hcju/isabelleps huggingface dataset:
one_closed: shows "\<one>\<^bsub>C\<^esub> \<in> carrier C"
lmeasurable_interval_1: fixes a :: "real"   and b :: "real" shows "{a..b} \<in> lmeasurable"
i\<^sub>\<UU>_def: shows "\<UU>.smallest_accepting_rank = Some i\<^sub>\<UU>"
norm_some_Basis: shows "norm (SOME i. i \<in> Basis) = 1"
eq_conv_compare: fixes eq :: "'d \<rightarrow> 'd \<rightarrow> tr"   and compare :: "'d \<rightarrow> 'd \<rightarrow> Ordering"   and x :: "'d"   and y :: "'d" assumes "class.Ord_linear eq compare" shows "eq\<cdot>x\<cdot>y = is_EQ\<cdot>(compare\<cdot>x\<cdot>y)"
indspaceD_supp: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> indspace G FH V" shows "supp f \<subseteq> FH"
logic_actual_nec_3: fixes \<phi> :: "'e \<Rightarrow> \<o>" shows "[[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha>)]]"
row_mat: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "i < nr" shows "row (mat nr nc f) i = vec nc (\<lambda>j. f (i, j))"
outer_measure_space_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" shows "outer_measure_space M f = (positive M f \<and> increasing M f \<and> countably_subadditive M f)"
invertible_modification_axioms: shows "invertible_modification (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F F.map\<^sub>0 (\<lambda>f. \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) \<tau>'\<tau>.map\<^sub>0 \<tau>'\<tau>.map\<^sub>1 unit"
a_closure: fixes x :: "'a" shows "antidomain_left_monoid.am_d antirange_op ar x = ar x"
vseq_implies_hpar_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.seq \<nu> \<mu>" shows "src \<nu> = src \<mu>"
fd: shows "\<forall>x y. dist (f x) (f y) = c * dist x y"
eq_refl: fixes x :: "'c list"   and y :: "'c list" assumes "x = y" shows "prefix x y"
goodstein_def': fixes c :: "nat"   and n :: "nat" assumes "c \<noteq> 0" shows "goodstein c n = goodsteinC c (N2H (c + 1) n)"
col_space_eq: fixes A :: "(('a, 'c) vec, 'b) vec" shows "col_space A = {y. \<exists>x. A *v x = y}"
l_delete_shadow_root_get_child_nodes_get_child_nodes_locs: shows "l_delete_shadow_root_get_child_nodes Shadow_DOM.get_child_nodes_locs"
assms: shows "qGoodAbs qA \<or> qGoodAbs qB"
safe_reach_suc: shows "safe_reach_upto (Suc n) safe_free_flowing c\<^sub>0"
add_frac_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b"   and w :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "plus (divide x y) (divide w z) = divide (plus (times x z) (times w y)) (times y z)"
eq_fst_iff: fixes a :: "'c"   and p :: "'c \<times> 'd" shows "(a = fst p) = (\<exists>b. p = (a, b))"
iter_hcomp_union: fixes vs :: "'a fset"   and ws :: "'a fset"   and p :: "'a \<Rightarrow> interface" assumes "vs |\<inter>| ws = {||}" shows "iter_hcomp (vs |\<union>| ws) p = (iter_hcomp vs p \<otimes> iter_hcomp ws p)"
assms_2: shows "abs_conv_abscissa f \<le> s0"
len_empty_on_subview1: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real"   and v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" assumes "sensors sensors" shows "\<parallel>sensors.len sensors v ts c\<parallel> = 0 \<and> v=v1\<parallel>v2 \<longrightarrow> \<parallel>sensors.len sensors v1 ts c\<parallel> = 0"
sum_list_length_card_dom_map_of_concat: fixes l :: "('a \<times> 'b) list list" assumes "ht_hash l"   and "ht_distinct l" shows "sum_list (map length l) = card (dom (map_of (concat l)))"
objI_trg: fixes a :: "'a" assumes "B.arr a"   and "trg\<^sub>B a = a" shows "B.obj a"
ex_is_poincare_line_points: fixes H :: "circline" assumes "is_poincare_line H" shows "\<exists>u v. u \<in> unit_disc \<and> v \<in> unit_disc \<and> u \<noteq> v \<and> {u, v} \<subseteq> circline_set H"
simps_1: fixes x :: "'a" shows "vars_term_ms (Var x) = {#x#}"
is_unit_iff_degree: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "is_unit p = (degree p = 0)"
prefix_app_Cons_elim: fixes xs :: "'a list"   and y :: "'a"   and z :: "'a"   and zs :: "'a list" assumes "prefix (xs @ [y]) (z # zs)" obtains "xs = []" and "y = z"   | xs' :: "'a list" where "xs = z # xs'" and "prefix (xs' @ [y]) zs"
hcmod_unit_one: fixes a :: "real star" shows "hcmod (HComplex ((*f* cos) a) ((*f* sin) a)) = 1"
T_def: shows "T = apply_chart d1 ` (domain d1 \<inter> domain c1) \<inter> inv_chart d1 -` (src.carrier \<inter> f -` domain d2)"
weight_simp: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "weight_fun.weight w w0 scf (Fun f ts) = w (f, length ts) + sum_list (map (weight_fun.weight w w0 scf) (scf_list (scf (f, length ts)) ts))"
sequentially_imp_eventually_nhds_within: fixes "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set"   and a :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.first_countable_topology open"   and "\<forall>f. (\<forall>n. f n \<in> s) \<and> filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (inf (topological_space.nhds open a) (principal s))"
image_compactin: fixes X :: "'c topology"   and S :: "'c set"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "compactin X S"   and "continuous_map X Y f" shows "compactin Y (f ` S)"
length_pos_if_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x mem xs" shows "0 < length xs"
as0_19: shows "\<forall>i<n. CAs0 ! i = Cs0 ! i + poss (AAs0 ! i)"
c13_inner: shows "h ::= Ct 1 ;; l ::= Ct 2 \<approx>wT l ::= Ct 2"
isNSUContD: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c star"   and y :: "'c star" assumes "isNSUCont f"   and "x \<approx> y" shows "(*f* f) x \<approx> (*f* f) y"
alphaAbs_qAbs_iff_alphaAbs_all_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_all_equal_or_qFresh xs x X xs' x' X'"
doesnt_read_or_modify_def: fixes c :: "'Com"   and x :: "'Var" shows "doesnt_read_or_modify c x \<equiv> doesnt_read_or_modify_vars c ({x} \<union> \<C>_vars x)"
simps_1: fixes K :: "'b set" shows "all_formulas_of_size K 0 = {\<bottom>} \<union> Atom ` K"
apex_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "span_in_category.apex C S \<equiv> partial_magma.dom C (Leg0 S)"
id: shows "adjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\<^sub>m n"
PO_refines_implies_Range_invariant: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "reach Tc \<subseteq> Range R"
map2_val_compat_def: fixes f :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list" shows "map2_val_compat f = (\<forall>zs. oalist_inv_raw zs \<longrightarrow> oalist_inv_raw (f zs) \<and> fst ` set (f zs) \<subseteq> fst ` set zs)"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.rank_skew_invar [t] = True"
atUV_def: fixes U :: "'f set"   and V :: "'f set"   and c :: "'f list multiset" shows "atUV U V c \<equiv> \<exists>su sv. {#su#} + {#sv#} \<subseteq># c \<and> atU_s U su \<and> atU_s V sv"
set_local: fixes vi :: "v list"   and j :: "nat"   and s :: "s"   and v :: "v"   and vs :: "v list"   and v' :: "v"   and i :: "nat" assumes "length vi = j" shows "\<lparr>s;vi @ [v] @ vs;[$C v', $Set_local j]\<rparr> \<leadsto>_ i \<lparr>s;vi @ [v'] @ vs;[]\<rparr>"
triangular_to_triangular: fixes A :: "'a mat" shows "upper_triangular A = triangular_to (dim_row A) A"
polynomial_incl: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "polynomial\<^bsub>R\<^esub> K p" shows "set p \<subseteq> K"
nonpos_Ints_of_int: fixes n :: "int" assumes "n \<le> 0" shows "of_int n \<in> \<int>\<^sub>\<le>\<^sub>0"
set_val_get_child_nodes: fixes ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
left_neutral: fixes a :: "'b \<Rightarrow> bool" shows "(\<box> \<and>* a) = a"
sorted_spvec_cons2: fixes a :: "nat \<times> 'a"   and b :: "nat \<times> 'a"   and t :: "(nat \<times> 'a) list" assumes "sorted_spvec (a # b # t)" shows "sorted_spvec (a # t)"
monoid_axioms: shows "monoid N (\<cdot>) \<one>"
closed_euler_imp_eq_degree: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "euler_trail u p u"   and "v \<in> verts G" shows "in_degree G v = out_degree G v"
mon_c_union_conc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and a :: "'b list multiset"   and b :: "'b list multiset" shows "mon_c fg (a + b) = mon_c fg a \<union> mon_c fg b"
simps_1: fixes k :: "udom \<rightarrow> udom\<cdot>maybe" shows "bindU\<cdot>Nothing\<cdot>k = Nothing"
map_converter_id_move_right: fixes f :: "'i \<Rightarrow> 'm"   and g :: "'n \<Rightarrow> 'j"   and f' :: "'m \<Rightarrow> 'k"   and g' :: "'l \<Rightarrow> 'n" shows "map_converter f g f' g' 1\<^sub>C = map_converter id id (f' \<circ> f) (g \<circ> g') 1\<^sub>C"
s0_in_S: shows "s0\<^bsub>SES\<^esub> \<in> S\<^bsub>SES\<^esub>"
afSize_def: fixes afile :: "nat \<times> (nat \<Rightarrow> byte)" shows "afSize afile \<equiv> fst afile"
nonempty_Y: shows "(Y::'c::type set) \<noteq> {}"
vectorize_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly (- p) = - vectorize_poly p"
ring_simprules_4: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus> y \<in> carrier R"
return_state_def: fixes return :: "'f \<times> 'd \<Rightarrow> 'e"   and x :: "'f" shows "return_state return x = StateT (\<lambda>s. return (x, s))"
Cnext_solve_implies_not: fixes K\<^sub>1 :: "clock"   and n :: "nat"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and K\<^sub>2 :: "clock"   and \<Phi> :: "'c TESL_atomic list" shows "{K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>, K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"
maybe_or_false: fixes x :: "trilean"   and y :: "trilean" shows "(x \<or>? y = false) = (x = false \<and> y = false)"
var2: fixes env :: "v sem_env"   and n :: "(char list, char list) id0"   and ck :: "bool"   and s :: "'b state" assumes "nsLookup (sem_env.v env) n = None" shows "evaluate ck env s (Var n) (s, Rerr (Rabort Rtype_error))"
nn_integral_snd: fixes f :: "real \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and t :: "real" assumes "exp_esc.random_variable x t borel f" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>distr (exponential (escape_rate x)) borel ((+) t) \<partial>measure_pmf (J x) = integral\<^sup>N (distr (exponential (escape_rate x)) borel ((+) t) \<Otimes>\<^sub>M measure_pmf (J x)) f"
sym: fixes x :: "'c"   and y :: "'c" shows "invert_order (c2 x y) = c2 y x"
AbstrLevels_S5_A32: fixes i :: "AbstrLevelsID" assumes "sS5 \<in> AbstrLevel i" shows "sA32 \<notin> AbstrLevel i"
insert_eq: shows "insert x (insert y (set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs))) = set vs"
FAccThrow: fixes e :: "char list exp"   and F :: "char list"   and D :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>throw e\<bullet>F{D},s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
UNION_empty_conv_1: fixes B :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "({} = \<Union> (B ` A)) = (\<forall>x\<in>A. B x = {})"
q'_def: shows "q' \<equiv> \<lambda>(x, y). (x, \<lambda>i. y i - g' x i)"
pp_total_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c" assumes "matrix_p_algebra.dense (f \<odot> mtop)" shows "\<exists>j. f (i, j) \<noteq> bot"
UNIV_2: shows "UNIV = insert 1 (\<eta> 2)"
cond_spmf_fst_def: fixes p :: "('i \<times> 'h) spmf"   and a :: "'i" shows "cond_spmf_fst p a = map_spmf snd (cond_spmf p ({a} \<times> UNIV))"
hyps_4: shows "AEeq M (\<lambda>w. \<Sum>i\<in>F. f i w) (\<lambda>w. \<Sum>i\<in>F. g i w)"
cpr: shows "lookup (p + r) (t \<oplus> lt f) \<noteq> (0::'b)"
finite_atLeastZeroLessThan_int: fixes u :: "int" shows "finite {0..<u}"
nn_integral_completion: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (completion M) f = integral\<^sup>N M f"
cvalidD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P Call p Q,A"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t"   and "s \<in> P"   and "t \<notin> Fault ` F" shows "t \<in> Normal ` Q \<union> Abrupt ` A"
mult_oner: fixes x :: "('a \<times> 'a set) set" shows "x \<parallel> 1\<^sub>\<pi> = x"
PPs_def: fixes X :: "'c set" shows ".[X] = {t. keys t \<subseteq> X}"
zero_le_dist: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "0 \<le> dist x y"
d_bot_only: fixes x :: "'a" assumes "- - x = bot" shows "x = bot"
hom_into_subgroup: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and G' :: "('c, 'e) monoid_scheme"   and H :: "'c set" assumes "h \<in> hom G G'"   and "h ` carrier G \<subseteq> H" shows "h \<in> hom G (subgroup_generated G' H)"
step_Normal_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Normal s) \<rightarrow> u" obtains "u = (nrms, css, Normal s)"
decomp_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "decomp P c c' Q \<equiv> \<lambda>t s'. \<exists>s t'. P s s' \<and> (c, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t' \<and> Q t t'"
measure_lborel_cbox: fixes l :: "'a"   and u :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "content (cbox l u) = prod ((\<bullet>) (u - l)) Basis"
child_parent_dual: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr"
Urysohn_local: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set"   and a :: "'b"   and b :: "'b" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "S \<inter> T = {}" obtains f :: "'a \<Rightarrow> 'b" where "continuous_on U f" and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b" and "\<And>x. x \<in> S \<Longrightarrow> f x = a" and "\<And>x. x \<in> T \<Longrightarrow> f x = b"
valid_Exit_path_cases: fixes n :: "'node"   and as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)"   and "as \<noteq> []" shows "(\<exists>a' as'. as = a' # as' \<and> intra_kind (kind a')) \<or> (\<exists>a' as' Q p f. as = a' # as' \<and> kind a' = Q\<hookleftarrow>\<^bsub>p\<^esub>f) \<or> (\<exists>as' as'' n'. as = as' @ as'' \<and> as' \<noteq> [] \<and> n -as'\<rightarrow>\<^bsub>sl\<^esub>* n')"
map_disj_empty_right: fixes h :: "'c \<Rightarrow> 'd option" shows "h \<bottom> Map.empty"
transpos_inj: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "inj_on (transpos i j) {i. i \<le> n}"
angle_construction_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "A' \<noteq> B'" shows "\<exists>C'. A B C CongA A' B' C'"
closed_segment_is_geodesic_2: fixes x :: "'b"   and y :: "'b" shows "geodesic_segment_between (closed_segment x y) x y"
card_1_singleton_iff: fixes A :: "'a set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
merge_true_star_ctx: fixes P :: "assn" shows "true * (true * P) = true * P"
assms_1: fixes F :: "'a set" assumes "finite F"   and "F \<subseteq> S" shows "(\<Sum>x\<in>F. norm (f x)) \<le> B"
axioms_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexFolding X f"
bind_eq_join_pmf: fixes M :: "'c pmf"   and f :: "'c \<Rightarrow> 'b pmf" shows "M \<bind> f = join_pmf (map_pmf f M)"
list_all2_conv_all_nth: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
Fun_mkArr'_1: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "\<guillemotleft>mkArr A B F : mkIde A \<rightarrow> mkIde B\<guillemotright>"
simps_6: fixes e :: "cexpr"   and t :: "pdf_type" shows "free_vars (\<integral>\<^sub>c e \<partial>t) = Suc -` free_vars e"
SigId: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and B :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> Vx : B" shows "A = B"
intros_1: fixes t :: "('b, 'c, 'd) Tree"   and tset :: "('b, 'c, 'd) Tree set['b]" assumes "t \<in> set_bset tset" shows "(t, tConj tset) \<in> Tree_wf"
fin: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "finite (((*) (1 / pi) \<circ> (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x) \<circ> (\<lambda>t. 2 * x_coord (1 - t))) -` {x})"
mod_type_axioms: shows "mod_type (int CARD('a bit1)) Rep_bit1 Abs_bit1"
product_run_finite_fst: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "finite (range (run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w))"
snd_eqD: fixes x :: "'c"   and y :: "'b"   and a :: "'b" assumes "final_b (x, y) = a" shows "y = a"
permute_bset_def: shows "permute \<equiv> (id ---> set_bset ---> Abs_bset) permute"
of_nat_eq_enat: fixes n :: "nat" shows "of_nat n = enat n"
relAPP_def: fixes f :: "('f \<times> 'g) set \<Rightarrow> 'e"   and x :: "('f \<times> 'g) set" shows "\<langle>x\<rangle>f \<equiv> f x"
sp_so: fixes p :: "'b upred"   and C :: "'b hrel"   and Q :: "'b upred" shows "`p sp C \<Rightarrow> Q` = \<lbrace>p\<rbrace> C \<lbrace>Q\<rbrace>\<^sub>u"
simps_1: fixes uu :: "'d" shows "funas_term (Var uu) = {}"
if_P: fixes P :: "bool"   and x :: "'a"   and y :: "'a" assumes "P" shows "(if P then x else y) = x"
steps'_take: fixes n :: "nat"   and qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and init :: "'c list" assumes "n < length qs"   and "length qs = length Strat" shows "steps' init (take n qs) (take n Strat) n = steps' init qs Strat n"
ELam_1: fixes \<rho> :: "(nat \<times> exp) list"   and v :: "exp"   and x :: "nat" assumes "closed_env \<rho>"   and "FV v = {}" shows "subst x v (psubst ((x, EVar x) # \<rho>) e) = psubst ((x, v) # \<rho>) e"
prems: shows "\<forall>e a. (e, a) \<in># add_mset x M \<longrightarrow> a = f e"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "dim_row (Matrix.mat nr nc f) = nr"
overflow_plus_one_self: fixes p :: "'b word" shows "(1 + p \<le> p) = (p = - 1)"
S_def: fixes m :: "((nat \<times> nat) \<times> transition) fset" shows "EFSM.S m = (\<lambda>((s, s'), t). s) |`| m |\<union>| (\<lambda>((s, s'), t). s') |`| m"
mult_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "Mp f * g =m f * g"
not_less_Least: fixes k :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "k < (LEAST x. P x)" shows "\<not> P k"
strict_mono_on_eqD: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "strict_mono_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "y = x"
wrap_Id_injective: fixes f :: "recf"   and a :: "nat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat" assumes "recfn a f"   and "wrap_Id f n\<^sub>1 = wrap_Id f n\<^sub>2" shows "n\<^sub>1 = n\<^sub>2"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "('c, 'd) RBT.rbt" shows "RBT.fold x xa = RBT_Impl.fold x (rbt.impl_of xa)"
dghm_singleton_ObjMap_vdomain: fixes j :: "V"   and \<CC> :: "V" shows "\<D>\<^sub>\<circ> (dghm_singleton j \<CC>\<lparr>ObjMap\<rparr>) = \<CC>\<lparr>Obj\<rparr>"
coprime_divisors: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a c"   and "dvd.dvd times b d"   and "algebraic_semidom.coprime times one c d" shows "algebraic_semidom.coprime times one a b"
ln_series_quadratic: fixes x :: "real" assumes "0 < x" shows "(\<lambda>n. 2 * ((x - 1) / (x + 1)) ^ (2 * n + 1) / real (2 * n + 1)) sums ln x"
map_pred_def: fixes f :: "'c \<Rightarrow> 'b" shows "map_pred f = f ---> id"
FL_bisimilar_symp: fixes F :: "'effect fs_set" shows "symp (FL_bisimilar F)"
simps_15: fixes x2 :: "'d" shows "set_option (Some x2) = {x2}"
f_the_inv_into_f_bij_betw: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and x :: "'c" assumes "bij_betw f A B"   and "bij_betw f A B \<Longrightarrow> x \<in> B" shows "f (the_inv_into A f x) = x"
measure_mon_conv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and A :: "nat \<Rightarrow> 'b set"   and B :: "'b set" assumes "measure_space M"   and "\<And>n. A n \<in> measurable_sets M"   and "A\<up>B" shows "(\<lambda>n. Measure.measure M (A n)) \<longlonglongrightarrow> Measure.measure M B"