hcju/isabelleps huggingface dataset:
exhaust: fixes y :: "GatewayStatus" obtains "y = init_state"   | "y = call"   | "y = connection_ok"   | "y = sending_data"   | "y = voice_com"
rtranclp_is_Sup_relpowp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "P\<^sup>*\<^sup>* = Sup (range ((^^) P))"
Red1BinOpFail: fixes bop :: "bop"   and v1 :: "'addr val"   and v2 :: "'addr val"   and a :: "'addr"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and s :: "'heap \<times> 'addr val list" assumes "binop bop v1 v2 = \<lfloor>Inr a\<rfloor>" shows "uf,P,t \<turnstile>1 \<langle>Val v1 \<guillemotleft>bop\<guillemotright> Val v2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
Cast: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and s2 :: "state"   and T :: "ty" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "s2 = abupd (raise_if (\<not> G,snd s1\<turnstile>v fits T) ClassCast) s1" shows "G\<turnstile>Norm s0 \<midarrow>Cast T e-\<succ>v\<rightarrow> s2"
strong_induct: fixes P :: "'e \<Rightarrow> 'd action \<Rightarrow> bool"   and z :: "'e"   and action :: "'d action" assumes "\<And>a1 a2 z. P z (a1\<lparr>a2\<rparr>)"   and "\<And>a1 list a2 z. P z (a1\<lparr>\<nu>*list\<rparr>\<langle>a2\<rangle>)"   and "\<And>z. P z (\<tau>)" shows "P z action"
hseqI': fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>"   and "S.arr \<nu>"   and "S.src \<nu> = S.trg \<mu>" shows "S.hseq \<nu> \<mu>"
numadd: fixes bs :: "real list"   and t :: "Ferrack.num"   and s :: "Ferrack.num" shows "Inum bs (numadd t s) = Inum bs (Add t s)"
idx_pm_of_pm_of_idx_pm: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd" assumes "distinct xs"   and "keys f \<subseteq> {0..<length xs}" shows "idx_pm_of_pm xs (pm_of_idx_pm xs f) = f"
cl_Un: fixes L :: "'b set set"   and r :: "'b set"   and s :: "'b set" assumes "lattice L" shows "cl L (r \<union> s) = cl L r \<union> cl L s"
vpeq_s_t: shows "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s t"
vseq_implies_hpar_2: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.seq \<nu> \<mu>" shows "trg\<^sub>D \<nu> = trg\<^sub>D \<mu>"
prodrg_sameTr5: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_tOp I A = prod_tOp I B"
fi: fixes x :: "'a" shows "RETURN (fid x) \<le> fi x"
i_Exec_Stream_causal: fixes n :: "nat"   and input1 :: "nat \<Rightarrow> 'd"   and input2 :: "nat \<Rightarrow> 'd"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "input1 \<Down> n = input2 \<Down> n" shows "i_Exec_Comp_Stream trans_fun input1 c \<Down> n = i_Exec_Comp_Stream trans_fun input2 c \<Down> n"
SigmaD1: fixes a :: "'f"   and b :: "'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
oalist_inv_list_of_oalist: fixes xs :: "('a, 'b) oalist" shows "ko.oalist_inv (list_of_oalist xs)"
simps_19: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a" shows "local.less_aexp_aux (Minus e1 e2) (V v) = False"
take_hd_drop: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
in_minus_fset: fixes F :: "'b fset"   and S :: "'b fset"   and x :: "'b" assumes "x |\<in>| F - S" shows "x |\<notin>| S"
PREFER_sv_D: fixes x :: "('a \<times> 'b) set" assumes "PREFER single_valued x" shows "single_valued x"
cd_closed: fixes n' :: "'node"   and n\<^sub>1 :: "'node"   and n\<^sub>2 :: "'node"   and S :: "'node set" assumes "n' \<longrightarrow>\<^sub>w\<^sub>o\<^sub>d n\<^sub>1,n\<^sub>2"   and "n\<^sub>1 \<in> wod_backward_slice S"   and "n\<^sub>2 \<in> wod_backward_slice S" shows "n' \<in> wod_backward_slice S"
f: fixes i :: "nat"   and c :: "bool"   and x :: "'a" shows "fst (f i c x) = (c \<or> x \<noteq> snd (f i c x))"
A3': fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2" assumes "a b \<congruent>\<^sub>K c c" shows "a = b"
mod_ring_finite_field_ops_integer: fixes p :: "int" assumes "mod_ring_locale TYPE('a) p" shows "mod_ring_gen (finite_field_ops_integer (integer_of_int p)) (mod_ring_locale.mod_ring_rel_integer p) p"
A: shows "h \<in> carrier P \<and> f = X_poly R \<otimes>\<^bsub>P\<^esub> h"
take_tl: fixes n :: "nat"   and xs :: "'b list" shows "take n (tl xs) = tl (take (Suc n) xs)"
ss_def: shows "ss = [ideal {f} \<inter> P[X], P, N]"
div_add1_eq1: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m + b mod m < m" shows "(a + b) div m = a div m + b div m"
inverse_transformations_sym: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<sigma>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<sigma> \<sigma>'" shows "inverse_transformations A B G F \<sigma>' \<sigma>"
assms_2: shows "snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
inject_1: fixes x1 :: "String.literal"   and y1 :: "String.literal" shows "(Class_type x1 = Class_type y1) = (x1 = y1)"
eta_induct: fixes x :: "ValD"   and x' :: "('o ValK \<rightarrow> 'o) \<rightarrow> 'o"   and P :: "ValD \<Rightarrow> (('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<Rightarrow> bool" assumes "\<eta>: x \<mapsto> x'"   and "\<lbrakk>x = \<bottom>; x' = \<bottom>\<rbrakk> \<Longrightarrow> P x x'"   and "\<And>n. \<lbrakk>x = ValTT; x' = unitK\<cdot>ValKTT\<rbrakk> \<Longrightarrow> P x x'"   and "\<And>n. \<lbrakk>x = ValFF; x' = unitK\<cdot>ValKFF\<rbrakk> \<Longrightarrow> P x x'"   and "\<And>n. \<lbrakk>x = ValN\<cdot>n; x' = unitK\<cdot>(ValKN\<cdot>n)\<rbrakk> \<Longrightarrow> P x x'"   and "\<And>f f'. \<lbrakk>x = ValF\<cdot>f; x' = unitK\<cdot>(ValKF\<cdot>f'); \<theta>: f \<mapsto> f'\<rbrakk> \<Longrightarrow> P x x'" shows "P x x'"
makeFaceFinals_preserve_finals: fixes f :: "face"   and g :: "graph"   and f' :: "face" assumes "f \<in> set (finals g)" shows "f \<in> set (finals (makeFaceFinal f' g))"
ord_eq_le_trans: fixes a :: "'a"   and b :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
entangled2_def: fixes u :: "complex Matrix.mat" shows "entangled2 u \<equiv> \<not> prod_state2 u"
assms: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3 {wp\<^sub>3 c Q} c { Q}"
polynate_stupid: fixes t :: "poly"   and bs :: "'b list" assumes "polynate t = 0\<^sub>p" shows "\<lparr>t\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'b)"
simps_7: fixes r :: "WS1S_Prelim.order \<Rightarrow> nat fset \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "interp"   and k :: "WS1S_Prelim.order"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" shows "WS1S.satisfies_gen r \<AA> (FAll k \<phi>) = (\<forall>P. r k P (Length \<AA>) \<longrightarrow> WS1S.satisfies_gen r (Extend k 0 \<AA> P) \<phi>)"
column_leading_coefficient_component_eq: fixes A :: "(('a, 'b) vec, 'c) vec"   and v :: "('a, 'c) vec"   and x :: "'c"   and y :: "'c" assumes "reduced_row_echelon_form A"   and "v \<in> {column (LEAST n. A $ i $ n \<noteq> (0::'a)) A |i. row i A \<noteq> 0}"   and "v $ x \<noteq> (0::'a)"   and "v $ y \<noteq> (0::'a)" shows "x = y"
the_elem_eq: fixes x :: "'c" shows "the_elem {x} = x"
assms_2: shows "\<forall>x\<in>MPLS_charles PROB. x < k"
assms_3: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<diamond>Q) s)) \<sigma>"
simps_2: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and x1 :: "'h"   and x2 :: "'i" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
insert_3: assumes "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and> continuous_on (\<Union> \<F> - C) g \<and> g ` (\<Union> \<F> - C) \<subseteq> T \<and> (\<forall>x\<in>(\<Union> \<F> - C) \<inter> K. g x = h x)"
exec_move_newArrayI: 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 T :: "ty" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (newA T\<lfloor>e\<rceil>) h s ta h' s'"
Un_upper2: fixes B :: "'d set"   and A :: "'d set" shows "B \<subseteq> A \<union> B"
relation_rules_14: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(P \<and> Q) x" obtains "P x" and "Q x"
erf_remainder_conv_Suc: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "erf_remainder n x = (- 1) ^ n * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * exp (- x\<^sup>2) / x ^ (2 * n + 1) + erf_remainder (Suc n) x"
split_with_invar_def: fixes i :: "'d \<Rightarrow> 'c set"   and X :: "'c set" shows "split_with_invar i X = SPEC (\<lambda>((Y, sctn), YS). X = Y \<union> YS \<and> Y \<subseteq> i sctn)"
i4: fixes x :: "('a, 'n) vec" assumes "inner x x = (0::'a)" shows "x = 0"
Not2E: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" shows "\<^bold>\<not> (\<^bold>\<not> F) \<triangleright> \<Gamma> \<turnstile> F"
out_to_bet: fixes A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and B :: "'p"   and A :: "'p"   and C :: "'p" assumes "Col A' B' C'"   and "(B Out A C) = (B' Out A' C')"   and "Bet A B C" shows "Bet A' B' C'"
lift_resumption_bind: fixes r :: "('g, 'e, 'f) Resumption.resumption"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) Resumption.resumption" shows "lift_resumption (r \<bind> f) = lift_resumption r \<bind> lift_resumption \<circ> f"
f_join_take: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" shows "xs \<down> n \<Join>\<^sub> I = xs \<Join>\<^sub> (I \<down>< n)"
sat_fv_cong: fixes \<phi> :: "Formula.formula"   and v :: "event_data list"   and v' :: "event_data list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat" assumes "\<forall>x\<in>fv \<phi>. v ! x = v' ! x" shows "Formula.sat \<sigma> V v i \<phi> = Formula.sat \<sigma> V v' i \<phi>"
backtrack_nsc: fixes nsc :: "('i \<times> 'a ns_constraint) list"   and J :: "'i set"   and s :: "'s"   and c :: "'c"   and K :: "'i set"   and s' :: "'s"   and s'' :: "'s" assumes "checked_nsc nsc J s"   and "checkpoint_nsc s = c"   and "invariant_nsc nsc K s'"   and "backtrack_nsc c s' = s''"   and "J \<subseteq> K" shows "invariant_nsc nsc J s''"
forward_ix_def: fixes os :: "'t list" shows "forward_ix os = forward_ix_rec (IArray os)"
action_simple: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and ct :: "'a \<Rightarrow> 'a pmf" assumes "Finite_Markov_Decision_Process K S"   and "s \<in> S" shows "action (Finite_Markov_Decision_Process.simple K S ct s) = ct s"
lim: shows "LIM x F. S x \<bullet> (1::'a) :> at_top"
max_ideal_UNIV_radical: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "generates_max_ideal UNIV F" shows "\<surd>ideal F = ideal F"
fsmult_zero: fixes r :: "'f" assumes "r \<in> UNIV" shows "r \<sharp>\<cdot> (0::'v) = (0::'v)"
Diff: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> null_sets M"   and "b \<in> null_sets M" shows "a - b \<in> null_sets M"
assms_1: shows "(f expands_to MS (MSLCons (MS ys c, e) xs) g) (b # basis)"
dflt_arity_4: shows "case_prod \<equiv> \<lambda>x. (#\<lambda>xa. (#SP case_prod $ (\<lambda>xa. (#\<lambda>xb. (#x $ xa $ xb#)#)) $ xa#)#)"
ga_gstep_eq: shows "ga_gstep = pid_gstep"
map_def: fixes f :: "'j" shows "Fun_g.map f \<equiv> if J.arr f then g else C.null"
approx_hnorm: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y" shows "hnorm x \<approx> hnorm y"
monom_poly_is_Zp_continuous: fixes c :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp"   and "f = monom (UP Zp) c n" shows "is_Zp_continuous (to_fun f)"
prod_mset_subset_imp_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "class.comm_monoid_mult times one"   and "A \<subseteq># B" shows "dvd.dvd times (comm_monoid_mult.prod_mset times one A) (comm_monoid_mult.prod_mset times one B)"
covov: shows "ov O ov \<subseteq> b \<union> m \<union> ov"
set_plus_mono3_b: fixes a :: "'a"   and C :: "'a set"   and x :: "'a"   and D :: "'a set" assumes "a \<in> C"   and "x \<in> a +o D" shows "x \<in> C + D"
assms_5: fixes m :: "'a" assumes "m \<in> B" shows "\<exists>n\<in>A. f n \<le>\<^sub>o\<^sub>w g m"
MapsToE: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "f maps\<^bsub>C\<^esub> X to Y" obtains "f \<in> mor\<^bsub>C\<^esub>" and "dom\<^bsub>C\<^esub> f = X" and "cod\<^bsub>C\<^esub> f = Y"
fold_insert2: fixes A :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c set"   and z :: "'c set" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) z (insert x A) = Finite_Set.fold (\<lambda>\<phi>. (\<union>) (f \<phi>)) (f x \<union> z) A"
menger: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and n :: "nat" assumes "v0_v1_Digraph G v0 v1"   and "\<And>S. Separation G v0 v1 S \<Longrightarrow> n \<le> card S" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = n"
dependent_def: fixes P :: "'b set" shows "cdependent P = (\<exists>a\<in>P. a \<in> cspan (P - {a}))"
induct_conj_2: fixes C :: "bool"   and A :: "bool"   and B :: "bool" shows "HOL.induct_implies C (HOL.induct_conj A B) = HOL.induct_conj (HOL.induct_implies C A) (HOL.induct_implies C B)"
subst_simps_1: fixes x :: "'b"   and t :: "('a, 'b) Term.term" shows "subst x t x = t"
partition_P: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and yes :: "'b list"   and no :: "'b list" assumes "partition P xs = (yes, no)" shows "(\<forall>p\<in>set yes. P p) \<and> (\<forall>p\<in>set no. \<not> P p)"
assms_2: shows "Domainp cr\<^sub>2 = (\<lambda>x. x \<in> \<UU>\<^sub>2)"
case_asset: fixes asset :: "'a" assumes "asset \<in> stocks Mkt" shows "asset = stk \<or> asset = risk_free_asset"
acyclic_3a_3b: fixes x :: "'a" shows "acyclic_3a x = acyclic_3b x"
gauss_int_norm_of_nat: fixes n :: "nat" shows "gauss_int_norm (of_nat n) = n\<^sup>2"
bounded_linear_right: fixes a :: "'b \<Rightarrow>\<^sub>L 'c" shows "bounded_linear (blinfun_apply a)"
in_sccs_verts_conv_reachable: fixes S :: "'a set" shows "(S \<in> H.sccs_verts) = (S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> u))"
exhaust: fixes y :: "igraph" obtains x1 :: "(location \<times> location) set" and x2 :: "location \<Rightarrow> char list set" and x3 :: "actor \<Rightarrow> char list set \<times> char list set" and x4 :: "location \<Rightarrow> char list \<times> ((actor \<times> actor set) \<times> nat) set" where "y = Lgraph x1 x2 x3 x4"
set_integrable_norm: fixes M :: "'a measure"   and k :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "set_integrable M k f" shows "set_integrable M k (\<lambda>x. norm (f x))"
assms_1: shows "cri.InvCapsNonneg (exchange_config c)"
B: shows "unitary_schur_decomposition A es = (B, P, Q)"
inangle_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.InAngle Bet P A B C" shows "A \<noteq> B \<and> C \<noteq> B \<and> P \<noteq> B"
mHom_to_zero: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and e :: "'i" assumes "Ring R"   and "R module M" shows "mHom R M (Zm R e) = {mzeromap M (Zm R e)}"
simps_1: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and c :: "float"   and t :: "taylor_model" shows "eval_poly_at_tm prec ord I a (c)\<^sub>p t = tm_const c"
BC: shows "\<forall>j\<ge>0. (0::'a) \<le> MAX_STR \<longrightarrow> length [] = 0 \<longrightarrow> List.member (fin_string_upto j) (Abs_fin_string [])"
arcs_blocks_simple: fixes bl :: "'a set" assumes "bl \<in># arcs_blocks" shows "multiplicity bl = 1"
assms_1: shows "U \<in> carrier_mat (prod_list (nths dims vars)) (prod_list (nths dims vars))"
assms_2: shows "\<A> \<Turnstile> \<Phi> \<Pi> 0"
big_small_mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> O[F](g1)"   and "f2 \<in> o[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
chamber_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp Y (f ` C)"
snd: shows "s = fst (splitAt ram2 s) @ [ram2] @ snd (splitAt ram2 s)"
Max_wrt_preorder: shows "Max_wrt (map_relation (inv f) le) = {x \<in> f ` carrier. \<forall>y\<in>f ` carrier. map_relation (inv f) le x y \<longrightarrow> map_relation (inv f) le y x}"
openin_clauses_3: fixes K :: "'b set set"   and U :: "'b topology" assumes "\<forall>S\<in>K. openin U S" shows "openin U (\<Union> K)"
LocalSecretsComposition1: fixes ls :: "KS"   and P :: "specID"   and PQ :: "specID"   and Q :: "specID" assumes "ls \<in> LocalSecrets P"   and "subcomponents PQ = {P, Q}" shows "ls \<in> LocalSecrets PQ"
ag_pOp_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier M"   and "y \<in> carrier M"   and "z \<in> carrier M" shows "x \<plusminus> y \<plusminus> z = x \<plusminus> (y \<plusminus> z)"
csol_2: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "{t0--t} \<subseteq> T"
I_2: shows "wf_interp_for_formula (w, I) \<phi>"
mult_carrier_mat: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B \<in> carrier_mat nr nc"
Rep_eint_inject: fixes x :: "eint"   and y :: "eint" shows "(Rep_eint x = Rep_eint y) = (x = y)"
abs_eq: fixes x :: "'a \<Rightarrow> 'b" assumes "eq_onp bounded_clinear x x" shows "CBlinfun x* = CBlinfun (x\<^sup>\<dagger>)"
Iff_Imp: fixes p :: "'a pfm"   and p' :: "'a pfm"   and q :: "'a pfm"   and q' :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"   and "\<turnstile>\<^sub>! (q \<^bold>\<longleftrightarrow>\<^sub>! q')" shows "\<turnstile>\<^sub>! ((p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! p' \<^bold>\<longrightarrow>\<^sub>! q')"
lin_add: fixes t :: "Reflection_Examples.num"   and s :: "Reflection_Examples.num"   and bs :: "nat list" shows "Inum (lin_add t s) bs = Inum (Add t s) bs"
det_fold_sorted_map: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c' :: "'g \<Rightarrow> bool"   and f' :: "'f \<times> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma>' :: "'g"   and result :: "('f \<Rightarrow> 'h option) \<Rightarrow> 'g"   and Rk :: "('i \<times> 'f) set"   and Rv :: "('j \<times> 'h) set"   and Rm :: "('i \<times> 'f) set \<Rightarrow> ('j \<times> 'h) set \<Rightarrow> ('k \<times> ('f \<Rightarrow> 'h option)) set"   and tsl :: "'k \<Rightarrow> ('i \<times> 'j) list"   and m :: "'k"   and m' :: "'f \<Rightarrow> 'h option"   and c :: "'l \<Rightarrow> bool"   and R\<sigma> :: "('l \<times> 'g) set"   and f :: "'i \<times> 'j \<Rightarrow> 'l \<Rightarrow> 'l"   and \<sigma> :: "'l" assumes "det_fold_map ordR c' f' \<sigma>' result"   and "is_map_to_sorted_list ordR Rk Rv Rm tsl"   and "(m, m') \<in> \<langle>Rk, Rv\<rangle>Rm"   and "(c, c') \<in> R\<sigma> \<rightarrow> bool_rel"   and "(f, f') \<in> Rk \<times>\<^sub>r Rv \<rightarrow> R\<sigma> \<rightarrow> R\<sigma>"   and "(\<sigma>, \<sigma>') \<in> R\<sigma>" shows "(foldli (tsl m) c f \<sigma>, result m') \<in> R\<sigma>"
slist_apply_filter_slist: fixes P :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "(nat, 'd) slist"   and i :: "nat" shows "slist_apply (filter_slist P xs) i = (if P i (slist_apply xs i) then slist_apply xs i else (0::'d))"
refine: fixes G :: "('b \<Rightarrow> bool, 'b \<Rightarrow> 'b list, 'b list, 'b \<Rightarrow> bool, 'c) gen_bg_impl_scheme" shows "RETURN (code_blue_dfs G) \<le> impl_blue_dfs G"
Pd'_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and ds :: "'e set"   and Pd' :: "'e \<Rightarrow> ('d \<times> 'd) set"   and d :: "'e" assumes "TruePrefs Xd Xh Pd Ch x X ds Pd'" shows "Field (Pd' d) \<subseteq> {x. Xd x = d}"
AbstrLevels_A1_A12: fixes i :: "AbstrLevelsID" assumes "sA1 \<in> AbstrLevel i" shows "sA12 \<notin> AbstrLevel i"
converse_meet: fixes r :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(inf r s)\<inverse>\<inverse> = inf r\<inverse>\<inverse> s\<inverse>\<inverse>"
alg_multE_2: fixes x :: "'a list"   and p :: "'a list list"   and n :: "nat" assumes "x \<in> carrier (poly_ring R)"   and "p \<in> carrier (poly_ring (poly_ring R))"   and "p \<noteq> []"   and "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> p" shows "n \<le> UP.alg_mult p x"
Append_rexp_rhs_def: fixes rhs :: "'b trm set"   and rexp :: "'b rexp" shows "Append_rexp_rhs rhs rexp \<equiv> Append_rexp rexp ` rhs"
inject_2: fixes x2 :: "tyvrs"   and x1 :: "ty"   and y2 :: "tyvrs"   and y1 :: "ty" shows "(TVarB x2 x1 = TVarB y2 y1) = (x2 = y2 \<and> x1 = y1)"
scopeExtParLeft: fixes x :: "name"   and P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "x \<sharp> P"   and "Id \<subseteq> Rel"   and "eqvt Rel"   and "\<And>R S y. y \<sharp> R \<Longrightarrow> (<\<nu>y>(R \<parallel> S), R \<parallel> <\<nu>y>S) \<in> Rel"   and "\<And>R S y z. y \<sharp> R \<Longrightarrow> (<\<nu>y><\<nu>z>(R \<parallel> S), <\<nu>z>(R \<parallel> <\<nu>y>S)) \<in> Rel" shows "<\<nu>x>(P \<parallel> Q) \<leadsto>[Rel] P \<parallel> <\<nu>x>Q"
fds_ln_def: fixes l :: "'a"   and f :: "'a fds" shows "fds_ln l f = fds_integral l (fds_deriv f / f)"
in_charts_in_atlas: fixes x :: "('a, 'b) chart" assumes "x \<in> charts_submanifold S" shows "x \<in> sub.atlas"
x0_hb_bound7: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "x\<^sub>0 < bs ! i * x + (hs ! i) x"
isPrefixPrefixToLevel: fixes l :: "nat"   and t :: "('b \<times> bool) list" shows "isPrefix (prefixToLevel l t) t"
ope: shows "openin (top_of_set (rel_frontier U)) S"
ideal_prod_el2: fixes I :: "'a set"   and J :: "'a set"   and a :: "'a" assumes "ideal R I"   and "ideal R J"   and "a \<in> I \<diamondsuit>\<^sub>r J" shows "a \<in> J"