hcju/isabelleps huggingface dataset:
refine: fixes R_Iec :: "('c \<times> unit) set" shows "(impl_blue_dfs_hash, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
card_of_vimage: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "B \<subseteq> range f" shows "|B| \<le>o |f -` B|"
le_cINF_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and u :: "'d multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_below (f ` A)" shows "(u \<subseteq># Inf (f ` A)) = (\<forall>x\<in>A. u \<subseteq># f x)"
nonzero_Ax: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "A *v x \<noteq> 0"
assms: fixes a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'value list" assumes "step s a = (ou, s')"   and "\<not> T (Trans s a ou s')"   and "consume (Trans s a ou s') vl vl'" shows "match \<Delta> s s1 vl1 a ou s' vl' \<or> ignore \<Delta> s s1 vl1 a ou s' vl'"
success_of_listI: fixes xs :: "'a list"   and h :: "heap" shows "success (Array.of_list xs) h"
F_def: shows "F = cis (- \<phi>) *k (inv_D ** cA ** D)"
LN_4: fixes es' :: "e list"   and les' :: "e list"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "Lfilled k l es' les'"   and "\<S>\<bullet>\<C> \<turnstile> lfilledk : ts _> ts'"   and "\<And>arb_labs ts ts'. \<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es : ts _> ts' \<Longrightarrow> \<S>\<bullet>\<C>\<lparr>label := arb_labs @ label \<C>\<rparr> \<turnstile> es' : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> les' : ts _> ts'"
nonces_def: fixes X :: "msg" shows "nonces X = \<Union> (freenonces ` Rep_Msg X)"
blocking_onI: fixes X'' :: "'a set"   and h :: "'c"   and X :: "'a set"   and ds :: "'b set" assumes "X'' \<noteq> Ch h X"   and "X'' = Ch h (X \<union> X'')"   and "\<And>x. x \<in> X'' \<Longrightarrow> x \<in> CD_on ds (X \<union> X'')" shows "blocking_on ds X h X''"
inv_i_iso_\<iota>_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and \<iota>\<^sub>1 :: "'a"   and f :: "'a" assumes "monoidal_category_with_alternate_unit C T \<alpha> \<iota> \<iota>\<^sub>1"   and "partial_magma.in_hom C f (monoidal_category.unity C \<iota>) (monoidal_category.unity C \<iota>\<^sub>1)"   and "category.iso C f"   and "C f \<iota> = C \<iota>\<^sub>1 (T (f, f))" shows "C (C (category.inv C (monoidal_category_with_alternate_unit.i C T \<alpha> \<iota> \<iota>\<^sub>1)) f) \<iota> = C \<iota> (T (C (category.inv C (monoidal_category_with_alternate_unit.i C T \<alpha> \<iota> \<iota>\<^sub>1)) f, C (category.inv C (monoidal_category_with_alternate_unit.i C T \<alpha> \<iota> \<iota>\<^sub>1)) f))"
nth_xs: shows "xs ! adjust_idx i j = (if adjust_idx i j < i then take i xs ! adjust_idx i j else (xs ! i # drop (Suc i) xs) ! (adjust_idx i j - i))"
c2_2: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>R2 x\<bar> \<le> c2 * x powr (- \<alpha> - 1)"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "L.seq g f" shows "L.dom (L.comp g f) = L.dom f"
deg_reduced_GB_univariate_le: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "card X \<le> 1"   and "F \<subseteq> P[X]"   and "f \<in> ideal F"   and "f \<noteq> 0"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> poly_deg f"
distinct_186: fixes x13 :: "Syntax.term"   and x71 :: "Syntax.term"   and x72 :: "Syntax.term" shows "Roll x13 \<noteq> Syntax.Pair x71 x72"
integrable_const_bound: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and B :: "real" assumes "finite_measure M"   and "AE x in M. norm (f x) \<le> B"   and "f \<in> borel_measurable M" shows "integrable M f"
set_prems_of: fixes \<iota> :: "'a inference" shows "set (prems_of \<iota>) = (if prems_of \<iota> = [] then {} else {main_prem_of \<iota>} \<union> set (side_prems_of \<iota>))"
pran_res_zero: fixes A :: "'b set" shows "{}\<^sub>p \<rhd>\<^sub>p A = {}\<^sub>p"
ideduct_mono: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term"   and M' :: "('a, 'b) Term.term set" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct arity public Ana M t"   and "M \<subseteq> M'" shows "intruder_model.intruder_deduct arity public Ana M' t"
Cons_in_lists_iff: fixes x :: "'e"   and xs :: "'e list"   and A :: "'e set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
pdevs_val_pdevs_scaleR: fixes e :: "nat \<Rightarrow> real"   and X :: "real pdevs"   and y :: "'a" shows "pdevs_val e (pdevs_scaleR X y) = pdevs_val e X *\<^sub>R y"
simps_20: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> x41 ;; x42"
botE: fixes x :: "'a"   and P :: "bool" assumes "pred.eval bot x" shows "P"
intro: fixes M :: "'b measure" assumes "sets M = sets borel" shows "finite_borel_measure'_axioms M"
wordinterval_intersection'_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_intersection' r1 r2) = wordinterval_to_set r1 \<inter> wordinterval_to_set r2"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "P x x"
append_eq_Cons_conv: fixes ys :: "'b list"   and zs :: "'b list"   and x :: "'b"   and xs :: "'b list" shows "(ys \<cdot> zs = x # xs) = (ys = \<epsilon> \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' \<cdot> zs = xs))"
pcp_lim_def: fixes C :: "'b set set"   and S :: "'b set" shows "pcp_lim C S \<equiv> \<Union> {pcp_seq C S n |n. True}"
spmf_eqI: fixes p :: "'b spmf"   and q :: "'b spmf" assumes "\<And>i. spmf p i = spmf q i" shows "p = q"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "b \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
frontier_halfspace_le: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b) \<or> b \<noteq> 0" shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
nonintDfmla_def: shows "nonintDfmla \<equiv> Fall2 p0 p1 (Imp (Alw (EqButGH p0 p1)) (Alw (EqOnGL p0 p1)))"
pathEntry': shows "(_Entry_) -a # asx\<rightarrow>* n"
goal_of_def: shows "goal_of \<equiv> id \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst strips_problem_ext_Tuple_Iso"
single_numeral: fixes n :: "num" shows "monomial (numeral n) (0::'a) = numeral n"
io_init_inclusion: shows "io_init \<in> L M1 \<and> io_init \<in> L M2"
lookup_def: fixes A :: "'a tensor" shows "lookup A = lookup_base (dims A) (Tensor.vec A)"
Gamma_times_hurwitz_zeta_integrable: fixes z :: "complex"   and a :: "real" assumes "1 < Re z"   and "0 < a" shows "complex_set_integrable lebesgue {0<..} (\<lambda>x. complex_of_real x powr (z - 1) * complex_of_real (exp (- a * x) / (1 - exp (- x))))"
assms_2: shows "gen P I \<inter> J \<noteq> {}"
uminus_add_conv_diff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "plus (uminus a) b = minus b a"
mig_componentwise_inner_Basis: fixes i :: "'b"   and a :: "'b"   and b :: "'b" assumes "i \<in> Basis" shows "mig_componentwise a b \<bullet> i = mig_real (a \<bullet> i) (b \<bullet> i)"
case_distrib: fixes h :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and prod :: "'f \<times> 'g" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
surjective: fixes r :: "'a State_scheme" shows "r = \<lparr>getF = getF r, getM = getM r, \<dots> = more r\<rparr>"
transpose_mat_eq: fixes A :: "'a mat"   and B :: "'a mat" shows "(A\<^sup>T = B\<^sup>T) = (A = B)"
Block_self: fixes a :: "'a set \<times> 'b"   and A :: "('a set \<times> 'b) set" assumes "a \<in> A"   and "A \<in> rel.Partition" shows "rel.Class a = A"
prems_2: shows "position_of (subst \<rho> w) ((a # p) @ [d])"
jumpNestingOk_eval: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and jmps :: "jump set"   and Env :: "env"   and T :: "tys" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "jumpNestingOk jmps t"   and "Env\<turnstile>t\<Colon>T"   and "wf_prog G"   and "prg Env = G"   and "\<forall>j. fst s0 = Some (Jump j) \<longrightarrow> j \<in> jmps" shows "(\<forall>j. fst s1 = Some (Jump j) \<longrightarrow> j \<in> jmps) \<and> (normal s1 \<longrightarrow> (\<forall>w upd. v = In2 (w, upd) \<longrightarrow> (\<forall>s j val. fst s \<noteq> Some (Jump j) \<longrightarrow> fst (upd val s) \<noteq> Some (Jump j))))"
above_sctn_def: fixes X :: "'b set"   and sctn :: "'b sctn" shows "above_sctn X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> sabove_halfspace sctn)"
\<psi>_in_terms_of_\<Psi>': fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = (\<Phi>\<Psi>.\<psi>C (F y, x) \<circ> \<Phi>\<Psi>.\<Psi>.FUN (y, x) \<circ> \<phi>\<psi>.inD) g"
dom_def: fixes f :: "'a arrow_of_spans_data" shows "V.dom f = (if V.domains f \<noteq> {} then SOME a. a \<in> V.domains f else V.null)"
fds_nth_0: fixes f :: "'a fds" shows "fds_nth f 0 = (0::'a)"
len_lexext_rev_wf: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. len_lexext_rev gt ys xs)"
dc_prop1: fixes x :: "'a" shows "d x \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi>"
assms_4: shows "\<I> \<turnstile>g gpv \<surd>"
False: shows "\<not> (fst instr = shift_type SLL \<and> get_operand_w5 (snd instr ! 3) \<noteq> 0)"
Units_not_right_zero_divisor: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Units R"   and "b \<in> carrier R"   and "a \<otimes> b = \<zero>" shows "b = \<zero>"
sort_of_Rep_perm: fixes p :: "perm"   and a :: "atom" shows "sort_of (Rep_perm p a) = sort_of a"
linear_iff: fixes f :: "'a \<Rightarrow> 'b" shows "linear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x))"
uve: shows "r\<^sup>2 + N * s\<^sup>2 = (u\<^sup>2 + N * v\<^sup>2) * (p\<^sup>2 + N * q\<^sup>2) \<and> r = p * u + e * N * q * v \<and> s = p * v - e * q * u \<and> \<bar>e\<bar> = 1"
assms_3: shows "path_conforms_with_strategy p P well_ordered_strategy"
frontier_halfspace_ge: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b) \<or> b \<noteq> 0" shows "frontier {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
assms_3: fixes p :: "'a"   and k :: "nat"   and x :: "'a" assumes "prime p"   and "0 < k"   and "\<not> p dvd x"   and "P x" shows "P (p ^ k * x)"
coeff_simp: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "coeff (UP R) f = f"
commute: fixes a :: "'b"   and b :: "'b" shows "a * b = b * a"
converse_tranclE: fixes x :: "'b"   and z :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, z) \<in> r\<^sup>+" obtains "(x, z) \<in> r"   | y :: "'b" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>+"
length_insort_wrt: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "length (insort_wrt R xs) = length xs"
primitive_encoding_1: shows "\<forall>k>0. \<not> coprime k n \<longrightarrow> gauss_sum k = 0"
span_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> cspan S" shows "sum f A \<in> cspan S"
comm_semiring_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "comm_semiring_ow U plus times \<equiv> ab_semigroup_add_ow U plus \<and> ab_semigroup_mult_ow U times \<and> comm_semiring_ow_axioms U plus times"
mult_less_0_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 abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
map_poly_def: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly" shows "map_poly f p = Poly (map f (coeffs p))"
boundedD: fixes step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and p :: "nat"   and q :: "nat"   and t :: "'b"   and xs :: "'b" assumes "bounded step n"   and "p < n"   and "(q, t) \<in> set (step p xs)" shows "q < n"
per_positive: fixes w :: "'b list"   and n :: "nat" assumes "periodN w n" shows "n \<noteq> 0"
run_state_def: fixes stateT :: "('a, 'b) stateT" shows "run_state stateT = (case stateT of StateT x \<Rightarrow> x)"
t_gt_chop_t: fixes t :: "('s, 'v) tm" assumes "is_App t" shows "t >\<^sub>t chop t"
leg1_composite: shows "H\<nu>H\<pi>\<rho>.leg1 = \<nu>.leg1 \<cdot> \<nu>_\<pi>\<rho>.prj\<^sub>1"
connected_Un: fixes s :: "'a set"   and t :: "'a set" assumes "connected s"   and "connected t"   and "s \<inter> t \<noteq> {}" shows "connected (s \<union> t)"
dim_eq_span: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "dim T \<le> dim S" shows "span S = span T"
PR3: fixes p :: "'b pfm"   and r :: "'b pfm" assumes "\<turnstile>\<^sub>! p" shows "\<turnstile>\<^sub>! [r]\<^sub>! p"
block_design_multiple: fixes n :: "nat" assumes "0 < n" shows "block_design \<V> (multiple_blocks n) \<k>"
min_int_poly_content: fixes x :: "'a" shows "Polynomial.content (min_int_poly x) = 1"
valid_init: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'g list \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list \<Rightarrow> bool" assumes "initial\<^sub>d ts \<S> valid" shows "valid (map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, [], \<D>, \<O>, \<R>)) ts)"
locally_essentially_surjective: fixes a :: "'a"   and b :: "'a"   and g :: "'b" assumes "B.obj a"   and "B.obj b"   and "\<guillemotleft>g : F.map\<^sub>0 a \<rightarrow>\<^sub>C F.map\<^sub>0 b\<guillemotright>"   and "C.ide g" shows "\<exists>f. B.in_hhom f a b \<and> B.ide f \<and> C.isomorphic (F f) g"
finite_permutations_of_set: fixes A :: "'b set" shows "finite (permutations_of_set A)"
in_var_def: fixes x :: "'a \<Longrightarrow> 'b" shows "$x = x ;\<^sub>L fst\<^sub>L"
has_integral_exp_minus_to_infinity: fixes a :: "real"   and c :: "real" assumes "0 < a" shows "((\<lambda>x. exp (- a * x)) has_integral exp (- a * c) / a) {c..}"
\<a>_Nil: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "ps\<^sub>+ = []" shows "\<a> ps = 0"
keys_less_poly_mapping_size: fixes k :: "'a"   and m :: "'a \<Rightarrow>\<^sub>0 'b" assumes "k \<in> keys m" shows "f k + g (lookup m k) < local.poly_mapping_size m"
at_within_Icc_at: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and x :: "'b"   and b :: "'b" assumes "class.order_topology less_eq less open"   and "less a x"   and "less x b" shows "topological_space.at_within open x (ord.atLeastAtMost less_eq a b) = topological_space.at_within open x UNIV"
dim_eq_card: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> U\<^sub>V\<^sub>S"   and "V \<subseteq> U\<^sub>V\<^sub>S"   and "local.span B = local.span V"   and "\<not> local.dependent B" shows "local.dim V = card B"
dis_log_def: fixes \<G> :: "'b cyclic_group" shows "dis_log \<G> \<equiv> 0 < order \<G>"
hyps_3: shows "\<forall>cn. mon_ww fg w \<inter> mon_c fg cn = {} \<longrightarrow> (\<exists>csp. c' = csp + ch \<and> mon_c fg csp = {} \<and> ((sh, cn), w, s', csp + cn) \<in> trcl (ntrs fg))"
fvs_simp_7: fixes P :: "'b trm" shows "fvs (Snd P) = fvs P"
assms_2: shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
st_4: shows "\<forall>x\<in>i - T. g differentiable at x within i"
\<G>_semantics': fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G> \<phi> w = subformulas\<^sub>\<nu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>}"
enc_satisfies_pred_impl_indRelR_satisfies_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))" shows "\<forall>(P, Q)\<in>indRelR. Pred (P, Q)"
associatedI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b"   and "b divides\<^bsub>G\<^esub> a" shows "a \<sim>\<^bsub>G\<^esub> b"
lim_i_1: shows "Liminf_upto_llist (lmap fst TNs) (enat i) = lnth (lmap fst TNs) i"
intro: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complemented_lattice minus uminus inf less_eq less sup bot top"   and "class.orthocomplemented_lattice_axioms uminus less_eq" shows "class.orthocomplemented_lattice minus uminus inf less_eq less sup bot top"
gc_step_intros_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e" shows "(\<Gamma>, Var x, S) \<Rightarrow>\<^sub>G (delete x \<Gamma>, e, Upd x # S)"
ide_def: fixes a :: "'a2" shows "C2.ide a \<equiv> a \<cdot>\<^sub>2 a \<noteq> C2.null \<and> (\<forall>f. (f \<cdot>\<^sub>2 a \<noteq> C2.null \<longrightarrow> f \<cdot>\<^sub>2 a = f) \<and> (a \<cdot>\<^sub>2 f \<noteq> C2.null \<longrightarrow> a \<cdot>\<^sub>2 f = f))"
pseudo_simplex_like_below_in_above_pseudominimal_is_top: fixes x :: "'a"   and P :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and A :: "'b set"   and t :: "'a" assumes "x \<in> P"   and "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "t \<in> P .\<^bold>\<le> x"   and "\<And>y. pseudominimal_in (P .\<^bold>\<le> x) y \<Longrightarrow> t \<succeq> y" shows "t = x"
renaming_par_beta: fixes t :: "sterm"   and t' :: "sterm"   and s :: "char list"   and sa :: "char list" assumes "t => t'" shows "[s \<rightarrow> Fvar sa] t => [s \<rightarrow> Fvar sa] t'"
comp_eqI: fixes f :: "'a"   and g :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "seq f g"   and "f = f'"   and "g = g'" shows "f \<cdot> g = f' \<cdot>\<^sub>B g'"
simps_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "bst_wrt P \<langle>\<rangle> = True"
keys_map_indets: fixes f :: "'b \<Rightarrow> 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "inj_on f (indets p)" shows "keys (map_indets f p) = (\<lambda>t. \<Sum>x\<in>keys t. monomial (lookup t x) (f x)) ` keys p"
not_evaluate_prog_timeout: fixes env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list" assumes "\<forall>res. \<not> evaluate_prog False env s prog res" shows "\<exists>r. evaluate_prog True env s prog r \<and> snd r = Rerr (Rabort Rtimeout_error)"
mult2_alt_ns_conv: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_alt_ns ns s = mult2_alt_s ns s \<union> multpw ns"
A2bRedundant: fixes y :: "i" assumes "AxiomsSet6" shows "dom cod y \<cong> cod y"
assms_2: shows "(\<^bold>\<not> \<^bold>\<forall> p) \<in> S"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'c list" shows "rotate n xs = rotate (n mod |xs|) xs"
borel_measurable_ereal_times: fixes f :: "'b \<Rightarrow> ereal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
assms_2: fixes P :: "ccs" assumes "(P, Q) \<in> Rel" shows "F !P"
otherassms_5: shows "(\<sigma>, pp) \<in> oreachable A S U"
list_rel_unsorted_term_poly_list_relD: fixes p :: "(char list list \<times> int) list"   and y :: "(char list multiset \<times> int) list" assumes "(p, y) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel" shows "mset y = {#(mset a, y). (a, y) \<in># mset p#} \<and> (\<forall>x\<in>mononoms p. distinct x)"
cfaI: fixes c :: "'a set" assumes "c \<in> T" shows "c face_of S"
continuous_map_eq_image_closure_subset: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
step_2: shows "b \<in> proj_arcs_H ` H.face_cycle_set (proj_arcs_G a)"
log: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow> Undecided"
br'_inner_step_def: shows "br'_inner_step \<equiv> \<lambda>r (Q, W, rcm). let c = the (rcm r) in (if c \<le> 1 then insert (lhs r) Q else Q, if c \<le> 1 \<and> lhs r \<notin> Q then insert (lhs r) W else W, rcm(r \<mapsto> c - 1))"
distinct_13: fixes x5 :: "char list" shows "Boolean \<noteq> Class x5"
leD: fixes y :: "'c list"   and x :: "'c list" assumes "suffix y x" shows "\<not> strict_suffix x y"
echelon_form_of_upt_k_det_iarrays_def: fixes A' :: "'d \<times> 'c iarray iarray"   and k :: "nat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" shows "echelon_form_of_upt_k_det_iarrays A' k bezout = (let A = snd A'; f = foldl echelon_form_of_column_k_det_iarrays (1::'c, A, 0, bezout) [0..<Suc k] in (fst f, fst (snd f)))"
simps: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'d pmf" shows "rel_pmf R p q = (\<exists>pq. (\<forall>x xa. (x, xa) \<in> set_pmf pq \<longrightarrow> R x xa) \<and> map_pmf fst pq = p \<and> map_pmf snd pq = q)"
eqr1: shows "st2.pmat_extension hadamard = st2.ptensor_mat hadamard (1\<^sub>m st2.d2)"
Inf_closed: shows "\<Sqinter>\<^sub>o\<^sub>w ` Pow U \<subseteq> U"
root_powr_inverse: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "root n x = x powr (1 / real n)"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H H'" shows "Group.group (H'\<lparr>carrier := H\<rparr>)"
assms_2: shows "ipresIGWls h' MOD' MOD''"
distinct_indices_state_def: fixes s :: "('b, 'c) state" shows "distinct_indices_state s = (\<forall>i x b x' b'. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x' = Some (i, b') \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x' = Some (i, b') \<longrightarrow> x = x' \<and> b = b')"
geometric_spmf_conv_while: shows "geometric_spmf p = map_spmf snd (local.while (True, 0))"
vfsequenceE: fixes xs :: "V" assumes "vfsequence xs" obtains "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
e_recurrence: fixes k :: "nat" assumes "0 < k" shows "e k = - (\<Sum>i = 1..k. (- 1) ^ i * e (k - i) * p i) / of_nat k"
pbij_vwb_is_bij_lens: fixes X :: "'b \<Longrightarrow> 'c" assumes "pbij_lens X"   and "vwb_lens X" shows "bij_lens X"