hcju/isabelleps huggingface dataset:
c_prod_comm: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<parallel> y = y \<parallel> x"
ListMem_iff: fixes x :: "'a"   and xs :: "'a list" shows "ListMem x xs = (x \<in> set xs)"
map_of_concat': fixes k :: "'a"   and l :: "('a \<times> 'b) list list"   and i :: "nat" assumes "k \<in> dom (map_of (l ! i)) \<and> i < length l" shows "k \<in> dom (map_of (concat l))"
Trans: shows "<\<nu>y>P \<longmapsto> a<\<nu>x> \<prec> RP'"
bot_real_max_bot_def: shows "bot \<equiv> MInfty"
count_prime_factorization: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "count (factorial_semiring.prime_factorization zero times one normalize x) p = (if normalization_semidom.prime zero times one normalize p then factorial_semiring.multiplicity times one p x else 0)"
less_ncols: fixes n :: "nat"   and A :: "'b matrix" shows "(n < ncols A) = (\<exists>j i. n \<le> i \<and> Rep_matrix A j i \<noteq> (0::'b))"
weaken: fixes a :: "'b seq_action" assumes "I' a" shows "I a"
eq: shows "ideal_decomp_aux F f = (S, map (apfst ((*) f)) (snd (split 0 X L')))"
intro: fixes n :: "nat" assumes "\<not> Carmichael_number n \<or> n < 3" shows "fermat_test_not_Carmichael_number n"
image_ring_zero': fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set" assumes "weak_ring_morphism f I R" shows "the_elem (f ` \<zero>\<^bsub>R Quot I\<^esub>) = \<zero>\<^bsub>image_ring f R\<^esub>"
LeftDerivationLadder_cut_appendix: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"   and "ladder_i L 0 < length \<alpha>" shows "\<exists>E F \<gamma>1 \<gamma>2 L'. D = E @ F \<and> \<gamma> = \<gamma>1 @ \<gamma>2 \<and> LeftDerivationLadder \<alpha> E L' \<gamma>1 \<and> derivation_ge F (length \<gamma>1) \<and> LeftDerivation \<delta> (derivation_shift F (length \<gamma>1) 0) \<gamma>2 \<and> length L' = length L \<and> ladder_i L' 0 = ladder_i L 0 \<and> ladder_last_j L' = ladder_last_j L"
a_d_mult_closure: fixes x :: "'a"   and y :: "'a" shows "d (ad x \<cdot> ad y) = ad x \<cdot> ad y"
uwalk_length_conv: fixes p :: "nat list" shows "uwalk_length p = length p - 1"
toEnum_def: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and n :: "nat" assumes "class.enum enum enum_all enum_ex" shows "enum.toEnum enum n \<equiv> if n < length enum then enum ! n else the None"
EphK_LtK: fixes X :: "ephkey"   and X' :: "ltkey" shows "EphK X \<noteq> LtK X'"
ide_in_hom: fixes a :: "('o, 'a) arr" shows "ide a = in_hom a a a"
ltake_ltake: fixes n :: "enat"   and m :: "enat"   and xs :: "'a llist" shows "ltake n (ltake m xs) = ltake (min n m) xs"
f_shrink_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" shows "xs \<up> (n * k) \<div> k = xs \<div> k \<up> n"
effectE: fixes c :: "'c Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'c" assumes "effect c h h' r" obtains "r = fst (the (execute c h))" and "h' = snd (the (execute c h))" and "success c h"
normalize_rules_preserves: fixes rs :: "'b rule list"   and P :: "'b match_expr \<Rightarrow> bool"   and f :: "'b match_expr \<Rightarrow> 'b match_expr list" assumes "\<forall>r\<in>set rs. P (get_match r)"   and "\<forall>m. P m \<longrightarrow> (\<forall>m'\<in>set (f m). P m')" shows "\<forall>r\<in>set (normalize_rules f rs). P (get_match r)"
list_renaming_perm: fixes bs :: "atom list"   and p :: "perm" shows "\<exists>q. (\<forall>b\<in>set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> p \<bullet> set bs"
While_backwards_frontier_both_ir_valid: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<And>n. ir_valid (\<lambda>s s'. P n s s' \<and> bval b s \<and> bval b' s') c c' (P (Suc n))"   and "ir_valid (\<lambda>s s'. \<exists>n. P n s s') (WHILE b DO c) (WHILE b' DO c') Q" shows "ir_valid (P 0) (WHILE b DO c) (WHILE b' DO c') Q"
prime_factorsI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p x" shows "p \<in># factorial_semiring.prime_factorization zero times one normalize x"
fdr_step_def: fixes v :: "nat"   and c :: "nat" shows "local.fdr_step v c = (if v = 0 then return_pmf None else let x = 2 ^ nat \<lceil>log 2 (real (max 1 n)) - log 2 (real v)\<rceil> in map_spmf (\<lambda>bs. (x * v, x * c + bs)) (spmf_of_set {..<x}))"
even_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) zero"
has_derivative_sum: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and f' :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and F :: "'d filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
None: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue (Suc l) xs" shows "is_binqueue l (None # xs)"
subset_imp: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "\<Sqinter> B \<preceq> \<Sqinter> A"
bind_is_OK_I2: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog" assumes "h \<turnstile> ok f"   and "\<And>x h'. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> h' \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
select_convs_1: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and gbg_F :: "'a set set"   and more :: "'b" shows "gbg_F \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, gbg_F = gbg_F, \<dots> = more\<rparr> = gbg_F"
all_bounds_def: fixes Q :: "'a set" shows "all_bounds Q = {Q\<^sub>b. is_bound Q\<^sub>b Q}"
simps_1: fixes n :: "String.literal"   and S :: "variable set" shows "variant_variable (Free n) S = (let s' = fresh_name' (name ` S) in (Free s', insert (Free s') S))"
dim_col_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b vec" shows "dim_col (mat\<^sub>r nr nc g) = nc"
sum0: shows "(\<Sum>i\<in>{0..<n - 1} - {k - 1}. mat_delete (make_mat n k B) 0 k $$ (i, 0) * Determinant.cofactor (mat_delete (make_mat n k B) 0 k) i 0) = (0::'a)"
is_path_f: assumes "infinite (deriv s)" shows "\<forall>n. f n \<in> deriv s \<and> fst (f n) = n \<and> snd (f (Suc n)) \<in> set (subs (snd (f n))) \<and> infinite (deriv (snd (f n)))"
L1: fixes P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'" shows "(Q, P') \<notin> weakBisim"
proj_poly_syz_mult_scalar: fixes n :: "nat"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly_syz n (q \<odot> p) = q \<odot> proj_poly_syz n p"
HMA_submatrix: fixes I :: "nat set"   and J :: "nat set" assumes "card {i. i < CARD('b) \<and> i \<in> I} = CARD('c)"   and "card {i. i < CARD('d) \<and> i \<in> J} = CARD('e)" shows "rel_fun HMA_M HMA_M (\<lambda>A. submatrix A I J) (\<lambda>A. submatrix_hma A I J)"
hyps: fixes P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<turnstile>\<^sub>p {wlp S P} S {P}"
find_eigenvector: fixes A :: "'c mat"   and n :: "nat"   and e :: "'c" assumes "A \<in> carrier_mat n n"   and "eigenvalue A e" shows "eigenvector A (find_eigenvector A e) e"
emb_step_at_subst: fixes t :: "('c, 'd) tm"   and p :: "dir list"   and d :: "dir"   and \<rho> :: "'d \<Rightarrow> ('c, 'd) tm" assumes "position_of t (p @ [d])" shows "emb_step_at p d (subst \<rho> t) = subst \<rho> (emb_step_at p d t)"
bounded_lattice_bot_ow_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c" shows "bounded_lattice_bot_ow U inf le ls sup bot \<equiv> lattice_ow U inf le ls sup \<and> order_bot_ow U bot le ls"
facetrelI_cardSuc: fixes z :: "'b set"   and x :: "'b set" assumes "order.greater_eq x z"   and "card x = Suc (card z)" shows "z \<lhd> x"
VUnion_vsubsetI: fixes A :: "V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> \<exists>y. y \<in>\<^sub>\<circ> B \<and> x \<subseteq>\<^sub>\<circ> y" shows "\<Union>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> B"
LCD_def: fixes B :: "'c set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "LCD B D f \<equiv> (\<forall>x y z. x \<in> B \<longrightarrow> y \<in> B \<longrightarrow> z \<in> D \<longrightarrow> f x (f y z) = f y (f x z)) \<and> (\<forall>x y. x \<in> B \<longrightarrow> y \<in> D \<longrightarrow> f x y \<in> D)"
enabled_def: fixes SES :: "('a, 'b) SES_rec"   and s :: "'a"   and t :: "'b list" shows "enabled SES s t \<equiv> \<exists>s'. s t\<Longrightarrow>\<^bsub>SES\<^esub> s'"
dis_introL: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<phi> \<pi>l" shows "Shallow.dis \<phi> \<psi> \<pi>l"
is_valid_problem_strips_def: fixes \<Pi> :: "'c strips_problem" shows "is_valid_problem_strips \<Pi> \<equiv> let ops = \<Pi>\<^sub>\<O>; vs = \<Pi>\<^sub>\<V>; I = \<Pi>\<^sub>I; G = \<Pi>\<^sub>G in list_all (is_valid_operator_strips \<Pi>) ops \<and> (\<forall>v. (I v \<noteq> None) = ListMem v vs) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem v vs)"
assms_2: shows "\<forall>X\<in>P. card X = 1"
col_map_mat: fixes j :: "nat"   and A :: "'a mat"   and f :: "'a \<Rightarrow> 'b" assumes "j < dim_col A" shows "col (map_mat f A) j = map_vec f (col A j)"
G_nested_propos_alt_def: fixes \<phi> :: "'a ltl" shows "\<^bold>G \<phi> = nested_propos \<phi> \<inter> {\<psi>. \<exists>x. \<psi> = G x}"
power_strict_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less a b"   and "less_eq zero a" shows "0 < n \<longrightarrow> less (power.power one times a n) (power.power one times b n)"
functor_axioms: shows "functor (\<cdot>) (\<cdot>\<acute>) \<Phi>"
chambers_3: 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 "C0 \<noteq> g ` C0"
span_substd_basis: fixes d :: "'c set" assumes "d \<subseteq> Basis" shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
poly_rel_cCons: shows "(R ===> list_all2 R ===> list_all2 R) (cCons_i ops) cCons"
permutes_bij: fixes p :: "'e \<Rightarrow> 'e"   and s :: "'e set" assumes "p permutes s" shows "bij p"
B: assumes "D x \<notin> sources I D u (x # xs)" shows "ipurge_tr_rev_t_aux \<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set \<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr>"
cStatEq: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', R, S) \<in> Rel"   and "\<Psi>' \<simeq> \<Psi>''" shows "(\<Psi>'', R, S) \<in> Rel"
sum_le_suminf: fixes f :: "nat \<Rightarrow> 'a"   and I :: "nat set" assumes "summable f"   and "finite I"   and "\<And>n. n \<in> - I \<Longrightarrow> (0::'a) \<le> f n" shows "sum f I \<le> suminf f"
vimage_image_subset: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "A \<subseteq> f -` f ` A"
profile_bij_profile: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set"   and f :: "'d \<Rightarrow> 'd" assumes "profile A Is P"   and "bij_betw f Is Is" shows "profile A Is (P \<circ> f)"
24648: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y \<squnion> - (- x \<sqinter> - y) = top"
prefixes_prod_encode: fixes t :: "nat"   and j :: "nat" assumes "local.prefixes t j \<down>" obtains b :: "nat" where "local.prefixes t j \<down>= b"
assms_3: shows "\<bar>valof a * valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
CPRE: shows "P \<Longrightarrow>\<^sub>A P'"
distinct_19: fixes x2 :: "nat"   and x61 :: "msg"   and x62 :: "msg" shows "Number x2 \<noteq> \<lbrace>x61, x62\<rbrace>"
basis_indep: fixes X :: "(nat \<times> int \<times> nat) set" assumes "s.SpanningForest X" shows "forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
safe_delayedE: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "'c"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "safe_delayed (ts, m, \<S>)"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, x, \<D>, \<O>, \<R>)" shows "map owned ts,map released ts,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
nth_Cons: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
finite_vimageI: fixes F :: "'c set"   and h :: "'d \<Rightarrow> 'c" assumes "finite F"   and "inj h" shows "finite (h -` F)"
exec_move_FAccI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and F :: "String.literal"   and D :: "String.literal" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<bullet>F{D}) h s ta h' s'"
HPhase1or2Write_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv6 s"   and "HInv4a s p"   and "Inv2c s" shows "HInv6 s'"
unambiguous_routing_Cons: fixes r :: "('c, 'd) routing_rule_scheme"   and rtbl :: "('c, 'd) routing_rule_scheme list" assumes "unambiguous_routing (r # rtbl)" shows "unambiguous_routing rtbl"
TRANSFER_tag_def: fixes P :: "'a" shows "TRANSFER_tag P \<equiv> P"
nn_integral_cong': fixes N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space N =simp=> f x = g x" shows "integral\<^sup>N N f = integral\<^sup>N N g"
\<Lambda>_PBD_axioms: shows "\<Lambda>_PBD \<V> \<B> \<Lambda> \<K>"
A_def: shows "A = {r \<in> lotteries_on alts. q \<prec>[Pareto (SD \<circ> R)] r}"
fifo_push_relabel_def: shows "fifo_push_relabel \<equiv> let f = pp_init_f; l = pp_init_l in (spec l. distinct l \<and> set l = {v \<in> V - {s, t}. excess f v \<noteq> (0::'capacity)}) \<bind> (\<lambda>Q. while\<^sub>T (\<lambda>(f, l, Q). Q \<noteq> []) (\<lambda>(f, l, Q). fifo_discharge f l Q) (f, l, Q) \<bind> (\<lambda>(f, l, uu_). assert (Height_Bounded_Labeling c s t f l) \<bind> (\<lambda>_. return f)))"
gallery_in_subcomplex: fixes Y :: "'a set set"   and ys :: "'a set list" assumes "ChamberSubcomplex Y"   and "order.greater_eq Y (set ys)"   and "gallery ys" shows "SimplicialComplex.maxsimpchain Y ys"
bst_inter: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.bst (Set2_Join.inter join t1 t2)"
topspace_subtopology_subset: fixes S :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> topspace X" shows "topspace (subtopology X S) = S"
equation_minus_iff: fixes a :: "'a"   and b :: "'a" shows "(a = - b) = (b = - a)"
convex_rel_interior_closure: fixes S :: "'c set" assumes "convex S" shows "rel_interior (closure S) = rel_interior S"
chaos_par_top: shows "\<top> \<sqsubseteq> chaos \<parallel> \<top>"
a: shows "N[x\<turnstile>n>y] = Ax z c"
interp_factorization_reflect: fixes fctrs :: "'a \<times> ('a \<times> nat) list" assumes "(0::'a) \<notin> fst ` set (snd fctrs)" shows "reflect_poly (interp_factorization fctrs) = interp_alt_factorization fctrs"
per_supset_union: fixes p1 :: "('a \<times> 'a) set"   and p2 :: "('a \<times> 'a) set"   and x1 :: "'a"   and x2 :: "'a" assumes "(p1, p2) \<in> local.per_supset_rel"   and "x1 \<in> Domain p2"   and "x2 \<in> Domain p2" shows "(per_union p1 x1 x2, per_union p2 x1 x2) \<in> local.per_supset_rel"
onode_sos_dest_is_net_state': fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and p :: "'e"   and R :: "nat set"   and a :: "'f node_action"   and s' :: "(nat \<Rightarrow> 'd) \<times> 'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, NodeS i p R), a, s') \<in> onode_sos S" shows "\<exists>\<sigma>' \<zeta>' R'. s' = (\<sigma>', NodeS i \<zeta>' R')"
addsI: fixes a :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and k :: "'c" assumes "a = plus b k" shows "plus.adds plus b a"
codomains_def: fixes f :: "'a" shows "H.codomains f \<equiv> {b. H.ide b \<and> b \<star> f \<noteq> H.null}"
Fresh: fixes Y :: "char list"   and Xs :: "char list set" assumes "Y \<notin> Xs \<or> infinite Xs" shows "fresh_string Xs Y = Y"
card1_tr1: fixes x :: "'c" shows "constmap {0} {x} \<in> {0} \<rightarrow> {x} \<and> surj_to (constmap {0} {x}) {0} {x}"
mult_minus_left: fixes a :: "'a"   and b :: "'a" shows "- a * b = - (a * b)"
next_name_def: fixes xs :: "name" shows "next xs = Name.append xs default"
assms_3: shows "\<exists>n'. latestAct_cond uid t n n'"
cases: fixes x :: "'c tree" obtains "x = \<langle>\<rangle>"   | l :: "'c tree" and uu_ :: "'c" and r :: "'c tree" where "x = \<langle>l, uu_, r\<rangle>"
simps_14: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (Plus e1 e2) (Minus v va) = True"
select_convs_1: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "hosts \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = hosts"
real_of_int_floor_add_one_gt: fixes r :: "real" shows "r < real_of_int \<lfloor>r\<rfloor> + 1"
f_diff: fixes i :: "nat" assumes "i \<in> N" shows "diff_fun k charts (f i)"
fractrel_iff: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" shows "fractrel x y = (snd x \<noteq> (0::'a) \<and> snd y \<noteq> (0::'a) \<and> fst x * snd y = fst y * snd x)"
neutr_eq_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "({#} = a \<union># b) = (a = {#} \<and> b = {#})"
strict_trans: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" assumes "b \<subset># a"   and "c \<subset># b" shows "c \<subset># a"
conv_callee_parallel_id_right: fixes callee :: "'g \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'g, 'e, 'f) gpv"   and s :: "'g"   and s' :: "'h" shows "CNV (callee \<ddagger>\<^sub>I 1\<^sub>I) (s, s') = CNV callee s |\<^sub>= 1\<^sub>C"
norm_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and u :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "norm u = 1"
iterate_Suc: fixes n :: "nat"   and F :: "'c \<rightarrow> 'c"   and x :: "'c" shows "F\<^bsup>Suc n\<^esup>\<cdot>x = F\<cdot>(F\<^bsup>n\<^esup>\<cdot>x)"
harry_meas_False_end: fixes xs :: "bool list" shows "harry_meas (xs @ [False]) = harry_meas xs"
retract_inverse: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "\<pi>\<^sub>* (\<pi>\<^sup>* x) = x"
LIMSEQ_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat" assumes "f \<longlonglongrightarrow> a" shows "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
source: fixes S1 :: "'a"   and S2 :: "'a"   and SRel :: "('a \<times> 'a) set"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" assumes "(S1, S2) \<in> SRel" shows "(SourceTerm S1, SourceTerm S2) \<in> encoding.indRelSTEQ Enc SRel TRel"
map_le_upds: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option"   and as :: "'b list"   and bs :: "'c list" assumes "f \<subseteq>\<^sub>m g" shows "f(as [\<mapsto>] bs) \<subseteq>\<^sub>m g(as [\<mapsto>] bs)"
valid_abs_goal_1: fixes v :: "nat" assumes "abs_ast_goal v \<noteq> None" shows "v \<in> set abs_ast_variable_section"
t_ins_def: fixes x :: "'b"   and xt :: "'b bintree" shows "t_ins x xt \<equiv> t_ins_out (t_ins_aux (t_ins_in x xt))"
singletonD: fixes b :: "'b"   and a :: "'b" assumes "b \<in> \<eta> a" shows "b = a"
scale_eq_0_iff: fixes a :: "'b"   and x :: "('b, 'c) vec" shows "(a *s x = 0) = (a = (0::'b) \<or> x = 0)"