hcju/isabelleps huggingface dataset:
extend_set_Int_distrib: fixes A :: "'a set"   and B :: "'a set" shows "extend_set h (A \<inter> B) = extend_set h A \<inter> extend_set h B"
normal_mult: fixes f :: "real poly"   and g :: "real poly" assumes "normal_poly f"   and "normal_poly g" shows "normal_poly (f * g)"
listDiffRemoveAllNonMember: fixes x :: "'b"   and a :: "'b list"   and b :: "'b list" assumes "x \<notin> set a" shows "list_diff a b = list_diff a (removeAll x b)"
ideD_3: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and a :: "'ag" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.cod C a = a"
record_call_def: fixes x :: "'a" shows "record_call x = (\<Lambda> ce. (\<lambda>y. if x = y then two_pred\<cdot>(ce y) else ce y))"
that: shows "0 \<le> a \<and> a \<le> b \<or> odd n \<and> a \<le> b \<or> even n \<and> a \<le> 0 \<and> b \<le> a"
simple_path_linepath: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "simple_path (linepath a b)"
connected_append_edge: fixes u :: "nat"   and v :: "nat"   and w :: "nat" assumes "local.connected u v"   and "(v, w) \<in> E" shows "local.connected u w"
rAlpha: fixes \<alpha> :: "'a action"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d"   and p :: "(name \<times> name) list"   and A\<^sub>P :: "name list"   and P' :: "('a, 'b, 'c) psi" assumes "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* C"   and "bn \<alpha> \<sharp>* (p \<bullet> \<alpha>)"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C"   and "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))"   and "distinctPerm p"   and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* P'"   and "Prop C \<Psi> P \<alpha> P' A\<^sub>P \<Psi>\<^sub>P" shows "Prop C \<Psi> P (p \<bullet> \<alpha>) (p \<bullet> P') A\<^sub>P \<Psi>\<^sub>P"
pt_fresh_bij1: fixes a :: "'e"   and x :: "'d"   and pi :: "('e \<times> 'e) list" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)"   and "a \<sharp> x" shows "pi \<bullet> a \<sharp> pi \<bullet> x"
prv_all_eql: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "bprv (all x (imp (eql t1 t2) (imp (subst \<phi> t2 y) (subst \<phi> t1 y))))"
face_cycle_eq: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and b :: "'d"   and a :: "'d" assumes "digraph_map G M"   and "b \<in> pre_digraph_map.face_cycle_set M a" shows "pre_digraph_map.face_cycle_set M b = pre_digraph_map.face_cycle_set M a"
rel_S_def: fixes L1 :: "'n \<Rightarrow> 'u \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'v \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'r \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'s \<Rightarrow> 'z \<Rightarrow> bool" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 = vimage2p Rep_S Rep_S (rel_G L1 L2 Co1 Co2 Contra1 Contra2)"
prod_algebra_def: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) ` {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
termi_if_non0: fixes l :: "nat"   and c :: "com"   and s :: "char list \<Rightarrow> int" assumes "l \<turnstile> c"   and "l \<noteq> 0" shows "\<exists>t. (c, s) \<Rightarrow> t"
eq: fixes x :: "real" shows "g (iexp x) = (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))"
prWls_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'h \<Rightarrow> 'i) \<Rightarrow> 'i"   and SEM :: "('j, 'k, 'l, 'sort, 'm, 'i, 'n) semDom_scheme" shows "prWls g SEM \<equiv> \<forall>s X val. wls s X \<and> sWlsVal SEM val \<longrightarrow> sWls SEM s (g X val)"
rank_continuous: fixes x :: "nat"   and n :: "nat"   and i :: "nat"   and m :: "nat"   and j :: "nat"   and m' :: "nat" assumes "rank x n = Some i"   and "rank x (n + m) = Some j"   and "m' \<le> m" shows "\<exists>k. rank x (n + m') = Some k"
all_partitions_paper_equiv_alg: fixes xs :: "'b list" assumes "distinct xs" shows "set (map set (all_partitions_list xs)) = all_partitions (set xs)"
S2: shows "c_f_pair g h \<in> PrimRec1'"
simps_2: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmark.type" shows "\<down>\<^sub>\<tau> (Suc n) k T = \<down>\<^sub>\<tau> n k (T[k \<mapsto>\<^sub>\<tau> Top]\<^sub>\<tau>)"
mapping_of_inject: fixes x :: "'b mpoly"   and y :: "'b mpoly" shows "(mapping_of x = mapping_of y) = (x = y)"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "tabulation_data_axioms V H r \<rho> f g" shows "tabulation_data V H \<a> \<i> src trg r \<rho> f g"
Finite2: shows "finite {sat_models (Abs \<phi>) |\<phi>. nested_propos \<phi> \<subseteq> P}"
ras_length_refine: fixes R :: "('c \<times> 'd) set" shows "(ras_length, length) \<in> \<langle>R\<rangle>ras_rel \<rightarrow> nat_rel"
shift_def: fixes e :: "nat \<Rightarrow> 'a"   and i :: "nat"   and a :: "'a" shows "e\<langle>i:a\<rangle> = (\<lambda>j. if j < i then e j else if j = i then a else e (j - 1))"
comm_ruler: fixes r :: "'b list"   and s :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "r \<cdot> s \<le>p w1"   and "s \<cdot> r \<le>p w2"   and "w1 \<bowtie> w2" shows "r \<cdot> s = s \<cdot> r"
scalar_prod_split_head: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "0 < n" shows "row A 0 \<bullet> col B 0 = A $$ (0, 0) * B $$ (0, 0) + (\<Sum>i = 1..<n. A $$ (0, i) * B $$ (i, 0))"
half: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s (Abs_fv x \<tau>' t $ Fv x \<tau>')"
establish_invarI_ND_CB: fixes I :: "'v tarjan_state \<Rightarrow> bool" assumes "\<And>u v s. on_discover tarjan_params u v s = on_new_root tarjan_params v s"   and "\<And>u v s. on_back_edge tarjan_params u v s = on_cross_edge tarjan_params u v s"   and "on_init tarjan_params \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; v \<notin> dom (discovered s); discovered s' = discovered s(v \<mapsto> counter s); finished s' = finished s; counter s' = Suc (counter s); stack s' = v # stack s; back_edges s' = back_edges s; cross_edges s' = cross_edges s; tree_edges s \<subseteq> tree_edges s'; state.more s' = state.more s\<rbrakk> \<Longrightarrow> on_new_root tarjan_params v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish tarjan_params u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); discovered s' = discovered s; finished s' = finished s; stack s' = stack s; tree_edges s' = tree_edges s; counter s' = counter s; pending s' = pending s - {(u, v)}; cross_edges s' \<union> back_edges s' = cross_edges s \<union> back_edges s \<union> {(u, v)}; state.more s' = state.more s\<rbrakk> \<Longrightarrow> on_cross_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
hyp_raw: fixes cs2 :: "('a, 'b, 'c) com list"   and css2 :: "(('a, 'b, 'c) com list \<times> ('a, 'b, 'c) com list) list"   and s2 :: "('a, 'c) xstate"   and y :: "('a, 'b, 'c) com list \<times> (('a, 'b, 'c) com list \<times> ('a, 'b, 'c) com list) list \<times> ('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (cs1, css1, s1) \<rightarrow> (cs2, css2, s2)"   and "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* (cs2, css2, s2)"   and "y = (cs2, css2, s2)" shows "\<Gamma>\<turnstile>cs2,css2 \<Down> s2"
Collect_restrict: fixes X :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "{x \<in> X. P x} \<subseteq> X"
flow_to_path_image0_right_open: fixes t :: "real"   and t' :: "real"   and x :: "'a" assumes "t < t'" shows "flow_to_path x t t' ` {0..<1} = flow0 x ` {t..<t'}"
borel_measurable_le: 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" shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
valid_ownership_axioms: shows "valid_ownership \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
pell_power_def: fixes D :: "nat"   and z :: "'a \<times> 'a"   and n :: "nat" assumes "pell D" shows "pell.pell_power D z n = ((\<lambda>z'. pell.pell_mul D z' z) ^^ n) (1::'a, 0::'a)"
imbal_ins_Bal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and tw :: "bool"   and y :: "'a"   and s :: "'a tree" assumes "ins n d x t = Bal t'" shows "real (imbal (node tw t' y s)) - real (imbal (node tw t y s)) \<le> 1"
ik\<^sub>s\<^sub>tD: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S"
bij_betw_roots_unity: fixes n :: "nat" assumes "0 < n" shows "bij_betw (\<lambda>k. cis (2 * pi * real k / real n)) {..<n} {z. z ^ n = 1}"
finite_outgoing': fixes U :: "nat set" assumes "finite V" shows "finite (outgoing' U)"
weakCongruenceSubst_def: fixes P :: "pi"   and Q :: "pi" shows "P \<simeq>\<^sup>s Q \<equiv> \<forall>\<sigma>. P[<\<sigma>>] \<simeq> Q[<\<sigma>>]"
compare_pow_less_iff: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "nat"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 < p"   and "less_eq zero x"   and "less_eq zero y" shows "less (power.power one times x p) (power.power one times y p) = less x y"
coprime_mult_self_left_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one (times c a) (times c b) = (dvd.dvd times c one \<and> algebraic_semidom.coprime times one a b)"
finfun_apply_inject: fixes x :: "'d \<Rightarrow>f 'e"   and y :: "'d \<Rightarrow>f 'e" shows "(($) x = ($) y) = (x = y)"
fusion_part: fixes F :: "i \<Rightarrow> bool"   and x :: "i" assumes "F x" shows "P x (\<sigma>x. F x)"
StepRecCond2: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "if (b) c\<^sub>1 else prog \<turnstile> \<langle>c,s,l + #:c\<^sub>1 + 1\<rangle> \<leadsto> \<langle>c',s',l' + #:c\<^sub>1 + 1\<rangle>"
intros: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> A \<Gamma>"   and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "d_OUT f a < weight \<Gamma> a" shows "hindrance \<Gamma> f"
comparable: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" assumes "a + b = c + d" shows "vle_comparable a c"
verit_comp_simplify1_2: fixes a :: "'h" shows "a \<le> a"
Int_Inter_eq_1: fixes A :: "'b set"   and \<B> :: "'b set set" shows "A \<inter> \<Inter> \<B> = (if \<B> = {} then A else \<Inter> ((\<inter>) A ` \<B>))"
inducts: fixes P :: "'b prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" assumes "P \<turnstile> lub(U, V) = T" obtains "P \<turnstile> U \<le> T" and "P \<turnstile> V \<le> T" and "\<And>T'. \<lbrakk>P \<turnstile> U \<le> T'; P \<turnstile> V \<le> T'\<rbrakk> \<Longrightarrow> P \<turnstile> T \<le> T'"
simps: fixes v1 :: "'d"   and p1 :: "'c"   and v2 :: "'d"   and p2 :: "'b" shows "edge_begin ((v1, p1), v2, p2) = v1"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}deliver(fdata) . p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
NetsCollectedSorted: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "noDenyAll1 p"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted p l" shows "NetsCollected p"
float_sqrt_def: fixes a :: "('a, 'b) IEEE.float" shows "float_sqrt a = fsqrt To_nearest a"
freeTransition_def: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" shows "P \<Longrightarrow>\<alpha> \<prec> P' \<equiv> \<exists>P''' P''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> \<alpha> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P'"
dagger_slide_var: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<dagger> \<le> (x \<cdot> y)\<^sup>\<dagger> \<cdot> x"
unlink_computation: shows "rat_poly_plus (rat_poly_times (rat_poly_times A A) (rat_poly_times A A)) (rat_poly_plus (rat_poly_times 2 (rat_poly_times A (rat_poly_times A (rat_poly_times B B)))) (rat_poly_times (rat_poly_times B B) (rat_poly_times B B))) = rat_poly_plus (rat_poly_plus (A ^ 4) (B ^ 4)) 2"
chainTransitionAppend_2: fixes P :: "pi"   and P'' :: "pi"   and u :: "name"   and P''' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P''"   and "P'' \<Longrightarrow>\<^sub>lu in P'''\<rightarrow>a<x> \<prec> P'" shows "P \<Longrightarrow>\<^sub>lu in P'''\<rightarrow>a<x> \<prec> P'"
h: shows "set (take_indices (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (find_nonzeros_from_input_vec (solve_for_lhs p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))))))) = set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) (F \<circ> F'G'.FG.map \<circ> G) FG.FG.map (F \<circ> F'G'.\<epsilon> \<circ> G)"
arctan_conv_pre_arctan: fixes x :: "real" assumes "norm x < 1" shows "arctan x = x * powser (msllist_of_msstream arctan_series_stream) (x\<^sup>2)"
mult_not_zero: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
rtrancl_idemp_self_comp: fixes R :: "('a \<times> 'a) set" shows "R\<^sup>* ; R\<^sup>* = R\<^sup>*"
is_renaming_inv_renaming_cancel_cls: fixes \<rho> :: "'s"   and C :: "'a literal multiset" assumes "is_renaming \<rho>" shows "C \<cdot> \<rho> \<cdot> inv_renaming \<rho> = C"
Join_absorb1: fixes F :: "'c program"   and G :: "'c program" assumes "F \<le> G" shows "F \<squnion> G = G"
boundedI: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq a b"   and "less_eq a c" shows "less_eq a (f b c)"
lossless_protocol_OT12: fixes M :: "bool \<times> bool"   and C :: "bool" shows "lossless_spmf (protocol_OT12 M C)"
coprime_mult_right_iff: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "coprime c (a * b) = (coprime c a \<and> coprime c b)"
prob_space_axioms_def: fixes M :: "'b measure" shows "prob_space_axioms M \<equiv> emeasure M (space M) = 1"
VerduennungS: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<DD> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Theta>" shows "\<Gamma> \<Rightarrow> \<DD> # \<Theta>"
mem_simps_9: fixes A :: "'b"   and C :: "'b set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
weak_preservation_of_barbs_in_barbed_encoding: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and a :: "'barbs" assumes "rel_weakly_preserves_barbs Rel (STCalWB SWB TWB)"   and "(P, Q) \<in> Rel"   and "(\<exists>S. S \<in>S P \<and> S\<Down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<Down><TWB>a)" shows "(\<exists>S. S \<in>S Q \<and> S\<Down><SWB>a) \<or> (\<exists>T. T \<in>T Q \<and> T\<Down><TWB>a)"
one_poly_mapping_def: shows "1 \<equiv> Abs_poly_mapping (\<lambda>k. (1::'c) when k = (0::'b))"
elims_2: fixes x :: "'a infty"   and xa :: "'a infty" assumes "x \<le> xa" obtains uv_ :: "'a infty" where "x = uv_" and "xa = Infty"   | a :: "'a" and b :: "'a" where "x = Num a" and "xa = Num b" and "a \<le> b"
assms: shows "A \<inter> \<Union> (set ` set xss) = {}"
hs_pending_loaded_locs_def: shows "hs_pending_loaded_locs = prefixed ''hs_'' - {hs_load_pending}"
zero_eq: shows "\<zero>\<^bsub>R\<^esub> = 0"
ipurge_tr_rev_t_intro_1: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list"   and X :: "('d, 'c) ipurge_rec" assumes "ipurge_tr_rev_t_inv_1 I D u xs X"   and "ipurge_tr_rev_t_form X" shows "ipurge_tr_rev_t_out X = ipurge_tr_rev I D u xs"
under_Field: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "under r a \<subseteq> Field r"
Inr_in_eps_iff: fixes q :: "hf"   and Q :: "hf set" shows "(HF.Inr q \<in> ST.epsclo Q) = (HF.Inr q \<in> Q \<and> q \<in> dfa.states MT \<or> q = dfa.init MT \<and> (\<exists>qf\<in>dfa.final MS. HF.Inl qf \<in> Q))"
mset_ordering_add1: fixes E1 :: "'b multiset"   and E2 :: "'b multiset"   and r :: "('b \<times> 'b) set"   and a :: "'b" assumes "(E1, E2) \<in> mult r" shows "(E1, E2 + {#a#}) \<in> mult r"
assms_1: shows "\<forall>G F nP i. wf_graph G \<and> F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> i \<in> fst ` F \<longrightarrow> \<not> sinvar G (nP(i := otherbot))"
valid_path_same_level_path_Append: fixes as :: "'edge list"   and as' :: "'edge list" assumes "valid_path as"   and "same_level_path as'" shows "valid_path (as @ as')"
is_sublist_hd_eq: fixes a :: "'a"   and vs :: "'a list" shows "is_sublist [a] vs = (a \<in> set vs)"
A: fixes k :: "'c"   and v :: "'d"   and x :: "'e" shows "f k v x \<le> f' k v x"
Un_Diff_cancel: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> (B - A) = A \<union> B"
proj2_abs_def: fixes v :: "(real, 3) vec" shows "proj2_abs v \<equiv> Abs_proj2 (proportionality `` {v})"
prems_3: shows "fst ` set (tx # txs) \<subseteq> atrm"
scaleR_ccsubspace_def: shows "(*\<^sub>R) \<equiv> map_fun id (map_fun space_as_set Abs_clinear_space) (\<lambda>c. (`) ((*\<^sub>R) c))"
convex_hull_insert_alt: fixes a :: "'c"   and S :: "'c set" shows "convex hull insert a S = (if S = {} then {a} else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
comp_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VxVxV.comp g f = (if seq (fst g) (fst f) \<and> VxV.seq (snd g) (snd f) then (fst g \<cdot> fst f, VxV.comp (snd g) (snd f)) else (null, VxV.null))"
root_nat_floor_def: fixes p :: "nat"   and x :: "nat" shows "root_nat_floor p x = root_int_floor_pos p (int x)"
pred_set: fixes P :: "'d \<Rightarrow> bool" shows "pred_fmap P = (\<lambda>x. Ball (fmran' x) P)"
rep_eq: fixes x :: "bool"   and xa :: "bool"   and xb :: "bool"   and xc :: "bool"   and xd :: "bool"   and xe :: "bool"   and xf :: "bool"   and xg :: "literal" shows "explode (Literal x xa xb xc xd xe xf xg) = Char x xa xb xc xd xe xf False # explode xg"
in_lists_rev: fixes u :: "'b list"   and A :: "'b set" assumes "u \<in> lists A" shows "rev u \<in> lists A"
measurable_on_bilinear: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "bilinear h"   and "f measurable_on S"   and "g measurable_on S" shows "(\<lambda>x. h (f x) (g x)) measurable_on S"
simps: fixes C :: "'b literal set"   and s :: "('b \<times> 'b trm) list" shows "subst_cl C s = {L. \<exists>L'. L' \<in> C \<and> L = subst_lit L' s}"
biQuot_union: fixes a :: "'b"   and b :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "biQuot a b (A \<union> B) = biQuot a b A \<union> biQuot a b B"
ih: assumes "x \<sharp> M" shows "M[x::=P] = M"
size_21: fixes x101 :: "'b ltln"   and x102 :: "'b ltln" shows "size (x101 W\<^sub>n x102) = size x101 + size x102 + Suc 0"
lift_wf: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "lowered.is_pref_profile R" shows "is_pref_profile (lift R)"
isin_bst: fixes t :: "'a tree"   and x :: "'a" assumes "bst t" shows "isin t x = (x \<in> set_tree t)"
cod_cod: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.cod (Cop.cod f) = Cop.cod f"
FSwSbImorph_igWlsInp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'e option"   and h :: "'e \<Rightarrow> 'h"   and hA :: "'f \<Rightarrow> 'i"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'h, 'i, 'j) model_scheme" assumes "igWlsInp MOD delta inp"   and "FSwSbImorph h hA MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
irbd_rbd_inv2: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "(bd\<^sub>\<R> \<circ> bd\<^sup>-\<^sub>\<R>) \<phi> = \<phi>"
size_instance: fixes v :: "vbl"   and A :: "formula" shows "size (instanceF v A) = size A"
ideduct_synth_reduce: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "M \<union> M' \<turnstile>\<^sub>c t"   and "\<And>t'. t' \<in> M' \<Longrightarrow> M \<turnstile>\<^sub>c t'" shows "M \<turnstile>\<^sub>c t"
parallel_append: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list"   and d :: "'b list" assumes "a \<parallel> b" shows "a \<cdot> c \<parallel> b \<cdot> d"
const_poly_dvd_iff_dvd_content: fixes c :: "'a"   and p :: "'a poly" shows "([:c:] dvd p) = (c dvd Polynomial.content p)"
opeU: shows "openin (top_of_set (S \<inter> f -` T')) U"
canon_ang_uminus_pi: fixes \<alpha> :: "real" assumes "\<downharpoonright>\<alpha>\<downharpoonleft> = pi" shows "\<downharpoonright>- \<alpha>\<downharpoonleft> = \<downharpoonright>\<alpha>\<downharpoonleft>"