hcju/isabelleps huggingface dataset:
matrix_to_iarray_fst_Gauss_Jordan_in_ij_det_P: fixes A :: "(('e, 'f) vec, 'd) vec"   and j :: "'f"   and i :: "'d" assumes "\<exists>n. A $ n $ j \<noteq> (0::'e) \<and> i \<le> n" shows "fst (Gauss_Jordan_in_ij_det_P A i j) = fst (Gauss_Jordan_in_ij_det_P_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j))"
pre_merge_inv: fixes P :: "nat set"   and \<gamma> :: "nat \<Rightarrow> thread_state option"   and z :: "nat"   and z' :: "nat" assumes "pre_merge P \<gamma> z z'" obtains j :: "nat" where "z' = sh (the (\<gamma> j)) z" and "j \<in> P" and "z \<in> W (the (\<gamma> j))" and "dom \<gamma> = P"   | "z' = sh (the (\<gamma> 0)) z" and "\<forall>j\<in>P. z \<notin> W (the (\<gamma> j))" and "dom \<gamma> = P"
rel_U_pos_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q) itself" shows "rel_U_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1'. rel_U L1 Co1 Co2 Contra1 Contra2 OO rel_U L1' Co1' Co2' Contra1' Contra2' \<le> rel_U (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2'))"
disc_2: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "\<not> lnull (unfold_llist p g21 g22 a)"
ennreal_zero_neq_top: shows "0 \<noteq> top"
l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf set_val set_val_locs"
assms_3: shows "\<forall>\<^sub>F x in F. 0 \<le> g x"
pos_cost_mu_triangle: fixes c :: "'b \<Rightarrow> real"   and e :: "'b"   and u :: "'a"   and v :: "'a"   and s :: "'a" assumes "\<And>e. e \<in> arcs G \<Longrightarrow> 0 \<le> c e"   and "arc_to_ends G e = (u, v)"   and "e \<in> arcs G" shows "\<mu> c s v \<le> \<mu> c s u + ereal (c e)"
payload_inductive_cases_4: fixes X :: "msg"   and Y :: "msg" assumes "\<langle>X, Y\<rangle> \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
is_finite_weak_ranking_Nil: shows "is_finite_weak_ranking []"
lossless_iff_set_pmf_None: fixes p :: "'c spmf" shows "lossless_spmf p = (None \<notin> set_pmf p)"
alw_mono: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream"   and \<psi> :: "'c stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "alw \<psi> xs"
ce_rel_lm_16: fixes x :: "nat"   and r :: "(nat \<times> nat) set" assumes "x \<in> ce_rel_to_set r" shows "(c_fst x, c_snd x) \<in> r"
FL_equivalent_iff_not_distinguished: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c fs_set"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply" shows "indexed_effect_nominal_ts.FL_logically_equivalent TYPE('a) L satisfies transition effect_apply F P Q = (\<nexists>x. effect_nominal_ts.is_FL_formula L F x \<and> indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply x P Q)"
nest_def: fixes r :: "('b \<times> 'b) set"   and s :: "((accent \<times> 'b) list \<times> (accent \<times> 'b) list) set" shows "nest r s = {(a, b). (ms_of_greek a, ms_of_greek b) \<in> mult (letter_less r <*lex*> s)}"
Graph10: fixes M :: "'a list"   and i :: "nat"   and a :: "'a" assumes "M ! i = a" shows "M[i := a] = M"
simps_18: fixes uy :: "'a"   and v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Atom uy) (Times v va) = True"
continuous_within_Arccos: fixes z :: "complex"   and s :: "complex set" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "continuous (at z within s) Arccos"
diff_self: fixes x :: "'a" assumes "x \<in> V" shows "x - x = (0::'a)"
factorization_m_degree: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "pl.factorization_m f (c, fs)"   and "pl.Mp f \<noteq> 0" shows "pl.degree_m f = \<Sum>\<^sub># (image_mset pl.degree_m fs)"
\<xi>_in_hom: shows "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> trg f\<guillemotright>"
arr_char: fixes f :: "'c" shows "C'.arr f = Arr f"
prems_2: fixes x :: "'a set"   and y :: "'a" assumes "x \<in> set (x__ # xs)"   and "y \<in> x" shows "f y \<in> x"
"2_2": fixes t :: "'t" assumes "worklists_vacant_to c t"   and "0 < zcount M t" shows "\<exists>s\<le>t. s \<in>\<^sub>A frontier (c_imp c loc2)"
ee_sym: fixes as :: "'a list"   and bs :: "'a list" assumes "essentially_equal G as bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G bs as"
min_int_poly_real_alg_def: fixes x :: "real_alg" shows "min_int_poly_real_alg x = (case info_real_alg x of Inl r \<Rightarrow> poly_rat r | Inr (p, x) \<Rightarrow> p)"
floor_split: fixes P :: "int \<Rightarrow> bool"   and t :: "'a" shows "P \<lfloor>t\<rfloor> = (\<forall>i. of_int i \<le> t \<and> t < of_int i + (1::'a) \<longrightarrow> P i)"
unfold_token_succeeds_eq: fixes x :: "nat" shows "\<MM>.token_succeeds x = \<UU>.token_succeeds x"
word_arith_wis_7: shows "0 = word_of_int 0"
bilinear_form_scale_v1: fixes k :: "complex"   and v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "bilinear_form (k *\<^sub>s\<^sub>v v1) v2 H = cnj k * bilinear_form v1 v2 H"
Xequ4': fixes x :: "Value"   and s :: "Store"   and t :: "New" assumes "alive x s" shows "s \<equiv>[x] s\<langle>t\<rangle>"
f0: fixes x :: "'a"   and y :: "'b" assumes "x \<in> A - A'"   and "y \<in> B x" shows "f (x, y) = (0::'c)"
int1: shows "integrable M (\<lambda>x. f x * max (g x) 0)"
l: shows "(\<Sum>y\<in>{v. p v \<noteq> 0}. p y *R (if y = x then c else v y)) = p x *R c + (\<Sum>y\<in>{v. p v \<noteq> 0} \<inter> (UNIV - {x}). p y *R v y)"
Group_Gp: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "Group (\<natural>H)"
permutes_def: fixes p :: "'d \<Rightarrow> 'd"   and S :: "'d set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
node_lift_step: fixes T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "nat"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and R\<^sub>i :: "nat set" assumes "T \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). Q \<sigma> \<sigma>')"   and "\<And>\<sigma> \<sigma>'. other U {i} \<sigma> \<sigma>' \<Longrightarrow> Q \<sigma> \<sigma>'"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'" shows "\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (otherwith S {i} (oarrivemsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). Q \<sigma> \<sigma>')"
f_def_2: shows "\<forall>B\<in>P\<^sub>1. real c < W B + w (f B)"
random_serial_dictatorship_support: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set" assumes "pref_profile_wf agents alts R"   and "finite agents"   and "finite alts"   and "agents' \<subseteq> agents"   and "alts' \<noteq> {}"   and "alts' \<subseteq> alts" shows "set_pmf (random_serial_dictatorship agents' alts' R) \<subseteq> alts'"
before_def: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and N :: "'b set"   and M :: "'b set" assumes "class.arelations meets \<I> e m b ov d s f" shows "arelations.before \<I> e m b ov d s f N M \<equiv> arelations.NEST \<I> e m ov d s f N \<and> arelations.NEST \<I> e m ov d s f M \<and> (\<exists>n m. n \<in> N \<and> m \<in> M \<and> (n, m) \<in> b)"
zip_map2: fixes xs :: "'g list"   and f :: "'i \<Rightarrow> 'h"   and ys :: "'i list" shows "zip xs (map f ys) = map2 (\<lambda>x y. (x, f y)) xs ys"
simps_3: fixes \<sigma> :: "nat \<Rightarrow> val"   and e1 :: "expr"   and e2 :: "expr" shows "expr_sem_rf \<sigma> <e1 ,  e2> = <|expr_sem_rf \<sigma> e1, expr_sem_rf \<sigma> e2|>"
step_1: shows "q \<rightarrow> l qs \<in> \<delta>"
Union_upper: fixes B :: "'d set"   and A :: "'d set set" assumes "B \<in> A" shows "B \<subseteq> \<Union> A"
hyps: shows "bv_to_nat a2 = bv_to_nat (remzero a2)"
quorum_def: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and Q :: "'b set" assumes "stellar slices W" shows "stellar.quorum slices W Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>slices p. Sl \<subseteq> Q"
tan_square_neq_neg1: fixes z :: "complex" shows "(tan z)\<^sup>2 \<noteq> - 1"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr = delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t (cast (cast shadow_root_ptr))"
commute: fixes a :: "'c"   and b :: "'c" shows "inf a b = inf b a"
poly_entire_lemma2: fixes p :: "'a list"   and q :: "'a list" assumes "poly p \<noteq> poly []"   and "poly q \<noteq> poly []" shows "poly (p *** q) \<noteq> poly []"
map_add_def: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
perm_2: fixes pi :: "(name \<times> name) list"   and x2 :: "freeRes"   and x1 :: "pi" shows "pi \<bullet> x2 \<prec> x1 = pi \<bullet> x2 \<prec> pi \<bullet> x1"
Sup_Inf_le: fixes A :: "'a set set" shows "\<Squnion> (\<Sqinter> ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> \<Sqinter> (\<Squnion> ` A)"
assms_1: shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal (\<I>_common_core \<oplus>\<^sub>\<I> \<I>_common_rest) adv"
cases: fixes p :: "'node"   and R :: "'node set"   and a :: "'node" assumes "not_blocked p R a" obtains Sl :: "'node set" and q :: "'node" where "a = q__" and "Sl__ \<in> slices p" and "\<forall>q\<in>Sl__ \<inter> W. q \<notin> R \<and> \<not> blocking_min R q" and "q__ \<in> Sl__"   | p' :: "'node" and Sl :: "'node set" and q :: "'node" where "a = q" and "not_blocked p R p'" and "p' \<in> W" and "Sl \<in> slices p'" and "\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> blocking_min R q" and "q \<in> Sl"
elims_2: fixes x :: "('i \<times> constraint) list"   and xa :: "'i set"   and xb :: "'a ns_constraint list \<times> 's" assumes "invariant_cs x xa xb" obtains cs :: "('i \<times> constraint) list" and J :: "'i set" and ds :: "'a ns_constraint list" and s :: "'s" where "x = cs" and "xa = J" and "xb = (ds, s)" and "ds = map snd (to_ns cs) \<and> invariant_nsc (to_ns cs) J s"
scaleC_2: fixes x :: "'b" shows "2 *\<^sub>C x = x + x"
r_inv_ex: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"
uminus_dvd_conv_1: fixes d :: "int"   and t :: "int" shows "d dvd t \<equiv> - d dvd t"
initial_lower_bound_def: fixes n :: "nat"   and l :: "rat" shows "initial_lower_bound n l = (if l \<le> 1 then l else rat_of_int (root_rat_floor n l))"
empty_def: shows "FingerTree.empty \<equiv> Abs_FingerTree Empty"
SecurityInvariant_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d"   and receiver_violation :: "bool" shows "SecurityInvariant sinvar default_node_properties receiver_violation \<equiv> SecurityInvariant_preliminaries sinvar \<and> SecurityInvariant_axioms sinvar default_node_properties receiver_violation"
multiplicity_diff_pow_coprime: fixes n :: "nat" assumes "coprime p n" shows "multiplicity (int p) (x ^ n - y ^ n) = multiplicity (int p) (x - y)"
C_ccsw: shows "C = connected_component_set (- S) w"
induct: fixes P1 :: "('a, 'b, 'c) val \<Rightarrow> bool"   and P2 :: "('a, 'b, 'c) expr \<Rightarrow> bool"   and val :: "('a, 'b, 'c) val"   and expr :: "('a, 'b, 'c) expr" assumes "\<And>x. P1 (CV x)"   and "\<And>x. P1 (Var x)"   and "\<And>x. P1 (Loc x)"   and "\<And>x. P1 (Rid x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Lambda x1 x2)"   and "\<And>x. P1 x \<Longrightarrow> P2 (VE x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (Apply x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (Ite x1 x2 x3)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Ref x)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Read x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (Assign x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Rfork x)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Rjoin x)" shows "P1 val \<and> P2 expr"
P2_ideal_real_end_eq: fixes b0 :: "'a"   and b1 :: "'a"   and h0 :: "'a"   and h1 :: "'a"   and r :: "nat"   and x1 :: "'a"   and x0 :: "'a"   and s' :: "'f"   and \<A>3 :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> 'e spmf" assumes "b0 \<otimes> inv b1 = (h0 \<otimes> inv h1) [^] r"   and "h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>"   and "x1 \<in> carrier \<G>"   and "x0 \<in> carrier \<G>" shows "P2_ideal_model_end (x0, x1) (b0 \<otimes> inv (h0 [^] r)) ((h0, h1, \<^bold>g [^] r, b0, b1), s') \<A>3 = P2_real_model_end (x0, x1) ((h0, h1, \<^bold>g [^] r, b0, b1), s') \<A>3"
pseudo_ordered_set_axioms: shows "pseudo_ordered_set A (\<lambda>x y. y \<sqsubseteq> x)"
limit_mat_mult: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat"   and Y :: "nat \<Rightarrow> complex mat"   and B :: "complex mat" assumes "limit_mat X A m"   and "limit_mat Y B m" shows "limit_mat (\<lambda>k. X k * Y k) (A * B) m"
induct: fixes P :: "(event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> event_data option list set list list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "'a"   and a2 :: "event_data option list set list list"   and a3 :: "nat list" assumes "\<And>f z buf ts. (\<not> ([] \<in> set buf \<or> ts = []) \<Longrightarrow> P f (f (map hd buf) (hd ts) z) (map tl buf) (tl ts)) \<Longrightarrow> P f z buf ts" shows "P a0 a1 a2 a3"
mk_next_pow_release: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "suffix (min i j) w \<Turnstile>\<^sub>n X\<^sub>n' (i - j) \<phi> R\<^sub>n X\<^sub>n' (j - i) \<psi> = w \<Turnstile>\<^sub>n X\<^sub>n' i \<phi> R\<^sub>n X\<^sub>n' j \<psi>"
mkSPRDetAuto_implements: shows "MC.SPR.implements (mkSPRDetAuto agents jkbp envInit envAction envTrans envVal envObsC envObs)"
UP_a_closed: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "q \<in> carrier (UP Zp)" shows "p \<oplus>\<^bsub>UP Zp\<^esub> q \<in> carrier (UP Zp)"
join_None2: fixes x :: "'b option" shows "x \<squnion> None = x"
PInitRes: fixes evsPIRes :: "event list"   and M :: "agent"   and LID_M :: "nat"   and Chall_C :: "nat"   and Transaction :: "msg"   and C :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and P :: "agent"   and Chall_M :: "nat"   and XID :: "nat" assumes "evsPIRes \<in> set_pur"   and "Gets M \<lbrace>Number LID_M, Nonce Chall_C\<rbrace> \<in> set evsPIRes"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "Notes M \<lbrace>Number LID_M, Agent P, Transaction\<rbrace> \<in> set evsPIRes"   and "Nonce Chall_M \<notin> used evsPIRes"   and "Chall_M \<notin> range CardSecret"   and "Chall_M \<notin> range PANSecret"   and "Number XID \<notin> used evsPIRes"   and "XID \<notin> range CardSecret"   and "XID \<notin> range PANSecret" shows "Says M C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P (pubEK P) onlyEnc (priSK RCA)\<rbrace>) # evsPIRes \<in> set_pur"
bet__le1213: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Le Bet Cong A B A C"
collapse_4: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" assumes "is_Insert stateful_strand_step" shows "insert\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
coeffs_smult: fixes a :: "'b"   and p :: "'b poly" shows "coeffs (smult a p) = (if a = (0::'b) then [] else map ((*) a) (coeffs p))"
compose_eq: fixes x :: "'a"   and A :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" assumes "x \<in> A" shows "(g \<circ> f \<down> A) x = g (f x)"
OrdP_cases_E: fixes x :: "tm"   and H :: "fm set"   and A :: "fm"   and k :: "name" assumes "insert (x EQ Zero) H \<turnstile> A"   and "insert (x EQ SUCC (Var k)) (insert (OrdP (Var k)) H) \<turnstile> A"   and "atom k \<sharp> (x, A)"   and "\<forall>C\<in>H. atom k \<sharp> C" shows "insert (OrdP x) H \<turnstile> A"
hlit_of_flit_flit_of_hlit: fixes l :: "hterm literal" shows "hlit_of_flit (flit_of_hlit l) = l"
divide_less_cancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a c) (divide b c) = ((less zero c \<longrightarrow> less a b) \<and> (less c zero \<longrightarrow> less b a) \<and> c \<noteq> zero)"
Pab: shows "\<forall>i<length as. P (as ! i) (bs ! i)"
tBB: shows "trace (B * B) = (\<Sum>i = 0..<n. (B * B) $$ (i, i))"
injective_map_axioms_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "injective_map_axioms \<alpha> S \<equiv> inj_on \<alpha> S"
invariant_s_def: shows "invariant_s = Incremental_State_Ops_Simplex.invariant_s"
in_opt_map_eq: fixes f :: "'e \<Rightarrow> 'f option"   and g :: "'f \<Rightarrow> 'd option"   and s :: "'e"   and v :: "'d" shows "((f |> g) s = Some v) = (\<exists>v'. f s = Some v' \<and> g v' = Some v)"
cEnum_finite_4_def: shows "CENUM(Enum.finite_4) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
add_point_finite: fixes p :: "'a" shows "finite_incidence_system (add_point p) \<B>"
bisim: shows "weak_reduction_bisimulation (indRelTEQ TRel) (STCal Source Target)"
conforms_globsD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and r :: "oref"   and obj :: "obj" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "globs s r = Some obj" shows "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"
subst_self_idemp: fixes x :: "'d"   and t :: "('e, 'd) Term.term" assumes "x \<notin> vars_term t" shows "subst x t \<circ>\<^sub>s subst x t = subst x t"
arr_dom: fixes f :: "'b" assumes "\<phi>.B.arr f" shows "\<phi>.B.arr (\<phi>.B.dom f)"
takeWhile_eq_take: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "takeWhile P xs = take (length (takeWhile P xs)) xs"
hm_empty_const_def: shows "hm_empty_const = RBT_HM (HashMap_Impl.empty ())"
satisfiable_prefix_implies_\<nu>LTL: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<nu>LTL"   and "\<nexists>i. af \<phi> (prefix i w) \<sim> false\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
null_def: fixes xs :: "'k list" shows "List.null xs = (xs = [])"
0: shows "(\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f}) = (\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f})"
sorted_wrt_true: fixes xs :: "'b list" shows "sorted_wrt (\<lambda>_ _. True) xs"
fMax_boundedD_1: fixes P :: "'a fset"   and n :: "'a"   and x :: "'a" assumes "fMax P < n"   and "x |\<in>| P" shows "x < n"
flatten_matrix_def: fixes A :: "'a mat"   and k :: "nat" shows "flatten_matrix A k = A $$ (k div dim_col A, k mod dim_col A)"
empty_queue_def: shows "empty_queue \<equiv> Abs_queue ([], [])"
C1_a: fixes x :: "'a" shows "(-l x r\<rightarrow> (0::'a)) l\<rightarrow> x = (1::'a)"
pop_aux: fixes k :: "nat"   and i\<^sub>0 :: "nat" assumes "k < i\<^sub>0"   and "i\<^sub>0 < length xs"   and "local.left_spec i\<^sub>0 \<le> Some k" shows "local.pop (k # local.der_stack k) (xs ! i\<^sub>0) = local.der_stack i\<^sub>0"
negligible: fixes S :: "'a set" shows "Henstock_Kurzweil_Integration.negligible S = (\<forall>T. (indicat_real S has_integral 0) T)"
dbl_inc_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (uminus (numeral.numeral one plus k)) = uminus (neg_numeral.dbl_dec minus plus one (numeral.numeral one plus k))"
c1_2: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>R1 x\<bar> \<le> c1 * x"
map_le_produces_same_raw_match_term: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and lesubs :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'"   and "dom subs \<subseteq> tvs t"   and "lesubs \<subseteq>\<^sub>m subs" shows "raw_match_term t u lesubs = Some subs'"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "ide u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> r \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : r\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> r\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<theta>) \<cdot> \<a>[r, r\<^sub>0, w] \<cdot> (\<rho> \<star> w)) \<cdot> \<nu> = \<omega>"
A: shows "(u, con_comp_map D E p q x) \<notin> con_comp_pol I"
add_mset_add_single: fixes a :: "'b"   and A :: "'b multiset" shows "add_mset a A = A \<oplus> a"
elements_matD: fixes a :: "'a"   and A :: "'a mat" assumes "a \<in> elements_mat A" shows "\<exists>i j. i < dim_row A \<and> j < dim_col A \<and> a = A $$ (i, j)"