hcju/isabelleps huggingface dataset:
fls_deriv_mult_const_right: fixes f :: "'a fls"   and c :: "'a" shows "fls_deriv (f * fls_const c) = fls_deriv f * fls_const c"
gpv1'_def: shows "gpv1' = Generative_Probabilistic_Value.Done y"
liftET_def: shows "liftET = (\<Lambda> m. ErrorT\<cdot>(fmap\<cdot>Ok\<cdot>m))"
hyps_3: assumes "P \<noteq> P''" shows "[a\<noteq>b]P \<Longrightarrow>\<^sub>\<tau> P''"
subspace_imp_convex: fixes s :: "'a set" assumes "subspace s" shows "convex s"
brrr: shows "x \<in> set (fst (TSdet init h (as @ x # bs @ [x]) (Suc (length as + length bs))))"
rank_hinsert: fixes a :: "hf"   and b :: "hf" shows "rank (b \<triangleleft> a) = rank b \<squnion> succ (rank a)"
set_subset_Univ: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and b :: "'a"   and a :: "'a" assumes "hom_functor C S \<phi>"   and "partial_magma.ide C b"   and "partial_magma.ide C a" shows "hom_functor.set C \<phi> (b, a) \<subseteq> Collect (category.terminal S)"
load_word_mem_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and asi :: "word8"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "load_word_mem s addr asi = load_word_mem (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr asi"
bvo_to_fvo: fixes x :: "'d"   and ODE :: "('e, 'd) ODE" assumes "Inl x \<in> BVO ODE" shows "x \<in> FVO ODE"
psi_bounds_sustained_induct: fixes j :: "nat"   and d :: "real"   and c :: "real"   and k :: "nat"   and n :: "nat" assumes "4 * ln (1 + 2 ^ j) + 3 \<le> d * ln 2 * (1 + 2 ^ j)"   and "4 / (1 + 2 ^ j) \<le> d * ln 2"   and "0 \<le> c"   and "c / 2 + d + 1 \<le> c"   and "j \<le> k"   and "\<And>n. n \<le> 2 ^ k \<Longrightarrow> psi n \<le> c * ln 2 * real n"   and "n \<le> 2 ^ Suc k" shows "psi n \<le> c * ln 2 * real n"
open_generated_order: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool" assumes "class.order_topology less_eq less open" shows "open = generate_topology (range (ord.lessThan less) \<union> range (ord.greaterThan less))"
power_eq_1_iff: fixes w :: "'a"   and n :: "nat" assumes "w ^ n = (1::'a)" shows "norm w = 1 \<or> n = 0"
weakly_preserves_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "F (src\<^sub>C \<mu>) \<cong>\<^sub>D src\<^sub>D (F \<mu>)"
is_heap_list_append_dest_2: fixes l :: "'c"   and ts :: "('c, 'd) bintree list"   and rs :: "('c, 'd) bintree list" assumes "is_heap_list l (ts @ rs)" shows "is_heap_list l rs"
P''Trans: shows "P'' \<longmapsto> \<alpha> \<prec> P'''"
sets_PiM_single: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
greaterThan_iff: fixes i :: "'c"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c" shows "(i \<in> ord.greaterThan less k) = less k i"
pp_powerprod_def: fixes ord :: "('a, nat) pp \<Rightarrow> ('a, nat) pp \<Rightarrow> bool"   and ord_strict :: "('a, nat) pp \<Rightarrow> ('a, nat) pp \<Rightarrow> bool" shows "pp_powerprod ord ord_strict \<equiv> ordered_powerprod ord ord_strict"
Fix_def: fixes f :: "'a \<Rightarrow> 'a" shows "Fix f = {n. f n = n}"
lunit'_simps_4: fixes f :: "'b" assumes "D.ide f" shows "D.dom \<l>\<^sub>D\<^sup>-\<^sup>1[f] = f"
closing_subst_closed: fixes e :: "db"   and k :: "nat"   and \<Gamma> :: "nat \<Rightarrow> db" assumes "\<forall>v. freedb e v \<longrightarrow> v < k" shows "closing_subst e \<Gamma> k = e"
fact_ge_1: fixes n :: "nat" shows "(1::'b) \<le> fact n"
SN_iff_wf: fixes A :: "('d \<times> 'd) set" shows "SN A = wf (A\<inverse>)"
sylvester_mat_sub_def: fixes m :: "nat"   and n :: "nat"   and p :: "'b poly"   and q :: "'b poly" shows "sylvester_mat_sub m n p q \<equiv> mat (m + n) (m + n) (\<lambda>(i, j). if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'b) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'b))"
set_times_rearrange3: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "a *o B * C = a *o (B * C)"
CRR_infinite_cts_filtration: shows "infinite_cts_filtration p M nat_filtration"
mod_0: fixes a :: "'a" shows "(0::'a) mod a = (0::'a)"
plus_Mp_2: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (f + poly_mod.Mp m g) = poly_mod.Mp m (f + g)"
assms: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> set xs"   and "x' \<in> set ys" shows "f x x' = g x x'"
partition_on_in_the_unique_part: fixes A :: "'b set"   and P :: "'b set set"   and x :: "'b" assumes "partition_on A P"   and "x \<in> A" shows "x \<in> (THE X. x \<in> X \<and> X \<in> P)"
countable_rtrancl: fixes X :: "('c \<times> 'c) set"   and Y :: "'c set" assumes "\<And>Y. countable Y \<Longrightarrow> countable (X `` Y)"   and "countable Y" shows "countable (X\<^sup>* `` Y)"
tends_to_tight_bounds_3: fixes y :: "real_alg_3" shows "(\<lambda>x. get_itvl_3 ((tighten_bounds_3 ^^ x) y)) \<longlonglongrightarrow>\<^sub>i real_of_3 y"
extendRule_def: fixes forms :: "('a, 'b) sequent"   and R :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" shows "extendRule forms R \<equiv> (map (extend forms) (fst R), extend forms (snd R))"
drop_eq_ConsD: fixes n :: "nat"   and xs :: "'a list"   and x :: "'a"   and xs' :: "'a list" assumes "drop n xs = x # xs'" shows "drop (Suc n) xs = xs'"
geodesic_segment_endpoints_2: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "y \<in> G"
B: shows "D x \<notin> c_sources I D u (x # x' # xs)"
set_tag_name_is_l_set_tag_name: shows "l_set_tag_name ShadowRootClass.type_wf set_tag_name set_tag_name_locs"
rec_entry_def: shows "rec_entry = Cn 2 rec_lo [recf.id 2 0, Cn 2 rec_pi [Cn 2 s [recf.id 2 1]]]"
step1: shows "gdpr_scenario \<rightarrow>\<^sub>n gdpr_scenario'"
domain_functions_def: fixes R :: "('a, 'b) ring_scheme" shows "domain_functions R \<equiv> ring_functions R \<and> domain R"
numeral_le_one_iff: fixes n :: "num" shows "(numeral n \<le> (1::'b)) = (n \<le> num.One)"
conservation_const: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Flow c s t f" shows "\<forall>v\<in>Graph.V c - {s, t}. sum f (Graph.incoming c v) = sum f (Graph.outgoing c v)"
npeadd_def: fixes x :: "pexpr"   and y :: "pexpr" shows "npeadd x y = (case x of PExpr1 (PCnst c) \<Rightarrow> if c = 0 then y else case y of PExpr1 (PCnst d) \<Rightarrow> PExpr1 (PCnst (c + d)) | PExpr1 _ \<Rightarrow> PExpr1 (PAdd x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PAdd x y) | PExpr1 _ \<Rightarrow> case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then x else PExpr1 (PAdd x y) | PExpr1 _ \<Rightarrow> PExpr1 (PAdd x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PAdd x y) | PExpr2 pexpr2 \<Rightarrow> case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then x else PExpr1 (PAdd x y) | PExpr1 _ \<Rightarrow> PExpr1 (PAdd x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PAdd x y))"
assms_5: fixes s1 :: "'s1" assumes "bad1 s1" shows "callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"
assms: shows "\<forall>xs\<in>set xss. sorted (map key xs)"
R4: fixes x :: "'b"   and xt :: "'b bintree"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list"   and X :: "'b t_type" assumes "\<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> \<in> t_ins_set X"   and "x \<le> y" shows "\<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr> \<in> t_ins_set X"
lookupDef_path: fixes m :: "'node"   and g :: "'g"   and v :: "'var" assumes "m \<in> set (\<alpha>n g)" obtains ns :: "'node list" where "g \<turnstile> lookupDefNode g m v-ns\<rightarrow>m" and "\<forall>x\<in>set (tl ns). v \<notin> defs g x"
Diagonalize_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (free_monoidal_category.norm C t) = monoidal_language.Diagonalize C t"
inner_dirsum_el_decomp_def: fixes Gs :: "'b set list" shows "\<Oplus>Gs\<leftarrow> = (\<lambda>x. if x \<in> (\<Oplus>G\<leftarrow>Gs. G) then THE gs. gs \<in> listset Gs \<and> x = sum_list gs else [])"
execution_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.execution subst e s s' \<equiv> (\<exists>r. s = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<and> LID\<^sub>E e = {} \<and> RID\<^sub>E e = {}) \<and> (s, s') \<in> (substitution.steps subst)\<^sup>*"
cases: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and a1 :: "(status \<times> 'a) \<times> 'd"   and a2 :: "(status \<times> 'b) \<times> 'e" assumes "FWbisimulation_base.init_fin_bisim final1 final2 bisim t a1 a2" obtains x1 :: "'a" and m1 :: "'d" and x2 :: "'b" and m2 :: "'e" where "a1 = ((PreStart, x1__), m1__)" and "a2 = ((PreStart, x2__), m2__)" and "bisim t (x1__, m1__) (x2__, m2__)"   | x1 :: "'a" and m1 :: "'d" and x2 :: "'b" and m2 :: "'e" where "a1 = ((Running, x1a__), m1a__)" and "a2 = ((Running, x2a__), m2a__)" and "bisim t (x1a__, m1a__) (x2a__, m2a__)"   | x1 :: "'a" and m1 :: "'d" and x2 :: "'b" and m2 :: "'e" where "a1 = ((Finished, x1), m1)" and "a2 = ((Finished, x2), m2)" and "bisim t (x1, m1) (x2, m2)" and "final1 x1" and "final2 x2"
schur_decomposition: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1::'b:])"   and "schur_decomposition A es = (B, P, Q)" shows "similar_mat_wit A B P Q \<and> upper_triangular B \<and> diag_mat B = es"
cnj_one: shows "Octonions.cnj 1 = 1"
sgn2_nz_impl_arg_zero: fixes x :: "nat" assumes "sgn2 x \<noteq> 0" shows "x = 0"
obsE: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n' :: "'b"   and n :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "n' \<in> CFG.obs sourcenode targetnode valid_edge n S" obtains as :: "'a list" where "CFG.path sourcenode targetnode valid_edge n as n'" and "\<forall>nx\<in>set (CFG.sourcenodes sourcenode as). nx \<notin> S" and "n' \<in> S"
prepend_associative: fixes us :: "nat list"   and vs :: "nat list"   and f :: "nat \<Rightarrow> nat option" shows "us @ vs \<odot> f = us \<odot> vs \<odot> f"
abs_eq_iff: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "(Abs_Integ x = Abs_Integ y) = intrel x y"
Sup_distr: fixes X :: "('b \<times> 'c) set"   and y :: "'b \<times> 'c" shows "times_prod' (Sup_prod X) y = Sup_prod ((\<lambda>x. times_prod' x y) ` X)"
le_cInf_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_below less_eq S" shows "less_eq a (Inf S) = (\<forall>x\<in>S. less_eq a x)"
dim_span_eq_card_independent: fixes B :: "('b, 'c) vec set" assumes "vec.independent B" shows "vec.dim (vec.span B) = card B"
subset_eq: fixes A :: "'a set"   and B :: "'a set" shows "order.greater_eq B A = (\<forall>x\<in>A. x \<in> B)"
vimage_subsetI: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and A :: "'b set" assumes "inj f"   and "B \<subseteq> f ` A" shows "f -` B \<subseteq> A"
hpowr_0: fixes r :: "real star" shows "r ^ 0 = 1"
f: shows "has_bochner_integral (lebesgue_on {a..a + c}) f i"
LIMSEQ_INF: fixes X :: "nat \<Rightarrow> 'c" assumes "decseq X" shows "X \<longlonglongrightarrow> \<Sqinter> range X"
f2: shows "(r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = (s \<otimes> r' \<oplus> s' \<otimes> r |\<^bsub>rel\<^esub> s' \<otimes> s)"
has_integral_unique: fixes f :: "'b \<Rightarrow> 'c"   and k1 :: "'c"   and i :: "'b set"   and k2 :: "'c" assumes "(f has_integral k1) i"   and "(f has_integral k2) i" shows "k1 = k2"
reduce_spec_def: fixes reach_options_o :: "unit"   and X :: "'b set" shows "reduce_spec reach_options_o X = SPEC ((\<subseteq>) X)"
simps_4: fixes A :: "fm"   and i :: "name"   and x :: "tm" shows "(Neg A)(i::=x) = Neg (A(i::=x))"
ordLeq_Well_order_simp: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r \<le>o r'" shows "Well_order r \<and> Well_order r'"
not_empty_M: fixes i :: "'i" assumes "i \<in> I" shows "space (M i) \<noteq> {}"
l_get_root_node_wf_axioms: shows "l_get_root_node_wf heap_is_wellformed get_root_node DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_ancestors get_parent"
grev: fixes t :: "real"   and x :: "real"   and y :: "real" shows "t \<in> point_ivl (1 / 8) \<longrightarrow> (x, y) \<in> point_ivl (1, 0) \<longrightarrow> t \<in> g.rev.existence_ivl0 (x, y) \<and> g.rev.flow0 (x, y) t \<in> {(11 / 10, - (9 / 10\<^sup>2))..(12 / 10, - (8 / 10\<^sup>2))}"
scc_root_def: fixes E :: "('c \<times> 'c) set"   and s :: "('c, 'd) state_scheme"   and v :: "'c"   and scc :: "'c set" shows "scc_root' E s v scc = (is_scc E scc \<and> v \<in> scc \<and> v \<in> dom (discovered s) \<and> scc \<inter> dom (discovered s) \<subseteq> (tree_edges s)\<^sup>* `` {v})"
assms: shows "liftAll (\<lambda>v. \<forall>v'. phi v v') inp"
select_convs_4: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "mem \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = mem"
zero_neq_one: fixes one :: "'c"   and zero :: "'c" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
diagonal_to_Smith_PQ_JNF_internalized_first: fixes A :: "'e mat"   and S :: "'e mat"   and P :: "'e mat"   and Q :: "'e mat"   and bezout :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<times> 'e \<times> 'e \<times> 'e \<times> 'e" assumes "class.nontriv TYPE('d)"   and "A \<in> carrier_mat CARD('d) CARD('f)"   and "S \<in> carrier_mat CARD('d) CARD('f)"   and "P \<in> carrier_mat CARD('d) CARD('d)"   and "Q \<in> carrier_mat CARD('f) CARD('f)"   and "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)"   and "isDiagonal_mat A"   and "is_bezout_ext bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S"
infinite_hyp_wrong: fixes n :: "nat"   and j :: "nat" shows "\<exists>m>n. \<phi> (the (s (\<psi> j \<triangleright> m))) \<noteq> \<psi> j"
S_def: shows "S = (\<Union>x\<in>{a--b}. ball x d)"
smooth_on: fixes f :: "'c \<Rightarrow> 'd"   and k :: "enat"   and S :: "'c set" assumes "bounded_linear f" shows "k-smooth_on S f"
subring_props_6: fixes K :: "'a set"   and h1 :: "'a"   and h2 :: "'a" assumes "subfield K R"   and "h1 \<in> K"   and "h2 \<in> K" shows "h1 \<otimes> h2 \<in> K"
two_children_parent_mems_le_finite: fixes vs :: "'d set"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "vs \<subseteq> prob_dom PROB" shows "prob_dom (prob_proj PROB vs) = vs"
next_plane0_incr_except: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and v :: "nat" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "v \<in> \<V> g" shows "except g v \<le> except g' v"
max_add_distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus (ord.max less_eq x y) z = ord.max less_eq (plus x z) (plus y z)"
INF_lower2: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd"   and u :: "'c \<times> 'd" assumes "i \<in> A"   and "less_eq_prod (f i) u" shows "less_eq_prod (Inf_prod (f ` A)) u"
c_def: shows "c = s0 \<bullet> (1::'a)"
iSp_Append_ddep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n''"   and "n'' -V\<rightarrow>\<^sub>d\<^sub>d n'"   and "n'' \<noteq> n'" shows "n i-ns @ [n'']\<rightarrow>\<^sub>d* n'"
hyps_1: shows "l \<in> A\<^sup>\<star>"
runit_naturality: fixes \<mu> :: "'a" assumes "Right.arr \<mu>" shows "Right.comp (Right.runit (Right.cod \<mu>)) (Right.R \<mu>) = Right.comp \<mu> (Right.runit (Right.dom \<mu>))"
adjunction_data_in_bicategory_axioms_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" shows "adjunction_data_in_bicategory_axioms V H src f g \<eta> \<epsilon> \<equiv> (partial_magma.ide V f \<and> partial_magma.ide V g) \<and> partial_magma.in_hom V \<eta> (src f) (H g f) \<and> partial_magma.in_hom V \<epsilon> (H f g) (src g)"
2: fixes x :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F q x" shows "\<not> (sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p x"
a2b2relprime: shows "coprime (a\<^sup>2) (b\<^sup>2)"
IH: fixes t :: "'a trm" assumes "subterm t p' u" shows "subterm t (p' @ q) v"
tab_simps_4: shows "local.dom r\<^sub>0s\<^sub>1.tab = r\<^sub>0s\<^sub>1.p\<^sub>1"
nth_factorization_lattice: fixes i :: "nat"   and u :: "'a poly"   and d :: "nat"   and m :: "'a" assumes "i < degree u + d" shows "factorization_lattice u d m ! i = vec_of_poly_n (if i < d then u * monom (1::'a) (d - Suc i) else monom m (degree u + d - Suc i)) (degree u + d)"
addpreRT_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and npre :: "nat set" shows "addpreRT rt dip npre \<equiv> map_option (\<lambda>s. rt(dip \<mapsto> addpre s npre)) (rt dip)"
edge_preserving_Id: fixes y :: "'e set"   and x :: "('f \<times> 'e \<times> 'e) set" shows "edge_preserving (Id_on y) x x"
translation: fixes G :: "('a, 'b) labeled_graph"   and x :: "'b"   and y :: "'b"   and e :: "'a 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)"
bal_substitute_subtree: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree"   and c :: "'b btree" assumes "bal (Node (ls @ (a, b) # rs) t)"   and "height a = height c"   and "bal c" shows "bal (Node (ls @ (c, b) # rs) t)"
conjI_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p"   and "A \<turnstile> q" shows "A \<turnstile> p .\<and>. q"
fls_base_factor_of_int: fixes i :: "int" shows "fls_base_factor (of_int i) = of_int i"
times_assn_def: fixes P :: "assn"   and Q :: "assn" shows "P * Q \<equiv> Abs_assn (times_assn_raw (Rep_assn P) (Rep_assn Q))"
rel_inject_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x21 :: "'b"   and x22 :: "'b list"   and y21 :: "'c"   and y22 :: "'c list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
HcompNml_Trg_Nml: fixes t :: "'a E.term" assumes "E.Nml t" shows "E.Trg t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> t = t"
fixp_preserves_mcont1: fixes F :: "('c \<Rightarrow> 'd \<Rightarrow> 'e llist) \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. mono_llist_lift (\<lambda>f. F f x)"   and "f \<equiv> llist_lift.fixp_fun F"   and "\<And>f. mcont lubb ordb llist.lub_fun llist.le_fun f \<Longrightarrow> mcont lubb ordb llist.lub_fun llist.le_fun (F f)" shows "mcont lubb ordb llist.lub_fun llist.le_fun f"
lselect_lselect: fixes w :: "'b llist"   and s :: "nat set"   and t :: "nat set" shows "lselect t (lselect s w) = lselect (nth_least s ` {i \<in> t. enat i < esize s}) w"
inv_ide: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.inv a = a"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vclosed (Vabs cs \<Gamma>) = (pred_fmap id (fmmap vclosed \<Gamma>) \<and> list_all (\<lambda>(pat, t). closed_except t (fmdom \<Gamma> |\<union>| frees pat)) cs)"
sym_preproc_aux_term_wf: fixes d :: "'a \<Rightarrow> nat" assumes "dickson_grading d" shows "wf (sym_preproc_aux_term d)"
sum_rw: shows "column k (Gram_Schmidt_matrix A) \<bullet> (\<Sum>x\<in>{column i (Gram_Schmidt_matrix A) |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x) = 0"
iMOD_0: fixes r :: "nat" shows "[ r, mod 0 ] = [r\<dots>,0]"
preserves_hom: fixes f :: "'d \<times> 'c"   and a :: "'d \<times> 'c"   and b :: "'d \<times> 'c" assumes "DopxC.in_hom f a b" shows "\<guillemotleft>\<Phi>.map f : Hom_FopxC.map a \<rightarrow>\<^sub>S Hom_DopxG.map b\<guillemotright>"
add_closed: fixes g :: "'r"   and h :: "'r" assumes "g \<in> R"   and "h \<in> R" shows "g + h \<in> R"
if_eqvt: fixes pi :: "('e \<times> 'e) list"   and b :: "bool"   and c1 :: "'d"   and c2 :: "'d" shows "pi \<bullet> (if b then c1 else c2) = (if pi \<bullet> b then pi \<bullet> c1 else pi \<bullet> c2)"
arrE: fixes f :: "'a" assumes "arr f" shows "C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f)"
prio_def: shows "prio = (\<lambda>p. if list_all (\<lambda>x. x = dir.Left) (butlast p) then 0 else 1)"
induct: fixes P :: "Cooper.num \<Rightarrow> bool"   and a0 :: "Cooper.num" assumes "\<And>c. P (C c)"   and "\<And>n. P (Bound n)"   and "\<And>n i a. P a \<Longrightarrow> P (CN n i a)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; \<And>x xa y. \<lbrakk>x = zsplit0 a; (xa, y) = x\<rbrakk> \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; \<And>x xa y. \<lbrakk>x = zsplit0 a; (xa, y) = x\<rbrakk> \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>i a. P a \<Longrightarrow> P (Mul i a)" shows "P a0"
minus_in_interval: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x - y \<in>\<^sub>i X - Y"
iMin_le: fixes k :: "'b"   and I :: "'b set" assumes "k \<in> I" shows "iMin I \<le> k"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray" shows "fs_state (f, mu, d) = of_list_repr f"