hcju/isabelleps huggingface dataset:
set_intros_3: fixes x32 :: "'e"   and x31 :: "'f" shows "x32 \<in> set2_step (Yield x31 x32)"
rep_eq: fixes x :: "'b trm" shows "depth x = size (rep_trm x)"
proper_implies_distinct: fixes x :: "i"   and y :: "i" assumes "M"   and "PP x y" shows "x \<noteq> y"
assms_3: shows "insertion (nth_default 0 (xs' @ x # xs)) b = B"
finite_intersection_of_inc: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "P S" shows "(finite intersection_of P) S"
DERIV_divide: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)"   and "g x \<noteq> (0::'a)" shows "((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
weakOutputFreshChainDerivative_2: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "yvec \<sharp>* P"   and "xvec \<sharp>* yvec"   and "xvec \<sharp>* M"   and "distinct xvec" shows "yvec \<sharp>* P'"
distinct_1: fixes x2 :: "'c" shows "\<turnstile> \<noteq> \<lbrakk>x2\<rbrakk>"
Sup_distl: fixes x :: "'a"   and Y :: "'a set" shows "x \<cdot> \<Squnion>Y = \<Squnion>{x \<cdot> y |y. y \<in> Y}"
degree_Poly_eq: fixes xs :: "'a list" assumes "xs = [] \<or> last xs \<noteq> (0::'a)" shows "degree (Poly xs) = length xs - 1"
ba_less: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<squnion> y = y \<and> y \<squnion> x \<noteq> x)"
assms_1: shows "lookupDef g n v = (v', n', def)"
assms_1: shows "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H"
OAlist_tc_empty_def: shows "OAlist_tc_empty = OAlist_tc []"
fls_X_subdegree: shows "fls_subdegree fls_X = 1"
adjl_\<alpha>_def: fixes l :: "'c list \<times> ('c \<times> 'd \<times> 'c) list" shows "adjl_\<alpha> l \<equiv> let (nl, el) = l in \<lparr>nodes = set nl \<union> fst ` set el \<union> snd ` snd ` set el, edges = set el\<rparr>"
part_Un_singl: fixes K :: "'b set"   and P :: "'b set set"   and I0 :: "'b set" assumes "part K P"   and "\<And>I. I \<in> P \<Longrightarrow> I0 \<inter> I = {}" shows "part (I0 \<union> K) ({I0} \<union> P)"
basis_null_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "vec.independent (basis_null_space A) \<and> null_space A = vec.span (basis_null_space A)"
ValidI: fixes c :: "'b com"   and P :: "'b set"   and Q :: "'b set" assumes "\<And>s s'. \<lbrakk>Sem c s s'; s \<in> P\<rbrakk> \<Longrightarrow> s' \<in> Q" shows "\<turnstile> P c Q"
reflect_poly_simps_6: fixes p :: "'a poly"   and n :: "nat" shows "reflect_poly (p ^ n) = reflect_poly p ^ n"
set_integrable_Un: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "set_integrable M A f"   and "set_integrable M B f"   and "A \<in> sets M"   and "B \<in> sets M" shows "set_integrable M (A \<union> B) f"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "\<exists>b. b \<in> B"
split_mult_pos_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero" shows "less_eq zero (times a b)"
circuit_in_dep_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.circuit_in \<E> X" shows "\<not> \<E>.indep_in \<E> X"
add_le_same_cancel1: fixes b :: "'a"   and a :: "'a" shows "(b + a \<le> b) = (a \<le> (0::'a))"
simps_2: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" shows "BsSkewBinomialHeapStruc.bs_invar (Inr x) = BsSkewBinomialHeapStruc.elem_invar x"
open_generated_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool" assumes "class.order_topology less_eq less open" shows "open = generate_topology (range (ord.lessThan less) \<union> range (ord.greaterThan less))"
is_span: shows "C.span assoc\<mu>\<nu>\<pi>.dom.leg0 assoc\<mu>\<nu>\<pi>.dom.leg1"
that: fixes c :: "'b" assumes "c \<in> ideal B"   and "(\<Prod>i\<in>I. (f i + g i) ^ m i) = (\<Prod>i\<in>I. f i ^ m i) + c" shows "thesis"
ge: shows "\<forall>\<^sub>F x in F. ereal C < g x"
f'_def: shows "f' \<equiv> \<lambda>(k, p). if k = 0 then 2 * card (nodes A) else f (k, p)"
strategyproofness: shows "pmf (sds R39) d + pmf (sds R39) c \<le> pmf (sds R36) d + pmf (sds R36) c"
K: fixes a :: "'a"   and b :: "'b" shows "norm (a ** b) \<le> norm a * norm b * K"
mem_infmal_iff: fixes x :: "'c star" shows "(x \<in> Infinitesimal) = (x \<approx> 0)"
UnI2: fixes c :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "c \<in> B" shows "c \<in> A \<union> B"
assms: shows "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
is_process5_S: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and sa :: "'a event list"   and X :: "'a event set"   and Y :: "'a event set" assumes "is_process P"   and "(sa, X) \<in> FAILURES P"   and "\<forall>c. c \<in> Y \<longrightarrow> (sa @ [c], {}) \<notin> FAILURES P" shows "(sa, X \<union> Y) \<in> FAILURES P"
nMTcharn: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and p :: "'a list" shows "none_MT_rules C p = (\<forall>r\<in>set p. dom (C r) \<noteq> {})"
simps_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Some x2 = Some y2) = (x2 = y2)"
that: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "red F a b" shows "thesis"
cobounded1: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "less_eq_prod a (sup_prod a b)"
xy: shows "ideal_generated {u, v} = ideal_generated {1::'a}"
defAssUses_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and g :: "'a" shows "CFG_SSA_wf_base.defAssUses \<alpha>n invar inEdges' Entry defs uses phis g \<equiv> \<forall>n\<in>set (\<alpha>n g). \<forall>v\<in>CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n. CFG_SSA_base.defAss \<alpha>n invar inEdges' Entry defs phis g n v"
less_multiset\<^sub>D\<^sub>M: fixes M :: "'c multiset"   and N :: "'c multiset" shows "(M < N) = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
induct: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "(('c, 'b) fm list \<times> 'b) list"   and x4 :: "(nat \<times> nat) set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> (('c, 'b) fm list \<times> 'b) list \<Rightarrow> (nat \<times> nat) set \<Rightarrow> bool" assumes "descendants x1 x2 x3 x4"   and "\<And>branch v qs i v' k. \<lbrakk>branch !. v = Some (qs, i); qs !. v' = Some (\<^bold>\<diamond> Nom k)\<rbrakk> \<Longrightarrow> P k i branch {(v, v')}"   and "\<And>branch v qs a v' k p i xs w w' rs. \<lbrakk>branch !. v = Some (qs, a); qs !. v' = Some (\<^bold>\<not> (\<^bold>@ k p)); descendants k i branch xs; P k i branch xs; (w, w') \<in> xs; branch !. w = Some (rs, a); rs !. w' = Some (\<^bold>\<diamond> Nom k)\<rbrakk> \<Longrightarrow> P k i branch ({(v, v')} \<union> xs)"   and "\<And>branch v qs a v' p k i xs w w' rs b. \<lbrakk>branch !. v = Some (qs, a); qs !. v' = Some p; descendants k i branch xs; P k i branch xs; (w, w') \<in> xs; branch !. w = Some (rs, b); rs !. w' = Some p; Nom a at b in' set branch\<rbrakk> \<Longrightarrow> P k i branch ({(v, v')} \<union> xs)" shows "P x1 x2 x3 x4"
expr_typing_ifE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and b :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> IF b THEN e1 ELSE e2 : t" obtains "\<Gamma> \<turnstile> b : BOOL" and "\<Gamma> \<turnstile> e1 : t" and "\<Gamma> \<turnstile> e2 : t"
f_le_g: fixes X :: "nat set" shows "suminf (zero_on f (- X)) \<le> suminf (zero_on g (- R `` X))"
mem_correct_empty_default: shows "mem_correct_empty (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (m k))) (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (m(k \<mapsto> v)))) (\<lambda>_. True) Map.empty"
Span_eq_combine_set: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "Span K Us = {combine Ks Us |Ks. set Ks \<subseteq> K}"
scalar_prod_minus_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>3 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 - v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 - v\<^sub>1 \<bullet> v\<^sub>3"
induct: fixes x1 :: "Mode \<Rightarrow> 'Var set"   and x2 :: "('Var, 'AExp, 'BExp) Stmt"   and x3 :: "Mode \<Rightarrow> 'Var set"   and P :: "(Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> bool" assumes "\<turnstile> x1 { x2 } x3"   and "\<And>mds annos. P mds (Skip \<otimes> annos) (mds \<oplus> annos)"   and "\<And>x mds e annos. \<lbrakk>x \<notin> mds GuarNoWrite \<union> mds GuarNoReadOrWrite; aexp_vars e \<inter> mds GuarNoReadOrWrite = {}; \<forall>v. (x \<in> \<C>_vars v \<longrightarrow> v \<notin> mds GuarNoWrite \<union> mds GuarNoReadOrWrite) \<and> (\<C>_vars v \<inter> aexp_vars e \<noteq> {} \<longrightarrow> v \<notin> mds GuarNoReadOrWrite)\<rbrakk> \<Longrightarrow> P mds ((x \<leftarrow> e) \<otimes> annos) (mds \<oplus> annos)"   and "\<And>mds annos c\<^sub>1 mds'' c\<^sub>2 e. \<lbrakk>\<turnstile> mds \<oplus> annos { c\<^sub>1 } mds''; P (mds \<oplus> annos) c\<^sub>1 mds''; \<turnstile> mds \<oplus> annos { c\<^sub>2 } mds''; P (mds \<oplus> annos) c\<^sub>2 mds''; bexp_vars e \<inter> mds GuarNoReadOrWrite = {}; \<forall>v. \<C>_vars v \<inter> bexp_vars e \<noteq> {} \<longrightarrow> v \<notin> mds GuarNoReadOrWrite\<rbrakk> \<Longrightarrow> P mds (Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos) mds''"   and "\<And>mds' mds annos c e. \<lbrakk>mds' = mds \<oplus> annos; \<turnstile> mds' { c } mds'; P mds' c mds'; bexp_vars e \<inter> mds' GuarNoReadOrWrite = {}; \<forall>v. \<C>_vars v \<inter> bexp_vars e \<noteq> {} \<longrightarrow> v \<notin> mds' GuarNoReadOrWrite\<rbrakk> \<Longrightarrow> P mds (While e c \<otimes> annos) mds'"   and "\<And>mds c\<^sub>1 mds' c\<^sub>2 mds''. \<lbrakk>\<turnstile> mds { c\<^sub>1 } mds'; P mds c\<^sub>1 mds'; \<turnstile> mds' { c\<^sub>2 } mds''; P mds' c\<^sub>2 mds''\<rbrakk> \<Longrightarrow> P mds (c\<^sub>1 ;; c\<^sub>2) mds''"   and "\<And>mds\<^sub>2 c mds\<^sub>2' mds\<^sub>1 mds\<^sub>1'. \<lbrakk>\<turnstile> mds\<^sub>2 { c } mds\<^sub>2'; P mds\<^sub>2 c mds\<^sub>2'; mds\<^sub>1 \<le> mds\<^sub>2; mds\<^sub>2' \<le> mds\<^sub>1'\<rbrakk> \<Longrightarrow> P mds\<^sub>1 c mds\<^sub>1'" shows "P x1 x2 x3"
matrix_equation_helper_step_R: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "vec_of_list (mtx_row_R signs I) \<bullet> w_vec_R p qs signs = rat_of_int (int (card {x. poly p x = 0 \<and> (\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x})) - rat_of_int (int (card {x. poly p x = 0 \<and> (\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> poly (prod_list (retrieve_polys qs (snd I))) x < 0}))"
mult2_alt_sI: fixes X :: "'a multiset"   and X1 :: "'a multiset"   and X2 :: "'a multiset"   and Y :: "'a multiset"   and Y1 :: "'a multiset"   and Y2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "X = X1 + X2"   and "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "Y2 \<noteq> {#}"   and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)" shows "(X, Y) \<in> mult2_alt_s ns s"
qinc: shows "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"
coprime_dvd_mult: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "coprime a b"   and "a dvd c * b" shows "a dvd c"
0: shows "rawpsubst (rawpsubst (eqv \<phi>1 \<phi>2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = eqv (rawpsubst (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubst (rawpsubst \<phi>2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
simps_2: fixes uv :: "Unboxed.type option list" shows "Op_example.ubx OpMulNum uv = None"
b2: fixes x1 :: "'b"   and x2 :: "'b" shows "(A\<^sup>\<dagger>) (x1 + x2) - ((A\<^sup>\<dagger>) x1 + (A\<^sup>\<dagger>) x2) = (0::'a)"
tset_tllist_of_llist: fixes b :: "'d"   and xs :: "'c llist" shows "tset (tllist_of_llist b xs) = lset xs"
p_continuous: fixes T :: "real set"   and a :: "real"   and v :: "real"   and s0 :: "real" shows "continuous_on T (movement.p a v s0)"
distinct_71: fixes x5 :: "msg"   and x7 :: "rat" shows "Hash x5 \<noteq> Pri_AgrK x7"
representation_add: fixes basis :: "'d set"   and v :: "'d"   and u :: "'d" assumes "cindependent basis"   and "v \<in> cspan basis"   and "u \<in> cspan basis" shows "crepresentation basis (u + v) = (\<lambda>b. crepresentation basis u b + crepresentation basis v b)"
assms: shows "GEN_ALGO_tag (is_set_to_sorted_list ordR Rk Rs tsl)"
comm_monoid_mult_hom_1_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_mult_hom_1 hom \<equiv> monoid_mult_hom_1 hom"
assms_1: shows "x \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sup>s\<^sub>l P \<parallel> <\<nu>x>Q"
MinPredicate: fixes P :: "nat \<Rightarrow> bool" assumes "\<exists>n. P n" shows "\<exists>n0. P n0 \<and> (\<forall>n'. P n' \<longrightarrow> n0 \<le> n')"
inv_op_closed2: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and h :: "'e" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "h \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H"
frestrictionI: fixes a :: "V"   and A :: "V"   and b :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A"
obj_is_self_adjoint_1: fixes a :: "'a" assumes "B.obj a" shows "B.adjoint_pair a a"
mopup_halt: fixes n :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and f :: "nat \<Rightarrow> (nat \<times> cell list \<times> cell list) \<times> nat"   and P :: "(nat \<times> cell list \<times> cell list) \<times> nat \<Rightarrow> bool" assumes "n < length lm"   and "mopup_inv (Suc 0, l, r) lm n ires"   and "f = (\<lambda>stp. (steps0 (Suc 0, l, r) (mopup_a n @ shift mopup_b (2 * n)) stp, n))"   and "P = (\<lambda>(c, n). is_final c)" shows "\<exists>stp. P (f stp)"
Src_Trg: fixes t :: "'a local.term" assumes "Arr t" shows "Src (Trg t) = Trg t"
span_is_subspace: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and S :: "'d set" assumes "vectorspace K V"   and "S \<subseteq> carrier V" shows "subspace K (LinearCombinations.module.span K V S) V"
O_simps_3: fixes \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and trn :: "('a, 'b, 'c) trans"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and tr :: "('a, 'b, 'c) trans list" assumes "\<gamma> trn" shows "BD_Security.O \<gamma> g (trn # tr) = g trn # BD_Security.O \<gamma> g tr"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> SourceTerm S2"
AsAwaitThrow: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and r :: "'e set"   and ae :: "('e, 'd, 'f) ann"   and b :: "'e set"   and e :: "('e, 'd, 'f) com" shows "assertionsR \<Gamma> \<Theta> Q A (AnnRec r ae) (Await b e) A"
prop_atoms_notin_1: fixes \<phi> :: "'b ltln" shows "true\<^sub>n \<notin> prop_atoms \<phi>"
intro: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. f (x + y) = f x + f y" shows "additive f"
assms_2: shows "S \<inter> interior T \<noteq> {}"
ActualE: fixes \<phi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> in dw]" shows "[\<phi> in dw]"
ccw_ncoll_imp_ccw: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw a b c"   and "det3 a b c \<noteq> 0" shows "ccw' a b c"
Mojmir_rabin_smallest_accepting_rank: shows "smallest_accepting_rank = smallest_accepting_rank\<^sub>\<R>"
mult_mat_vec_of_list: fixes ls :: "'a list"   and nc :: "nat"   and rs :: "'a list list" assumes "length ls = nc"   and "list_all (\<lambda>r. length r = nc) rs" shows "mat_of_rows_list nc rs *\<^sub>v vec_of_list ls = vec_of_list (map (\<lambda>r. vec_of_list r \<bullet> vec_of_list ls) rs)"
cod_dom: fixes f :: "'a" assumes "S.arr f" shows "S.cod (S.dom f) = S.dom f"
V_1: fixes i :: "nat" shows "norm (V i) = 1"
avl_delete: fixes t :: "'b tree"   and x :: "'b" assumes "avl t" shows "avl (delete x t)"
V_reset_closed: fixes u :: "'c \<Rightarrow> real"   and r :: "'c list"   and d :: "nat" assumes "u \<in> V" shows "[r\<rightarrow>real d]u \<in> V"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'c"   and event :: "'d event" shows "P (case event of Broadcast x \<Rightarrow> f1 x | Deliver x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. event = Broadcast x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. event = Deliver x2 \<and> \<not> P (f2 x2))))"
bind_returns_heap_E3: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h' :: "'a"   and x :: "'d" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "pure f h" shows "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
ccSUP_le_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and u :: "'a" assumes "countable A" shows "(Sup (f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
the_inv_f_f: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g" assumes "inj f" shows "the_inv f (f x) = x"
finite_set_tree: fixes t :: "'a tree" shows "finite (set_tree t)"
nb: fixes f :: "sat_plan_variable formula" assumes "f \<in> set (map (\<lambda>op. Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) (filter (\<lambda>op. ListMem v (delete_effects_of op)) (\<Pi>\<^sub>\<O>)))" shows "is_lit_plus f"
cinner_real: fixes x :: "'b" shows "\<langle>x, x\<rangle> \<in> \<real>"
monad_fail_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" shows "monad_fail return bind fail \<equiv> monad return bind \<and> monad_fail_axioms bind fail"
downset_filter: fixes xss :: "'b list set"   and P :: "'b \<Rightarrow> bool" assumes "downset xss" shows "downset (filter P ` xss)"
rel_fun_assoc: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "class.rel_semigroup \<rho>" shows "rel_magma.times_rel_fun \<rho> (rel_magma.times_rel_fun \<rho> f g) h = rel_magma.times_rel_fun \<rho> f (rel_magma.times_rel_fun \<rho> g h)"
set_restriction_Int_conv: fixes f :: "'a set \<Rightarrow> 'a set" shows "set_restriction f = (\<exists>B. \<forall>A. f A = A \<inter> B)"
that: fixes C :: "real" assumes "0 \<le> C"   and "\<And>z. z \<notin> (\<Union>n. ball (complex_of_real (real_of_int (2 * n) * pi) * \<i>) r) \<Longrightarrow> cmod (H z) \<le> C" shows "thesis"
r_rev_aux_total: fixes i :: "nat"   and e :: "nat" shows "eval r_rev_aux [i, e] \<down>"
WTrt_elim_cases_5: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'b J_prog"   and h :: "'a"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (throw e) T" obtains T :: "ty" where "J_heap_base.WTrt typeof_addr P h E e T" and "P \<turnstile> T \<le> Class Throwable"
assign\<^sub>2: fixes x :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'AExp"   and t :: "Sec" assumes "x \<in> dom \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t" shows "\<turnstile> \<Gamma> {x \<leftarrow> e} \<Gamma>(x \<mapsto> t)"
group_integer_mod_group: fixes n :: "nat" shows "Group.group (integer_mod_group n)"
wp1_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp\<^sub>1 (c\<^sub>1;; c\<^sub>2) Q = wp\<^sub>1 c\<^sub>1 (wp\<^sub>1 c\<^sub>2 Q)"
f_def: shows "f = (\<lambda>(R, s). \<Prod>R * s\<^sup>2)"
del_height: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'b btree"   and x :: "'b" assumes "split split"   and "0 < k"   and "root_order k t"   and "bal t" shows "height (split.del split k x t) = height t"
tinst_iff_ex_raw_match_term: fixes t1 :: "Core.term"   and t2 :: "Core.term" shows "tinst t1 t2 = (\<exists>subs. raw_match_term t2 t1 Map.empty = Some subs)"
assms_3: fixes i :: "nat" assumes "k \<le> i"   and "i < l" shows "w !! i \<notin> visible"
cases: fixes a :: "Predicate_Compile_Tests.char" assumes "is_F_or_G a" obtains x :: "Predicate_Compile_Tests.char" where "a = x" and "x = F \<or> x = G"
disc_eq_case_1: fixes llist :: "'c llist" shows "lnull llist = (case llist of LNil \<Rightarrow> True | LCons uu_ uua_ \<Rightarrow> False)"
vertex_map: fixes x :: "'a" assumes "x \<in> \<Union> X" shows "f x \<in> \<Union> X"
foldl_distrstart: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and l :: "'a list" assumes "\<forall>a x y. f (g x y) a = g x (f y a)" shows "foldl f (g x y) l = g x (foldl f y l)"