hcju/isabelleps huggingface dataset:
first_countable_basisE: fixes x :: "'a" obtains \<A> :: "'a set set" where "countable \<A>" and "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" and "\<And>A. A \<in> \<A> \<Longrightarrow> open A" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
lincomb_dim: fixes X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite X"   and "X \<subseteq> carrier_vec n" shows "dim_vec (lincomb a X) = n"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "src f = null"
invertibleMtx1: shows "invertible_mat (M_mat_R signs1 subsets1)"
l11_29_a: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "\<exists>Q. C InAngle A B Q \<and> A B Q CongA D E F"
wf_prog_compPI: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and wf\<^sub>1 :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and wf\<^sub>2 :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>C M Ts T m. \<lbrakk>P \<turnstile> C sees M: Ts\<rightarrow>T = m in C; wf_mdecl wf\<^sub>1 P C (M, Ts, T, m)\<rbrakk> \<Longrightarrow> wf_mdecl wf\<^sub>2 (compP f P) C (M, Ts, T, f m)"   and "wf_prog wf\<^sub>1 P" shows "wf_prog wf\<^sub>2 (compP f P)"
max_perm_degree_def: fixes A :: "('c poly, 'd) sq_matrix" shows "max_perm_degree A = (MAX p\<in>{p. p permutes UNIV}. \<Sum>i\<in>UNIV. degree (to_fun A i (p i)))"
incseq_le: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c"   and n :: "nat" assumes "incseq X"   and "X \<longlonglongrightarrow> L" shows "X n \<le> L"
g: shows "g \<in> carrier G \<and> \<phi> g x = y"
fU0: fixes z :: "complex" assumes "z \<in> U" shows "f z = 0"
PerpAt_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D \<equiv> A \<noteq> B \<and> C \<noteq> D \<and> Tarski_neutral_dimensionless.Col Bet X A B \<and> Tarski_neutral_dimensionless.Col Bet X C D \<and> (\<forall>U V. Tarski_neutral_dimensionless.Col Bet U A B \<and> Tarski_neutral_dimensionless.Col Bet V C D \<longrightarrow> Tarski_neutral_dimensionless.Per Bet Cong U X V)"
Var_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and b :: "('e \<Rightarrow> 'f list) \<times> (('f list \<Rightarrow> 'f list) \<Rightarrow> 'e \<Rightarrow> 'e)"   and a :: "('d \<Rightarrow> 'f list) \<times> (('f list \<Rightarrow> 'f list) \<Rightarrow> 'd \<Rightarrow> 'd)" assumes "P \<preceq>S Q"   and "\<And>ff A. S (snd b ff A) = snd a ff (S A)" shows "Var a P \<preceq>S Var b Q"
rel_refl_strong: fixes x :: "'d cset"   and Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z. z \<in> rcset x \<Longrightarrow> Ra z z" shows "rel_cset Ra x x"
simps_2: fixes p :: "'c \<times> 'd"   and ps :: "('c \<times> 'd) list" shows "map_of (p # ps) = map_of ps(fst p \<mapsto> snd p)"
is_discrete: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a0 :: "'b"   and a1 :: "'b" assumes "binary_product_diagram C a0 a1" shows "partial_magma.ide C a0 \<and> partial_magma.ide C a1"
that: fixes r :: "real" assumes "0 < r"   and "ball \<xi> r \<subseteq> S"   and "\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0" shows "thesis"
part_circlepath_conv_subpath: fixes c :: "complex"   and r :: "real"   and a :: "real"   and b :: "real" shows "part_circlepath c r a b = subpath (a / (2 * pi)) (b / (2 * pi)) (circlepath c r)"
edom_Trivial_Aexp: fixes e :: "exp"   and n :: "Arity" shows "edom (Aexp_syn n e) = fv e"
map_take_subseq: fixes f :: "'d \<Rightarrow> 'c"   and n :: "nat"   and xs :: "'d list" shows "map f (take n xs) \<preceq> map f xs"
poly_mult_lead_coeff_aux: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p1 :: "'c list"   and p2 :: "'c list" assumes "domain R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p1"   and "polynomial\<^bsub>R\<^esub> K p2"   and "p1 \<noteq> []"   and "p2 \<noteq> []" shows "ring.coeff R (ring.poly_mult R p1 p2) (degree p1 + degree p2) = lead_coeff p1 \<otimes>\<^bsub>R\<^esub> lead_coeff p2"
CAs0'_CAs: shows "CAs0' \<cdot>\<cdot>cl \<eta>s0' = CAs"
subst: fixes P :: "'b \<Rightarrow> bool"   and t :: "'b"   and env :: "(name, 'b) fmap" assumes "subst_syntactic_and P"   and "P t"   and "fmpred (\<lambda>_. P) env" shows "P (subst t env)"
mndet_unit: fixes a :: "'a"   and P :: "'a \<Rightarrow> 'a process" shows "mndet {a} P = (a \<rightarrow> P a)"
less_max_iff_disj: fixes z :: "'b word"   and x :: "'b word"   and y :: "'b word" shows "(z <s signed.max x y) = (z <s x \<or> z <s y)"
methd_Some_dynmethd_Some: fixes G :: "prog"   and statC :: "qtname"   and sig :: "sig"   and statM :: "qtname \<times> methd"   and dynC :: "qtname" assumes "methd G statC sig = Some statM"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC"   and "is_class G statC"   and "ws_prog G" shows "\<exists>dynM. dynmethd G statC dynC sig = Some dynM"
tendsto_def: fixes f :: "'c \<Rightarrow> 'a"   and l :: "'a"   and F :: "'c filter" shows "(f \<longlongrightarrow> l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
hfun_sigma_ord_def: fixes r :: "hf" shows "hfun_sigma_ord r \<equiv> \<forall>z\<^bold>\<in>r. \<forall>z'\<^bold>\<in>r. \<exists>x y x' y'. z = \<langle>x, y\<rangle> \<and> z' = \<langle>x', y'\<rangle> \<and> Ord x \<and> Ord x' \<and> (x = x' \<longrightarrow> y = y')"
divide_numeral_1: fixes a :: "'a" shows "a / Numeral1 = a"
\<epsilon>_def: shows "\<epsilon> = Min {pmf p x / (pmf p x - pmf q x) |x. pmf q x < pmf p x}"
assms: shows "(x, y) \<in> matchrel\<^sup>*"
RDirProd_def: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "RDirProd R S = monoid.extend (R \<times>\<times> S) \<lparr>zero = \<one>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>, add = (\<otimes>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>)\<rparr>"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "F (g \<cdot>\<^sub>A f) = F g \<cdot>\<^sub>B F f"
hyps_3: assumes "y \<notin> xs"   and "\<And>x. x \<in> xs \<Longrightarrow> degree (f x) = 0" shows "coeff (prod f (insert y xs)) i = (\<Prod>x\<in>xs. coeff (f x) 0) * coeff (f y) i"
vsv_vconst_on: fixes A :: "V"   and c :: "V" shows "vsv (vconst_on A c)"
npeneg_correct: fixes xs :: "'a list"   and e :: "pexpr" shows "peval xs (npeneg e) = peval xs (PExpr1 (PNeg e))"
mdpt_gt: assumes "z < - b / (2 * a)" shows "a * z\<^sup>2 + b * z + c < 0"
homeomorphic_empty_2: fixes S :: "'b set" shows "{} homeomorphic S = (S = {})"
ANDConc: fixes a :: "('b set set, 'c) Combinators"   and p :: "('b set set, 'c) Combinators list" assumes "allNetsDistinct (a # p)" shows "allNetsDistinct p"
ran_process_mru_only_voters: fixes vs :: "nat \<Rightarrow> process \<Rightarrow> val option"   and Q :: "process set" shows "ran (process_mru vs |` Q) = ran (process_mru vs |` (Q \<inter> voters vs))"
well_def_wp_under: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "wp_under_wlp prog"
finite_Units: fixes m :: "int" assumes "residues m" shows "finite (Units (residue_ring m))"
assms: shows "tabulation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C r \<rho> f g"
elims: fixes x :: "real mpoly"   and y :: "real option" assumes "get_if_const x = y" obtains p :: "real mpoly" where "x = p" and "y = (if is_constant p then Some (MPoly_Type.coeff p 0) else None)"
1: shows "OpposedThinChamberComplexFoldings X f g' C0"
cons: fixes y :: "'a"   and ys :: "'a list" assumes "xs = y # ys"   and "P y"   and "list_all P ys" shows "thesis"
changes_Cons: fixes x :: "'a"   and xs :: "'a list" shows "changes (x # xs) = (if filter (\<lambda>x. x \<noteq> (0::'a)) xs = [] then 0 else if x * hd (filter (\<lambda>x. x \<noteq> (0::'a)) xs) < (0::'a) then 1 + changes xs else changes xs)"
accept: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Accept], Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
left_commute: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "b \<sqinter> (a \<sqinter> c) = a \<sqinter> (b \<sqinter> c)"
ResF: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and x :: "name" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "<\<nu>x>P \<Longrightarrow>\<^sup>^\<alpha> \<prec> <\<nu>x>P'"
select_in_set: fixes k :: "nat"   and xs :: "'a list" assumes "k < length xs" shows "select k xs \<in> set xs"
i_cut_absorb_disj: fixes cut_op :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and I :: "'a set"   and t :: "'a" assumes "cut_op = (\<down><) \<or> cut_op = (\<down>\<le>) \<or> cut_op = (\<down>>) \<or> cut_op = (\<down>\<ge>)" shows "cut_op (cut_op I t) t = cut_op I t"
f5: shows "arcs i j xs = arcs i (last xs) (butlast xs) @ [(last xs, j)]"
natLeq_UNIV_ofilter: shows "ofilter natLeq UNIV"
interfree_aux2: fixes c :: "'b ann_com option"   and s :: "'b"   and r :: "'b ann_com option"   and t :: "'b"   and q :: "'b set"   and a :: "'b ann_com option" assumes "(c, s) -1\<rightarrow> (r, t)"   and "interfree_aux (c, q, a)" shows "interfree_aux (r, q, a)"
birkhoff_sum_1_3: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "birkhoff_sum f (Suc 0) x = f x"
not_up_less_UU: fixes x :: "'b" shows "up\<cdot>x \<notsqsubseteq> \<bottom>"
inverse_inverse: fixes a :: "'ag" assumes "a \<in> U" shows "-\<^sub>o\<^sub>w (-\<^sub>o\<^sub>w a) = a"
basic_trans_rules_23: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
geq_arg_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "geq_arg f \<equiv> \<forall>x y. x \<le> f x y \<and> y \<le> f x y"
write_state_reg_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (write_state_reg_instr instr s1)) \<and> t2 = snd (fst (write_state_reg_instr instr s2))" shows "low_equal t1 t2"
sel: fixes x2 :: "'h" shows "ok (OK x2) = x2"
k2_def: shows "k2 = s2 / (1 - s1)"
r_prefixes_aux_def: shows "r_prefixes_aux \<equiv> Cn 3 r_ifeq [Cn 3 r_s [Cn 3 r_snoc [recf.Id 3 1, r_constn 2 0]], Cn 3 r_s [recf.Id 3 1], Cn 3 r_snoc [recf.Id 3 1, r_constn 2 1], Cn 3 r_snoc [recf.Id 3 1, r_constn 2 0]]"
VPow_le_VPow_iff: fixes a :: "V"   and b :: "V" shows "(VPow a \<le> VPow b) = (a \<le> b)"
path_from_to_last': fixes v :: "'a"   and xs :: "'a list"   and x :: "'a"   and xs' :: "'a list"   and w :: "'a" assumes "v \<leadsto>(xs @ x # xs')\<leadsto> w" shows "w \<notin> set xs"
MGT_def: fixes c :: "com" shows "MGT c = ((=), c, \<lambda>z. exec z c)"
ZassenhausTr6_2: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "Group (\<natural>\<^bsub>G\<^esub>H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K / (H1 \<diamondop>\<^bsub>G\<^esub> H \<inter> K1))"
submodule_sum_def: fixes N1 :: "'c set"   and N2 :: "'c set" shows "submodule_sum N1 N2 = (\<lambda>(x, y). x \<oplus>\<^bsub>M\<^esub> y) ` {(x, y). x \<in> N1 \<and> y \<in> N2}"
sum_upto_divisor_sum1: fixes f :: "nat \<Rightarrow> real"   and x :: "real" shows "sum_upto (\<lambda>n. \<Sum>d | d dvd n. f d) x = sum_upto (\<lambda>n. f n * real_of_int \<lfloor>x / real n\<rfloor>) x"
trg_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.hseq \<mu> \<nu>" shows "trg (\<mu> \<star> \<nu>) = trg \<mu>"
mult_Suc_right: fixes m :: "nat"   and n :: "nat" shows "m *n Suc n = m + (m *n n)"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and i :: "nat" shows "first_ind (x # xs) y i = (if x = y then i else first_ind xs y (Suc i))"
2: fixes a :: "'a" assumes "I +> a \<noteq> I"   and "a \<in> carrier R" shows "\<exists>y\<in>carrier R. I +> a \<otimes> y = I +> \<one>"
has_integral_0_eq: fixes i :: "'b"   and S :: "'a set" shows "((\<lambda>x. 0::'b) has_integral i) S = (i = (0::'b))"
intros: fixes rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m"   and y :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m" assumes "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y" shows "blinding_of_G rha boa rhb bob (G\<^sub>m x) (G\<^sub>m y)"
map_cong0: fixes x :: "'a stream"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in> sset x \<Longrightarrow> f z = g z" shows "smap f x = smap g x"
mult_le_cancel_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a c) (times b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
Diff_disjoint: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> (B\<setminus>A) = {}"
is_map_def: fixes M :: "('a \<times> 'b) set" shows "is_map M = (\<forall>(a\<^sub>1, b\<^sub>1)\<in>M. \<forall>(a\<^sub>2, b\<^sub>2)\<in>M. a\<^sub>1 = a\<^sub>2 \<longrightarrow> b\<^sub>1 = b\<^sub>2)"
countable_image: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "countable A" shows "countable (f ` A)"
pell_square_minus1: shows "pell (a\<^sup>2 - Suc 0)"
assms: shows "A \<subseteq>\<^sub>\<circ> set {x}"
append_eq_append_conv: fixes xs :: "'b list"   and ys :: "'b list"   and us :: "'b list"   and vs :: "'b list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
guha_equal_Action: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d"   and a :: "'e" assumes "no_overlaps \<gamma> ft"   and "OF_priority_match \<gamma> ft p = Action a" shows "guha_table_semantics \<gamma> ft p (Some a)"
fill_stable: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat"   and k :: "'d" assumes "index_less index key"   and "index_same index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "ns \<noteq> []"   and "offs_pred ns ub xs index key mi ma" shows "map the (filter (\<lambda>w. \<exists>x. w = Some x \<and> key x = k) (fill xs ns index key ub mi ma)) = filter (\<lambda>x. k = key x) xs"
intro: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B1 :: "'b set"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and B2 :: "'c set" assumes "finite_dimensional_vector_space s1 B1"   and "finite_dimensional_vector_space s2 B2" shows "finite_dimensional_vector_space_pair s1 B1 s2 B2"
Span_base_incl: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R" shows "set Us \<subseteq> ring.Span R K Us"
path_trans_single: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t :: "'d list"   and s' :: "'c"   and e :: "'d"   and sn :: "'c" assumes "s t\<Longrightarrow>\<^bsub>SES\<^esub> s'"   and "s' e\<longrightarrow>\<^bsub>SES\<^esub> sn" shows "s (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> sn"
complex_is_Real_iff: fixes z :: "complex" shows "(z \<in> \<real>) = is_real z"
llcp_lappend_same: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" shows "llcp (lappend xs ys) (lappend xs zs) = llength xs + llcp ys zs"
assms_1: assumes "\<paragraph>''then''"   and "bval b s" shows "wlp \<pi> c\<^sub>1 Q s"
absorb1: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" shows "a \<inter># b = a"
separation: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x * y\<^sup>\<circ>" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ>"
integral_cong_AE: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
NT_NF: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<notin> T P" shows "(s, {}) \<notin> F P"
agents_eq: shows "agents = {i. permute_profile (permutation_of_list xs) R i \<noteq> (\<lambda>_ _. False)}"
HL_is_L: shows "(\<mu> \<rho>''. (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>')(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>')) ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>'') = (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>')(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>'))"
eq_refl: fixes \<phi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>P \<phi>"
restrict_hom_iff: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "((\<lambda>x. if x \<in> carrier G then f x else g x) \<in> hom G H) = (f \<in> hom G H)"
vec_smult_Cons: fixes \<alpha> :: "'b"   and a :: "'b"   and as :: "'b list" shows "vec_smult \<alpha> (a # as) = \<alpha> * a # vec_smult \<alpha> as"
bisimReflexive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<sim> P"
pt_set_bij: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(pi \<bullet> x \<in> pi \<bullet> X) = (x \<in> X)"
comp1: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'b" assumes "bounded_cbilinear prod"   and "bounded_clinear g" shows "bounded_cbilinear (\<lambda>x. prod (g x))"
DSourcesA23_L1: shows "DSources level1 sA23 = {sA11}"
direct_decomp_cone_insert: fixes x :: "'x"   and U :: "'x set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "x \<notin> U" shows "direct_decomp (cone (h, insert x U)) [cone (h, U), cone (monomial (1::'b) (monomial (Suc 0) x) * h, insert x U)]"
aset_sum_def: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and K :: "'c set" shows "H \<minusplus>\<^bsub>A\<^esub> K = H \<diamondop>\<^bsub>b_ag A\<^esub> K"
set_mset_intersect: fixes M :: "'a multiset"   and A :: "'a set" shows "set_mset (M \<inter>s A) = set_mset M \<inter> A"
prems_1: fixes w :: "('a, 'b) tm" assumes "s \<unrhd>\<^sub>e\<^sub>m\<^sub>b w"   and "subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "\<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u" shows "thesis"
swap: fixes p :: "perm"   and a :: "atom"   and b :: "atom" assumes "P p"   and "a \<noteq> b"   and "sort_of a = sort_of b" shows "P ((a \<rightleftharpoons> b) + p)"
ge_plus_game_right: fixes y :: "game"   and z :: "game"   and x :: "game" shows "ge_game (y, z) = ge_game (plus_game y x, plus_game z x)"
pmf_integral_pmf_set_integral: fixes p :: "'e pmf"   and f :: "'e \<Rightarrow> real" shows "pmf_integral p f = pmf_set_integral p f (set_pmf p)"
wfPUNIVI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>P x. \<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wfP r"
elims_2: fixes x :: "'a extT"   and xa :: "'a extT" assumes "x < xa" obtains x :: "'a extT" and y :: "'a extT" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
add_transl: fixes x :: "'a"   and y :: "'a" shows "!x + !y = !x \<oplus> !y"
before_between: fixes f :: "face"   and x :: "nat"   and y :: "nat"   and z :: "nat" assumes "before (verticesFrom f x) y z"   and "distinct (vertices f)"   and "x \<in> \<V> f"   and "x \<noteq> y" shows "y \<in> set (between (vertices f) x z)"
remdups_filter: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "remdups (List.filter P xs) = List.filter P (remdups xs)"
cimage_ident: fixes Y :: "'f cset" shows "cimage (\<lambda>x. x) Y = Y"
jumpF_polyR_inverse_add: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" assumes "coprime p q" shows "jumpF_polyR q p x + jumpF_polyR p q x = jumpF_polyR 1 (q * p) x"
pair_acc_modi_def: fixes p :: "'a \<times> 'b" shows "accmodi p = accmodi (snd p)"
entt_fr_drop: fixes F :: "assn"   and F' :: "assn"   and A :: "assn" assumes "F \<Longrightarrow>\<^sub>t F'" shows "F * A \<Longrightarrow>\<^sub>t F'"
Safe_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "Safe P = (\<lambda>s. P s \<or> M\<^sub>p P s)"
le_max_word_ucast_id: fixes x :: "'c word" assumes "x \<le> UCAST('d \<rightarrow> 'c) (- 1)" shows "UCAST('d \<rightarrow> 'c) (UCAST('c \<rightarrow> 'd) x) = x"
sterm_to_pterm_all_frees: fixes t :: "sterm" shows "all_frees (sterm_to_pterm t) = all_frees t"
empty_def: shows "SkewBinomialHeap_loc.empty \<equiv> Abs_SkewBinomialHeap BsSkewBinomialHeapStruc.bs_empty"
uint_shiftl: fixes n :: "'a word"   and i :: "nat" shows "uint (n << i) = take_bit (size n) (uint n << i)"
closed_Collect_conj: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.topological_space open"   and "topological_space.closed open {x. P x}"   and "topological_space.closed open {x. Q x}" shows "topological_space.closed open {x. P x \<and> Q x}"
P_t_app: fixes t :: "'a"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b" assumes "P t (app u\<^sub>1 u\<^sub>2)" shows "\<exists>t\<^sub>1 t\<^sub>2. t = app t\<^sub>1 t\<^sub>2 \<and> P t\<^sub>1 u\<^sub>1 \<and> P t\<^sub>2 u\<^sub>2"
fmeasurable_eq_sets: shows "fmeasurable local.lim = P.events"
dg_Par_Hom_vsubset_dg_Rel_Hom: fixes a :: "V"   and \<alpha> :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> dg_Par \<alpha>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> dg_Par \<alpha>\<lparr>Obj\<rparr>" shows "Hom (dg_Par \<alpha>) a b \<subseteq>\<^sub>\<circ> Hom (dg_Rel \<alpha>) a b"
rbtreeify_g_conv_drop: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "snd (rbtreeify_g n kvs) = drop (n - 1) kvs"