hcju/isabelleps huggingface dataset:
add_diff_cancel_right: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and c :: "'k"   and b :: "'k" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
totient_dvd: fixes m :: "nat"   and n :: "nat" assumes "m dvd n" shows "totient m dvd totient n"
prv_eqv_refl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv \<phi> \<phi>)"
in_indets_poly_substE: fixes x :: "'b"   and f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<in> indets (poly_subst f p)" obtains y :: "'d" where "y \<in> indets p" and "x \<in> indets (f y)"
hyps: fixes tab :: "'a \<Rightarrow> 'b option"   and z :: "'b"   and ys :: "'b list" assumes "tab vn = Some z" shows "\<exists>z. (tab(xs [\<mapsto>] ys)) vn = Some z"
l6_3_2: fixes A :: "'p"   and P :: "'p"   and B :: "'p" assumes "A \<noteq> P"   and "B \<noteq> P"   and "\<exists>C. C \<noteq> P \<and> Bet A P C \<and> Bet B P C" shows "P Out A B"
l2_inv4D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "l2_state"   and gny :: "msg" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progress (progress x Ra) xsk"   and "guessed_frame Ra xgny = Some gny"   and "x \<in> l2_inv4" shows "guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))"
adjoin_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" assumes "finite_comm_group G"   and "subgroup H G"   and "a \<in> carrier G" shows "subgroup (adjoin G H a) G"
C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set" shows "(f C1_differentiable_on S) = (\<exists>D. (\<forall>x\<in>S. (f has_vector_derivative D x) (at x)) \<and> continuous_on S D)"
quotient_ring_axioms: shows "Comm_Ring.quotient_ring S R (+) (\<cdot>) \<zero> \<one>"
assms_2: fixes x :: "complex" assumes "x \<in> X" shows "\<beta> x \<noteq> 0"
ie5: shows "\<gamma>3 \<delta>S \<le> \<delta>"
assms_1: shows "rs \<turnstile>\<^sub>i t \<longrightarrow> t'"
bv_extend_norm_signed: fixes w :: "bit list"   and b :: "bit" assumes "bv_msb w = b" shows "bv_extend (length w) b (norm_signed w) = w"
\<F>_semantics: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<F> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<mu> \<phi>. w \<Turnstile>\<^sub>n F\<^sub>n \<psi>}"
eq_neutr_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<union># b = {#}) = (a = {#} \<and> b = {#})"
p_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "y \<le> t_stop" shows "p x \<le> p y"
assms_2: assumes "c \<in>\<^sub>\<circ> A"   and "c \<in>\<^sub>\<circ> B" shows "P"
power2_eq_square: fixes a :: "'a" shows "a\<^sup>2 = a * a"
in_quotient_imp_subset: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<subseteq> A"
case_1: fixes f1 :: "'b"   and f2 :: "'c set \<Rightarrow> 'b" shows "(case FAILi of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = f1"
Max_wrt_among_nonempty: fixes B :: "'a set" assumes "B \<inter> carrier \<noteq> {}"   and "finite (B \<inter> carrier)" shows "Max_wrt_among le B \<noteq> {}"
open_einterval: fixes a :: "ereal"   and b :: "ereal" shows "open (einterval a b)"
supplement1_Jacobi: fixes p :: "int" assumes "odd p"   and "1 < p" shows "Jacobi (- 1) p = (- 1) ^ nat ((p - 1) div 2)"
ortho_span: fixes W :: "'a vec set"   and X :: "'a vec set"   and w :: "'a vec"   and x :: "'a vec" assumes "W \<subseteq> carrier_vec n"   and "X \<subseteq> carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> X\<rbrakk> \<Longrightarrow> w \<bullet> x = (0::'a)"   and "w \<in> local.span W"   and "x \<in> X" shows "w \<bullet> x = (0::'a)"
\<delta>\<^sub>c_def: shows "(\<delta>\<^sub>c, c\<^sub>0, c\<^sub>1) = closest_pair_bf_code (p\<^sub>2 # p\<^sub>3 # ps)"
triangle_right: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (H g \<epsilon>) (V (\<a> g f g) (H \<eta> g)) = V (category.inv V (bicategory.runit V H \<a> \<i> src g)) (bicategory.lunit V H \<a> \<i> src trg g)"
finfun_apply: fixes x :: "'d \<Rightarrow>f 'e" shows "finfun_apply x \<in> finfun"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> zg"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> zg; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> zg; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 L C K m NRO A B. \<lbrakk>evs1 \<in> zg; P evs1; Nonce L \<notin> used evs1; C = Crypt K (Number m); K \<in> symKeys; NRO = Crypt (priEK A) \<lbrace>Number f_nro, Agent B, Nonce L, C\<rbrace>\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Number f_nro, Agent B, Nonce L, C, NRO\<rbrace> # evs1)"   and "\<And>evs2 B L C NRO A NRR. \<lbrakk>evs2 \<in> zg; P evs2; Gets B \<lbrace>Number f_nro, Agent B, Nonce L, C, NRO\<rbrace> \<in> set evs2; NRO = Crypt (priEK A) \<lbrace>Number f_nro, Agent B, Nonce L, C\<rbrace>; NRR = Crypt (priEK B) \<lbrace>Number f_nrr, Agent A, Nonce L, C\<rbrace>\<rbrakk> \<Longrightarrow> P (Says B A \<lbrace>Number f_nrr, Agent A, Nonce L, NRR\<rbrace> # evs2)"   and "\<And>evs3 C K M A B L NRO NRR sub_K. \<lbrakk>evs3 \<in> zg; P evs3; C = Crypt K M; K \<in> symKeys; Says A B \<lbrace>Number f_nro, Agent B, Nonce L, C, NRO\<rbrace> \<in> set evs3; Gets A \<lbrace>Number f_nrr, Agent A, Nonce L, NRR\<rbrace> \<in> set evs3; NRR = Crypt (priEK B) \<lbrace>Number f_nrr, Agent A, Nonce L, C\<rbrace>; sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>\<rbrakk> \<Longrightarrow> P (Says A TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> # evs3)"   and "\<And>evs4 K B L sub_K A con_K. \<lbrakk>evs4 \<in> zg; P evs4; K \<in> symKeys; Gets TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs4; sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>; con_K = Crypt (priEK TTP) \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K\<rbrace>\<rbrakk> \<Longrightarrow> P (Says TTP Spy con_K # Notes TTP \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K, con_K\<rbrace> # evs4)" shows "P x"
rexp_empty_iff: fixes r :: "'b rexp" shows "rexp_empty r = (lang r = {})"
closure_atLeastLessThan: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "closure {a..<b} = {a..b}"
tagged_division_ofD_1: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "s tagged_division_of i" shows "finite s"
simps_2: fixes Y :: "tyvrs"   and T :: "ty" shows "Top[Y \<mapsto> T]\<^sub>\<tau> = Top"
ket_canonical_basis: fixes x :: "'b" shows "ket x = canonical_basis ! enum_idx x"
intros_15: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and sx :: "sx"   and k :: "i32"   and off :: "nat"   and tp :: "tp"   and t :: "t"   and vs :: "v list"   and a :: "nat" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "load_packed sx m (nat_of_int k) off (tp_length tp) (t_length t) = None" shows "\<lparr>s;vs;[$C ConstInt32 k, $Load t (Some (tp, sx)) a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
invariant_steps: fixes a :: "'a"   and as :: "'a list" assumes "A.steps (a # as)"   and "PA a" shows "list_all PA as"
that: fixes tx :: "'a \<Rightarrow>\<^sub>0 nat"   and ty :: "'a \<Rightarrow>\<^sub>0 nat" assumes "tx \<in> .[X]"   and "ty \<in> .[Y]"   and "t = tx + ty" shows "thesis"
frechet_derivative_nth_derivative_commute: fixes i :: "nat"   and f :: "'b \<Rightarrow> 'a"   and h :: "'b"   and x :: "'b" shows "frechet_derivative (\<lambda>x. nth_derivative i f x h) (at x) h = nth_derivative i (\<lambda>x. frechet_derivative f (at x) h) x h"
intros: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs < length ys"   and "take (length xs) ys = xs" shows "xs \<prec> ys"
intro_elim_4_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]"   and "[\<phi> \<^bold>\<rightarrow> \<chi> in v]"   and "[\<psi> \<^bold>\<rightarrow> \<chi> in v]" shows "[\<chi> in v]"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "(x # xs) \<up> Suc n = xs \<up> n"
items_le_Scan: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "items_le k (Scan T k I) = Scan (empty_tokens T) k (items_le k I)"
app_eq2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "app xs ys zs" shows "zs = xs @ ys"
that: fixes g :: "complex \<Rightarrow> complex"   and g' :: "complex \<Rightarrow> complex" assumes "\<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (\<lambda>n. f' n x) sums g' x \<and> (g has_field_derivative g' x) (at x)" shows "thesis"
trg_dom: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "trg\<^sub>D (D.dom \<mu>) = trg\<^sub>D \<mu>"
image_mult_atLeastAtMost: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and d :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero d" shows "times d ` ord.atLeastAtMost less_eq a b = ord.atLeastAtMost less_eq (times d a) (times d b)"
Aexp_pre_simps_3: fixes e :: "exp"   and x :: "Vars.var"   and a :: "Arity" shows "Aexp_syn' a (App e x) = Aexp_syn' (inc\<cdot>a) e \<squnion> esing x\<cdot>(up\<cdot>0)"
matrix_inv_left: fixes M :: "(('b, 'c) vec, 'd) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'b)"
out_subseqs1_set_def: fixes prevs :: "(nat \<times> 'b) list" shows "out_subseqs1_set prevs \<equiv> (f head \<circ> snd) ` set prevs \<union> \<Union> {out_subseqs2_set v [0..<i] |v i. (i, v) \<in> set prevs}"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'b \<Rightarrow> 'a" assumes "\<exists>F \<eta> \<epsilon>. equivalence_of_categories C D F G \<eta> \<epsilon>" shows "equivalence_functor_axioms C D G"
fresh_star_atom_at_base: fixes as :: "atom set"   and b :: "'a" shows "as \<sharp>* atom b = as \<sharp>* b"
assms: fixes i :: "nat" assumes "\<not> fresh_floatariths f i" shows "xs ! i = ys ! i"
set_pmf_spmf_of_set: fixes A :: "'a set" shows "set_pmf (spmf_of_set A) = (if finite' A then Some ` A else {None})"
exec_meth_stk_offer: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and stk'' :: "'addr val list" assumes "exec_meth ci P ins xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_meth ci P ins (stack_xlift (length stk'') xt) t h (stk @ stk'', loc, pc, xcp) ta h' (stk' @ stk'', loc', pc', xcp')"
permutes_inv_eq: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and y :: "'b"   and x :: "'b" assumes "p permutes S" shows "(inv p y = x) = (p x = y)"
assms_3: shows "\<forall>x\<in>U\<^sub>2. g x \<in> U"
triangular_toD: fixes k :: "nat"   and A :: "'a mat"   and j :: "nat"   and i :: "nat" assumes "local.triangular_to k A"   and "j < k"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
size_wbt_eq_size: fixes t :: "('b \<times> nat) tree" assumes "wbt t" shows "size_wbt t = size t"
msgs_unchanged_if_snapshotted_RecvMarker_for_other_is: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and i' :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "ev = RecvMarker i p q"   and "ps c p \<noteq> None"   and "i' \<noteq> i" shows "msgs c i' = msgs c' i'"
equal_deg_sum': fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "f \<in> carrier (UP R)"   and "g \<in> carrier (UP R)"   and "deg R g < n"   and "deg R f = n" shows "deg R (f \<oplus>\<^bsub>UP R\<^esub> g) = n"
shift1: fixes bs :: "'b list"   and p :: "'b poly" shows "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
quasi_isometry_betweenD_2: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_between X Y f" shows "f ` X \<subseteq> Y"
hrp_imp_def: fixes RR :: "('c \<Rightarrow> 'd \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'd \<Rightarrow> assn)"   and RR' :: "('c \<Rightarrow> 'd \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'd \<Rightarrow> assn)" shows "hrp_imp RR RR' \<equiv> \<forall>a b. (fst RR' a b \<Longrightarrow>\<^sub>t fst RR a b) \<and> (snd RR a b \<Longrightarrow>\<^sub>t snd RR' a b)"
dconvfactor_inverse: fixes c :: "'c \<Rightarrow>\<^sub>U 'd"   and d :: "(int, sdim) dimvec" shows "dconvfactor c (inverse d) = inverse (dconvfactor c d)"
disjunctive_add: fixes a :: "'a"   and b :: "'a" assumes "\<And>n. \<not> bit a n \<or> \<not> bit b n" shows "a + b = a OR b"
assms_3: fixes n'' :: "nat" assumes "n \<le> n''"   and "n'' < n'" shows "eval the_singleton t t' n'' \<gamma>'"
flatten_def: fixes G :: "('d set, 'e) monoid_scheme"   and rep :: "'d set \<Rightarrow> 'c" shows "flatten G rep = \<lparr>carrier = rep ` carrier G, monoid.mult = \<lambda>x y. rep (the_inv_into (carrier G) rep x \<otimes>\<^bsub>G\<^esub> the_inv_into (carrier G) rep y), one = rep \<one>\<^bsub>G\<^esub>\<rparr>"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "B.arr f"   and "B.dom f = a" shows "f \<cdot>\<^sub>B a = f"
shared_weight_net_Pool_3: shows "count_weights True (remove_weights m1) = count_weights True (remove_weights m2)"
NewInitReds: fixes s :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and a :: "nat"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h'' :: "heap" assumes "\<nexists>sfs. shp s C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,s,False\<rangle> \<rightarrow>* \<langle>Val v',(h', l', sh'),b'\<rangle>"   and "new_Addr h' = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h'' = h'(a \<mapsto> blank P C)"   and "is_class P C" shows "P \<turnstile> \<langle>new C,s,False\<rangle> \<rightarrow>* \<langle>addr a,(h'', l', sh'),False\<rangle>"
that: fixes p1 :: "('v \<times> 'v) list"   and p2 :: "('v \<times> 'v) list" assumes "p = p1 @ p2"   and "is_path v p1 u"   and "u \<notin> int_vertices p1" shows "thesis"
Step: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "shift (Suc n) f = poly_shift (shift n f)"
frechet_derivative_times: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "f differentiable at a"   and "g differentiable at a" shows "frechet_derivative (\<lambda>x. f x * g x) (at a) = (\<lambda>x. f a * frechet_derivative g (at a) x + frechet_derivative f (at a) x * g a)"
mon_s_alt: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and s :: "'g list" shows "mon_s fg s \<equiv> \<Union> (mon fg ` proc_of fg ` set s)"
length_augment_list: fixes n :: "nat"   and f :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "length ((augment_list f ^^ n) p) = n + length p"
powr_def: fixes x :: "'a"   and a :: "'a" shows "x powr a \<equiv> if x = (0::'a) then 0::'a else exp (a * ln x)"
sets_vimage_algebra2: fixes f :: "'d \<Rightarrow> 'e"   and X :: "'d set"   and M :: "'e measure" assumes "f \<in> X \<rightarrow> space M" shows "sets (vimage_algebra X f M) = {f -` A \<inter> X |A. A \<in> sets M}"
I_def: shows "I = (\<lambda>x. integral {x..} r)"
if_eq_const_conv: fixes x :: "bool"   and y :: "'l"   and z :: "'l"   and w :: "'l" shows "((if x then y else z) = w) = (x \<and> y = w \<or> \<not> x \<and> z = w)"
lesssub_def: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" shows "(x \<sqsubset>\<^bsub>r\<^esub> y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> x \<noteq> y)"
i_expand_def: shows "(\<odot>) \<equiv> \<lambda>f k n. if k = 0 then NoMsg else if n mod k = 0 then f (n div k) else NoMsg"
nat_pow_Suc: fixes n :: "nat"   and x :: "int \<times> int" shows "[Suc n] \<cdot>\<^bsub>R\<^esub> x = [n] \<cdot>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x"
prems_2: shows "eqvt_at transform_sumC (PropLetBody (AnalLet as body a), body)"
Contra: fixes A :: "fmla"   and H :: "fmla set" assumes "insert (neg A) H \<turnstile> A" shows "H \<turnstile> A"
rel_mono: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_pmf R \<le> rel_pmf Ra"
limitin_def: fixes X :: "'a topology"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" shows "limitin X f l F \<equiv> l \<in> topspace X \<and> (\<forall>U. openin X U \<and> l \<in> U \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> U))"
bt_augment_v_commute_1: fixes t :: "('d, 'c) bintree"   and r :: "('d, 'c) bintree"   and q :: "('c, 'd) pq" shows "set |bt_augment t (bt_augment r q)| = set |bt_augment r (bt_augment t q)|"
eq_subst_scheme_list_eq_free: fixes S1 :: "nat \<Rightarrow> typ"   and A :: "type_scheme list"   and S2 :: "nat \<Rightarrow> typ"   and n :: "nat" assumes "$ S1 A = $ S2 A"   and "n \<in> free_tv A" shows "S1 n = S2 n"
span_explicit: fixes b :: "'b set" shows "local.span b = {\<Sum>a\<in>t. r a *s a |t r. finite t \<and> t \<subseteq> b}"
assms_5: shows "revision_step r s\<^sub>2' (s\<^sub>1(r' \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r'')]), r'' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e), r \<mapsto> x))"
trss_mon_e_no_ctx: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" shows "mon_e fg e \<inter> mon_c fg c = {}"
ennreal_divide_times: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "a / b * c = a * (c / b)"
simps_29: fixes x2 :: "'b"   and x31 :: "'b local.term"   and x32 :: "'b local.term" shows "\<^bold>\<langle>x2\<^bold>\<rangle> \<noteq> x31 \<^bold>\<star> x32"
local_determinism: fixes r :: "'r"   and s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s\<^sub>1 s\<^sub>2"   and "revision_step r s\<^sub>1 s\<^sub>2'" shows "s\<^sub>2 \<approx> s\<^sub>2'"
converse_unfold: fixes r :: "('d \<times> 'c) set" shows "r\<inverse> = _Collect (y, x) ((x, y) \<in> r)"
sum0: shows "(\<Sum>k\<in>UNIV - {a}. (if a = k then 1::'a else (0::'a)) * A $ k $ b) = (0::'a)"
convert_extTA_simps_5: fixes f :: "'i \<Rightarrow> 'f"   and ta :: "('e, 'c, 'i, 'g, 'd, 'h) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>w\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>"
floor_rat_def: fixes x :: "rat" shows "\<lfloor>x\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
simplex_segment_cases: fixes a :: "'b"   and b :: "'b" shows "(if a = b then 0 else 1) simplex closed_segment a b"
trans_O_iff: fixes A :: "('a \<times> 'a) set" shows "trans A = (A O A \<subseteq> A)"
routing_port_rangesI: fixes tbl :: "'b routing_rule list"   and k :: "'b word"   and output_port :: "char list"   and wi :: "'b wordinterval" assumes "valid_prefixes tbl"   and "output_iface (routing_table_semantics tbl k) = output_port"   and "k \<in> wordinterval_to_set wi" shows "\<exists>ip_range. (output_port, ip_range) \<in> set (routing_port_ranges tbl wi) \<and> k \<in> wordinterval_to_set ip_range"
monad_fail_alt_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and fail :: "'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_fail_alt return bind fail alt" shows "monad_fail_alt (return_state return) (bind_state bind) (fail_state fail) (alt_state alt)"
homology_excision_axiom: fixes X :: "'a topology"   and U :: "'a set"   and T :: "'a set"   and S :: "'a set"   and p :: "int" assumes "X closure_of U \<subseteq> X interior_of T"   and "T \<subseteq> S" shows "hom_induced p (subtopology X (S - U)) (T - U) (subtopology X S) T id \<in> Group.iso (relative_homology_group p (subtopology X (S - U)) (T - U)) (relative_homology_group p (subtopology X S) T)"
b_mutual_exclusion': fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat"   and t' :: "nat"   and tk :: "nat"   and tk' :: "nat" assumes "B.reachable s"   and "t < N"   and "t' < N"   and "t \<noteq> t'"   and "locks_ticket (tts s t) tk"   and "locks_ticket (tts s t') tk'" shows "False"
tab\<^sub>1_simps_3: fixes r :: "'a" assumes "ide r" shows "src (tab\<^sub>1 r) = src (tab\<^sub>0 r)"
lcm_abs1_int: fixes x :: "int"   and y :: "int" shows "lcm \<bar>x\<bar> y = lcm x y"
efsig: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> ef : B"
incl_gen: fixes I :: "'a set"   and P :: "'a set set" shows "I \<subseteq> gen P I"
set_tag_name_locs_def: shows "set_tag_name_locs \<equiv> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name_locs"
lift_p_def: fixes P :: "'a \<Rightarrow> bool"   and f :: "('a, 'b) state" shows "lift_p P f = (\<forall>heap. P heap \<longrightarrow> (case run_state f heap of (x, heap) \<Rightarrow> P heap))"
comp_assoc: fixes h :: "('a E.term, 'a) arr"   and g :: "('a E.term, 'a) arr"   and f :: "('a E.term, 'a) arr" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
iPP: shows "isTrue (wrepr.PP \<langle>\<phi>\<rangle>)"
filter_mono': fixes F1 :: "'a filter"   and F2 :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F1 \<le> F2" shows "L F2 f \<subseteq> L F1 f"
is_faithful: fixes f :: "'c"   and f' :: "'c" assumes "C.par f f'"   and "FG.map f = FG.map f'" shows "f = f'"