hcju/isabelleps huggingface dataset:
select_convs_4: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "cod \<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = cod"
set_integrable_norm: fixes M :: "'b measure"   and k :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "set_integrable M k f" shows "set_integrable M k (\<lambda>x. norm (f x))"
range_construct_eq_span: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "\<not> module.dependent s1 B" shows "range (vector_space_pair.construct s1 s2 B f) = module.span s2 (f ` B)"
restrict_sup: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list" shows "ks \<langle>f \<oplus> g\<rangle> ls = ks \<langle>f\<rangle> ls \<oplus> ks \<langle>g\<rangle> ls"
coef_hd_gt_0: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and \<zeta> :: "('s, 'v) hd"   and i :: "nat" assumes "legal_zpassign A" shows "0 < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> i))"
fimageE: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c fset" assumes "b |\<in>| f |`| A" obtains x :: "'c" where "b = f x" and "x |\<in>| A"
finite_field_ops_integer_def: fixes p :: "integer" shows "finite_field_ops_integer p \<equiv> Arith_Ops_Record 0 1 (plus_p_integer p) (mult_p_integer p) (minus_p_integer p) (uminus_p_integer p) (divide_p_integer p) (inverse_p_integer p) (\<lambda>x y. if y = 0 then x else 0) (\<lambda>x. if x = 0 then 0 else 1) (\<lambda>x. x) integer_of_int int_of_integer (\<lambda>x. 0 \<le> x \<and> x < p)"
arity_sym_def: fixes n :: "nat" shows "arity_sym n = \<infinity>"
od_fm_fun_carrier: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and H :: "'a set" shows "carrier (od_fm_fun R M N f H) = fsps R M N f H"
f: fixes n :: "nat"   and s :: "(nat \<times> form) list" shows "\<forall>x. ((n, x) \<in> deriv s) = (x \<in> set (f [s] n))"
liftWT_return: fixes x :: "'e" shows "liftWT\<cdot>(return\<cdot>x) = unitWT\<cdot>x"
P_LCons: shows "P = LCons (\<sigma> v0) (ltl P)"
r2f_def: fixes R :: "('a \<times> 'b) set" shows "\<F> R = bd\<^sup>-\<^sub>\<F> ((``) R)"
in_hom: shows "\<guillemotleft>f \<star> g : f \<star> g \<rightarrow> f \<star> g\<guillemotright>"
supr_greater: fixes B :: "'b set"   and b :: "'b" assumes "B \<subseteq> Field s"   and "s.Above B \<noteq> {}"   and "b \<in> B" shows "(b, s.supr B) \<in> s"
frac_non_empty: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R \<times> S" shows "valid_frac (a / b)"
invariant_other: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and i :: "'i"   and i'' :: "'i"   and t'' :: "'t"   and x'' :: "'x" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P i t x m"   and "P i'' t'' x'' m" shows "P i'' t'' x'' m'"
nstut_noteq: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "NSTUTINV F"   and "NSTUTINV G" shows "NSTUTINV F \<noteq> G"
mono_splitter: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<pi>' :: "(char list \<times> 'a list) prefix"   and k :: "'b" assumes "\<pi> \<le> \<pi>'" shows "splitter \<pi> k \<le> splitter \<pi>' k"
closedin_subset_trans: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "closedin (top_of_set T) S"
of_int_poly_eq_0_iff: fixes p :: "int poly" shows "(of_int_poly p = 0) = (p = 0)"
agree_on_def: fixes G :: "('b, 'c) labeled_graph"   and f\<^sub>1 :: "('c \<times> 'd) set"   and f\<^sub>2 :: "('c \<times> 'd) set" shows "agree_on G f\<^sub>1 f\<^sub>2 \<equiv> \<forall>v\<in>vertices G. f\<^sub>1 `` {v} = f\<^sub>2 `` {v}"
set_iterator_product_order_def: fixes R_a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R_b :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and ab :: "'d \<times> 'e"   and ab' :: "'d \<times> 'e" shows "set_iterator_product_order R_a R_b ab ab' = (if fst ab = fst ab' then R_b (fst ab) (snd ab) (snd ab') else R_a (fst ab) (fst ab'))"
simps_1: fixes var :: "nat" shows "find_eq var [] = ([], [])"
fromsip: fixes dsk1 :: "k"   and dsn2 :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and sip :: "nat"   and dsn1 :: "nat"   and flag1 :: "f"   and hops1 :: "nat"   and nhip1 :: "nat"   and pre1 :: "nat set"   and dsk2 :: "k"   and flag2 :: "f"   and hops2 :: "nat"   and nhip2 :: "nat"   and pre2 :: "nat set" assumes "dsk1 = unk \<or> Suc 0 \<le> dsn2" shows "\<pi>\<^sub>3 (the (update rt sip (dsn1, dsk1, flag1, hops1, nhip1, pre1) sip)) = unk \<or> Suc 0 \<le> sqn (update rt sip (dsn2, dsk2, flag2, hops2, nhip2, pre2)) sip"
arith_simps_49: fixes a :: "'a" shows "(0::'a) + a = a"
linear_inverse_left: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec"   and f' :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "Vector_Spaces.linear (*s) (*s) f'" shows "(f \<circ> f' = id) = (f' \<circ> f = id)"
schur: shows "unitary_schur_decomposition A es = (B, P, Q)"
swap: fixes n :: "nat"   and bs :: "'b list"   and m :: "nat"   and t :: "'b poly" assumes "n < length bs"   and "m < length bs" shows "Ipoly bs (swap n m t) = Ipoly (bs[n := bs ! m, m := bs ! n]) t"
spr_simAbs_inj: shows "inj spr_simAbs"
irreflexive_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<le> - top" shows "x \<sqinter> y \<le> - top"
star_dfa_in_ST: fixes ustar :: "'a list" assumes "ustar \<in> star (dfa.language MS)" shows "q0 \<in> ST.nextl {q0} ustar"
dconvfactor_convinv: fixes c :: "'d \<Rightarrow>\<^sub>U 'c"   and d :: "(int, sdim) dimvec" shows "dconvfactor (inv\<^sub>C c) d = inverse (dconvfactor c d)"
scalingD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and c :: "real"   and x :: "'b" assumes "scaling t"   and "sound P"   and "0 \<le> c" shows "c * t P x = t (\<lambda>x. c * P x) x"
lsl_comp_closed_var: fixes x :: "'b"   and y :: "'b" shows "\<nu> (\<nu> x \<cdot> \<nu> y) = \<nu> x \<cdot> \<nu> y"
filter_map_filter: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c Filters.filter" assumes "mono f"   and "\<forall>x y. f x \<le> y \<longrightarrow> (\<exists>z\<ge>x. y = f z)" shows "ord_class.filter (f ` Filters.filter.Rep_filter F)"
times_sqmatrix_def: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix" shows "A \<cdot> B = sqmatrix_of_matrix (matrix_of_sqmatrix A *\<^sub>M matrix_of_sqmatrix B)"
all_normalized_and_ex_tcsigs_iff_exe_all_normalized_and_ex_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss" shows "all_normalized_and_ex_tcsigs (set cs) (translate_ars arss) = exe_all_normalized_and_ex_tcsigs cs arss"
add: fixes x :: "'a"   and N :: "'a multiset" assumes "M = add_mset x N" shows "thesis"
length_rotate1: fixes xs :: "'c list" shows "length (rotate1 xs) = length xs"
L: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> P\<^sub>2 \<union> wrap (B\<^sub>2 \<union> {u})) = \<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> wrap {} \<union> P\<^sub>2 \<union> wrap B\<^sub>2) \<union> {u}"
rp_def: shows "rp = midpoint (flow0 x t1) (flow0 x t2)"
edges_it_def: fixes edges_list_it :: "'e \<Rightarrow> (('c \<times> 'd \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'd \<times> 'c \<Rightarrow> ('c \<times> 'd \<times> 'c) list \<Rightarrow> ('c \<times> 'd \<times> 'c) list) \<Rightarrow> ('c \<times> 'd \<times> 'c) list \<Rightarrow> ('c \<times> 'd \<times> 'c) list"   and g :: "'e" shows "graph_edges_it_defs.edges_it edges_list_it g \<equiv> it_to_it (edges_list_it g)"
nodes_trace: fixes p :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and r :: "(nat \<times> 'a) stream" assumes "p \<in> gnodes A w V"   and "grun A w V r p" shows "sset (gtrace r p) \<subseteq> gnodes A w V"
simps: fixes p :: "real poly"   and qs1 :: "real poly list"   and m1 :: "rat mat"   and sub1 :: "(nat list \<times> nat list) list"   and sgn1 :: "rat list list"   and qs2 :: "real poly list"   and m2 :: "rat mat"   and sub2 :: "(nat list \<times> nat list) list"   and sgn2 :: "rat list list" shows "combine_systems_R p (qs1, m1, sub1, sgn1) (qs2, m2, sub2, sgn2) = smash_systems_R p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
assms_2: shows "unat i * unat k < 2 ^ LENGTH('a)"
object_ptr_kinds_M_eq2_h3: shows "|h3 \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r"
RepFun_def: fixes A :: "hf"   and f :: "hf \<Rightarrow> hf" shows "RepFun A f = \<lbrace>y . x \<^bold>\<in> A, y = f x\<rbrace>"
subst_img_comp_subset': fixes t :: "('b, 'c) Term.term"   and \<theta>1 :: "'c \<Rightarrow> ('b, 'c) Term.term"   and \<theta>2 :: "'c \<Rightarrow> ('b, 'c) Term.term" assumes "t \<in> subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)" shows "t \<in> subst_range \<theta>2 \<or> (\<exists>t'\<in>subst_range \<theta>1. t = t' \<cdot> \<theta>2)"
assms: assumes "\<And>x. \<not> P x" shows "False"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) 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"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs" shows "l_set_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_shadow_root set_shadow_root_locs"
wellformed_induce_subgraph: fixes vs :: "'a set" shows "wf_digraph ((G \<restriction> verts H) \<restriction> vs)"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c Q,A"
foldl_Cons: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and a :: "'d"   and x :: "'e"   and xs :: "'e list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
norm_gt_square: fixes a :: "real"   and x :: "'b" shows "(a < norm x) = (a < 0 \<or> a\<^sup>2 < x \<bullet> x)"
N_def: shows "N \<equiv> LEAST n. x [^] n = \<one> \<and> 0 < n"
generate_empty: shows "generate (G\<lparr>carrier := generate G {a}\<rparr>) {} = {\<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>}"
sup_commute: fixes x :: "'b"   and y :: "'b" shows "sup x y = sup y x"
kronecker_product_def: fixes A :: "'a mat"   and B :: "'a mat" shows "kronecker_product A B = (let ra = dim_row A; ca = dim_col A; rb = dim_row B; cb = dim_col B in mat (ra * rb) (ca * cb) (\<lambda>(i, j). A $$ (i div rb, j div cb) * B $$ (i mod rb, j mod cb)))"
nths_project: fixes i :: "nat"   and xs :: "'b list"   and j :: "nat" assumes "i < length xs"   and "j < length xs"   and "i < j" shows "nths xs {i, j} ! 0 = xs ! i \<and> nths xs {i, j} ! 1 = xs ! j"
simps_3: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int" shows "aval (Plus a\<^sub>1 a\<^sub>2) s = aval a\<^sub>1 s + aval a\<^sub>2 s"
elims_2: fixes x :: "'b tree" assumes "pheap x" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "Ball (set_tree l) ((\<le>) xa) \<and> pheap l \<and> pheap r"
eval_cring_hom: fixes K :: "'a set"   and a :: "'a" assumes "subring K R"   and "a \<in> carrier R" shows "ring_hom_cring (K [X]) R (\<lambda>p. local.eval p a)"
join: fixes a :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m"   and b :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m" assumes "root_hash_F (root_hash_T rha) rhb a = root_hash_F (root_hash_T rha) rhb b"   and "a \<in> {x. set1_F\<^sub>m x \<subseteq> {x. set1_T\<^sub>m x \<subseteq> A} \<and> set3_F\<^sub>m x \<subseteq> B}" shows "\<exists>ab. merge_F (root_hash_T rha) (merge_T rha ma) rhb mb a b = Some ab \<and> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob a ab \<and> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob b ab \<and> (\<forall>u. blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob a u \<longrightarrow> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob b u \<longrightarrow> blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob ab u)"
ASSERT_same_eq_conv: fixes \<Phi> :: "bool"   and m :: "'a nres"   and n :: "'a nres" shows "(ASSERT \<Phi> \<bind> (\<lambda>_. m) = ASSERT \<Phi> \<bind> (\<lambda>_. n)) = (\<Phi> \<longrightarrow> m = n)"
rankeq: shows "vec_space.rank nc A\<^sup>H = card S"
blinfun_apply_inject: fixes x :: "'d \<Rightarrow>\<^sub>L 'e"   and y :: "'d \<Rightarrow>\<^sub>L 'e" shows "(blinfun_apply x = blinfun_apply y) = (x = y)"
Ej_def: fixes j :: "nat"   and x :: "nat list" shows "Ej j x = {local.eij i j - 1 |i. i < length x \<and> local.dij i j \<le> x ! i}"
stutter_equiv_trans: fixes \<rho> :: "nat \<Rightarrow> 'a"   and \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "\<rho> \<approx> \<sigma>"   and "\<sigma> \<approx> \<tau>" shows "\<rho> \<approx> \<tau>"
simps_2: fixes x2 :: "ltkey"   and y2 :: "ltkey" shows "(ltK x2 = ltK y2) = (x2 = y2)"
polyhedron_convex_hull: fixes S :: "'b set" assumes "finite S" shows "polyhedron (convex hull S)"
Pi'_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" shows "Pi' I A = {P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^sub>F i \<in> A i)}"
intros_1: fixes i :: "init_state" shows "Prepared \<le>\<^sub>i i"
euler_char_def: shows "GM.euler_char \<equiv> int (card (verts G)) - int (card (arcs G) div 2) + int (card GM.face_cycle_sets)"
set_less_eq_aux''_spec_2: shows "order_consistent {(A, B). A \<sqsubseteq>'' B} {(A, B). A \<sqsubseteq>''' B}"
CURRY_def: fixes R :: "(('a \<times> 'b \<Rightarrow> 'c) \<times> ('d \<times> 'e \<Rightarrow> 'f)) set" shows "CURRY R \<equiv> {(f, g). (uncurry f, uncurry g) \<in> R}"
tarjan_params_simps_7: shows "is_break tarjan_params = (\<lambda>s. False)"
triangle_in_hom_4: shows "\<guillemotleft>\<r>[g] \<cdot> (g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] : g \<Rightarrow> g\<guillemotright>"
winningAllocationIsMaximal: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b)) \<in> maximalStrictAllocations N (set \<Omega>) b"
invP: shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
extremum: fixes a :: "'d upred" shows "a \<sqsubseteq> false"
filter_cong: fixes xs :: "'k list"   and ys :: "'k list"   and P :: "'k \<Rightarrow> bool"   and Q :: "'k \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
assms_12: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "I (\<Gamma>, Bool b, Alts e1 e2 # S)" shows "P (\<Gamma>, Bool b, Alts e1 e2 # S) (\<Gamma>, if b then e1 else e2, S)"
fun_rel_id: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('b \<times> 'b) set" assumes "R1 = Id"   and "R2 = Id" shows "R1 \<rightarrow> R2 = Id"
d_dist1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d x \<cdot> (d y + d z) = d x \<cdot> d y + d x \<cdot> d z"
img_mkPoint_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "t \<in> local.set a" shows "img (mkPoint a t) = t"
s_3: shows "flow0 x ` {0..s} \<subseteq> M"
ex_inverse_of_nat_less: fixes x :: "'a" assumes "(0::'a) < x" shows "\<exists>n>0. inverse (of_nat n) < x"
sign_changes_neg_inf_def: fixes ps :: "'a poly list" shows "sign_changes_neg_inf ps = length (remdups_adj (filter (\<lambda>x. x \<noteq> (0::'a)) (map poly_neg_inf ps))) - 1"
P5_b: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x r\<rightarrow> y = (1::'a)"   and "y r\<rightarrow> z = (1::'a)" shows "x r\<rightarrow> z = (1::'a)"
mono_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "mono f"   and "mono g" shows "mono (f \<circ> g)"
snd_snd_A': shows "matrix_to_iarray (snd (snd A')) = snd (snd B)"
im_sheaf_def: fixes V :: "'a set" shows "imS.im_sheaf V \<equiv> \<FF> (identity S \<^sup>\<inverse> S V)"
inv_into_injective: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "inv_into A f x = inv_into A f y"   and "x \<in> f ` A"   and "y \<in> f ` A" shows "x = y"
poly_add_append_replicate: fixes p :: "'a list"   and q :: "'a list" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R" shows "poly_add (p @ replicate (length q) \<zero>) q = local.normalize (p @ q)"
\<PP>_are_doc_tokens: fixes p :: "(symbol \<times> character list) list" assumes "p \<in> \<PP>" shows "doc_tokens p"
cmp_fun: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f \<in> A \<rightarrow> C"
induct: fixes x :: "('a, 'b) fm"   and P :: "('a, 'b) fm \<Rightarrow> bool" assumes "\<turnstile> x"   and "\<And>p. tautology p \<Longrightarrow> P p"   and "\<And>p t. P (\<^bold>\<forall> p \<^bold>\<longrightarrow> p\<langle>t/0\<rangle>)"   and "\<And>p q. \<lbrakk>\<turnstile> p \<^bold>\<longrightarrow> q; P (p \<^bold>\<longrightarrow> q); \<turnstile> p; P p\<rbrakk> \<Longrightarrow> P q"   and "\<And>q p a. \<lbrakk>\<turnstile> q \<^bold>\<longrightarrow> p\<langle>\<^bold>\<dagger> a []/0\<rangle>; P (q \<^bold>\<longrightarrow> p\<langle>\<^bold>\<dagger> a []/0\<rangle>); a \<notin> params {p, q}\<rbrakk> \<Longrightarrow> P (q \<^bold>\<longrightarrow> \<^bold>\<forall> p)" shows "P x"
pinduct: fixes a0 :: "Core.term"   and P :: "Core.term \<Rightarrow> bool" assumes "eta_norm_dom a0"   and "\<And>T t. \<lbrakk>eta_norm_dom (Abs T t); P t; \<And>x51 x52 x3. \<lbrakk>eta_norm t = x51 $ x52; x52 = Bv x3; x3 = 0; \<not> is_dependent x51\<rbrakk> \<Longrightarrow> P x51\<rbrakk> \<Longrightarrow> P (Abs T t)"   and "\<And>t u. \<lbrakk>eta_norm_dom (t $ u); P t; P u\<rbrakk> \<Longrightarrow> P (t $ u)"   and "\<And>v va. eta_norm_dom (Ct v va) \<Longrightarrow> P (Ct v va)"   and "\<And>v va. eta_norm_dom (Fv v va) \<Longrightarrow> P (Fv v va)"   and "\<And>v. eta_norm_dom (Bv v) \<Longrightarrow> P (Bv v)" shows "P a0"
inf: shows "\<integral>\<^sup>+ x. ennreal (expl_cond_expect M Y X w * indicat_real (Y -` {Y w} \<inter> space M) x) \<partial>M < \<infinity>"
PP_oalist_def: fixes xs :: "('b, 'c) oalist_tc" shows "PP_oalist xs = pp_of_fun (OAlist_tc_lookup xs)"
IF2set_simps: fixes x :: "('d, 'd IF1, 'd IF2) F2" shows "IF2set (ctor2 x) = F2set1 x \<union> (\<Union> (IF1set ` F2set2 x) \<union> \<Union> (IF2set ` F2set3 x))"
ctxt_sub_closed: fixes e :: "db"   and C :: "db" assumes "closed e" shows "closed (C<e>)"
comp_x: shows "compile_env_entry \<A> \<O> x = Ok y"
assms_2: shows "finite {a. h a \<noteq> \<^bold>1}"
ae: shows "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x) - Pxy x * log b (Py (snd x)) = Pxy x * log b (Pxy x / Py (snd x))"
d_fixpoint: fixes x :: "'a" shows "(\<exists>y. x = ans_d y) = (x = ans_d x)"
hyps_1: shows "pi \<bullet> (\<parallel>(A)\<parallel>) = \<parallel>(A)\<parallel>"
P_i_first: shows "H.first_edge_of_walk P_i = (v0, v')"
multiset_partition: fixes M :: "'k multiset"   and P :: "'k \<Rightarrow> bool" shows "M = filter_mset P M + {#x \<in># M. \<not> P x#}"
A: shows "nat_to_pr n = nat_to_pr (pr_index_enumerator n na)"
seq_sentences_ref: fixes P :: "'c option process"   and xs :: "'c option list" assumes "ref_union_closed P"   and "sequential P"   and "xs \<in> sentences P" shows "(xs, {x. x \<noteq> None}) \<in> failures P"
iMODb_inext: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "t \<in> [ r, mod m, c ]"   and "t < r + m * c" shows "inext t [ r, mod m, c ] = t + m"