lbrakk>e\<rbrakk>) = (\<exists>f. graph_homomorphism (translation e) G f \<and> (0, x) \<in> f \<and> (1, y) \<in> f)"
kleene_fixed_point: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and k :: "nat" assumes "\<psi> \<in> \<P>\<^sup>2" obtains i :: "nat" where "k \<le> i" and "\<phi> i = \<psi> i"
subset_psubset_trans: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "A \<subseteq> B"   and "B \<subset> C" shows "A \<subset> C"
pwalk_verts_def: fixes v :: "'a" shows "pwalk_verts v = {u. u \<in> set (awalk_verts s (pwalk v))}"
satisfies_properties_sys1: shows "satisfies_properties p qs1 subsets1 signs1 matrix1"
strict_trans2: fixes a :: "nat list"   and b :: "nat list"   and c :: "nat list" assumes "a <\<^sub>v b"   and "b \<le>\<^sub>v c" shows "a <\<^sub>v c"
i_4: shows "\<forall>i\<in>insert x I. subgroup (S i) G"
TreeDecomposition_def: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" shows "TreeDecomposition G T bag \<equiv> Graph G \<and> Tree T \<and> TreeDecomposition_axioms G T bag"
true_alt_def: shows "true = \<guillemotleft>True\<guillemotright>"
dist_Pair_Pair: fixes a :: "'d"   and b :: "'e"   and c :: "'d"   and d :: "'e" shows "dist (a, b) (c, d) = sqrt ((dist a c)\<^sup>2 + (dist b d)\<^sup>2)"
fresh_at_snoc: fixes t :: "('form, 'rule, 'c, 'd) itnode rose_tree"   and "is" :: "nat list"   and i :: "nat" shows "fresh_at t (is @ [i]) = freshenLC (iAnnot (tree_at t is)) ` a_fresh (inPorts' (iNodeOf (tree_at t is)) ! i)"
ns_1: shows "g \<turnstile> Entry g-ns\<rightarrow>n"
assms_1: shows "\<forall>(j, p)\<in>(\<lambda>(t, s). (i, t, s)) ` set F' \<union> set D. \<forall>(k, q)\<in>(\<lambda>(t, s). (i, t, s)) ` set F' \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"