hcju/isabelleps huggingface dataset:
until_frmlsr_def: shows "until_frmlsr \<equiv> rec_ltlr {} {} (\<lambda>_. {}) (\<lambda>_. {}) (\<lambda>\<mu> \<psi>. (\<union>)) (\<lambda>\<mu> \<psi>. (\<union>)) (\<lambda>\<mu> \<mu>'. \<mu>') (\<lambda>\<mu> \<psi> \<mu>' \<psi>'. insert (\<mu>, \<psi>) (\<mu>' \<union> \<psi>')) (\<lambda>\<mu> \<psi>. (\<union>))"
flatten_plus: fixes p :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd" shows "flatten (p + q) = flatten p + flatten q"
rep_eq: fixes x :: "nat"   and xa :: "'c"   and xb :: "'c" shows "blinfun_apply (ode_d x xa xb) = (\<lambda>d. if xa \<in> Csafe then ode_d_raw x xa xb d else (0::'c))"
simps: fixes a :: "'form \<times> nat list"   and ps :: "('form \<times> nat list) \<times> ('form, 'var) antecedent" shows "(a \<in> scope ps) = (\<exists>v. a = v \<and> v |\<in>| vertices \<and> (\<forall>x xa. path v xa x \<longrightarrow> terminal_vertex xa \<longrightarrow> ps \<in> snd ` set x))"
distinguishing_weak_formula_card_aux: fixes P :: "'state"   and Q :: "'state" shows "|{- p \<bullet> (SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}| <o natLeq +c |UNIV|"
intro: fixes m :: "int" assumes "1 < m" shows "residues m"
distinct_7: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "UNIT \<noteq> PRODUCT x51 x52"
psubsetE: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater B A" obtains "order.greater_eq B A" and "\<not> order.greater_eq A B"
assms_1: fixes x :: "'a" shows "P x \<longrightarrow> Q x"
carrier_eq: shows "A - M = {x. of_weak_ranking (weak_ranking_aux (A - M)) x x}"
merge_heap: fixes l :: "'b tree"   and r :: "'b tree" assumes "heap l"   and "heap r" shows "heap (merge l r)"
assms_3: shows "set \<phi>s \<subseteq> fmla"
card_eq_0_iff: fixes A :: "'d set" shows "(card A = 0) = (A = {} \<or> infinite A)"
Cons_6: shows "getWatch1 stateA = getWatch1 stateB"
acas_imp_dcas: fixes v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and es :: "('c \<times> nat) edge list"   and v :: "'c \<times> nat" assumes "(v1, v2) \<in> subs"   and "SubRel.ces v2 es v subs" shows "SubRel.ces v1 es v subs"
map_poly_pCons: fixes f :: "'b \<Rightarrow> 'a"   and c :: "'b"   and p :: "'b poly" assumes "f (0::'b) = (0::'a)" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
fresh_def: fixes a :: "'c"   and x :: "'d" shows "a \<sharp> x = (a \<notin> supp x)"
induct: fixes H :: "'b pl set"   and x :: "'b pl"   and P :: "'b pl \<Rightarrow> bool" assumes "H |- x"   and "\<And>p. p \<in> H \<Longrightarrow> P p"   and "\<And>p q. P (p -> q -> p)"   and "\<And>p q r. P ((p -> q -> r) -> (p -> q) -> p -> r)"   and "\<And>p. P (((p -> false) -> false) -> p)"   and "\<And>p q. \<lbrakk>H |- p -> q; P (p -> q); H |- p; P p\<rbrakk> \<Longrightarrow> P q" shows "P x"
simps_2: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set"   and N :: "nat" shows "inferred_clause_sets R S (Suc N) = simplify (add_all_deducible_clauses R (inferred_clause_sets R S N))"
foldli_ball_aux: fixes l :: "'a list"   and P :: "'a \<Rightarrow> bool"   and b :: "bool" shows "foldli l (\<lambda>x. x) (\<lambda>x _. P x) b = (b \<and> Ball (set l) P)"
erlang_CDF_def: fixes k :: "nat"   and l :: "real"   and x :: "real" shows "erlang_CDF k l x = (if x < 0 then 0 else 1 - (\<Sum>n\<le>k. (l * x) ^ n * exp (- l * x) / fact n))"
Src_HcompNml: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Nml t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "E.Src (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = E.Src u"
no_msgs_change_if_no_channel: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "channel i = None" shows "msgs c i = msgs c' i"
suminf_minus: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "(\<Sum>n. - f n) = - (\<Sum>n. f n)"
eval_rewrite: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p = ring.eval (K [X]\<^bsub>K [X]\<^esub>) (map UP.poly_of_const p) X\<^bsub>K [X]\<^esub>"
fixing_ide_gives_functor_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd"   and a2 :: "'c" assumes "binary_functor A1 A2 B F"   and "partial_magma.ide A2 a2" shows "functor A1 B (\<lambda>f1. F (f1, a2))"
congSubstBisimSubst: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq>\<^sup>s Q" shows "P \<approx>\<^sup>s Q"
OrdLeastI: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "P (LEAST n. Ord n \<and> P n)"
prems: shows "\<Gamma>,T \<turnstile> while (b) c'"
supp_deltafun: fixes b :: "'a"   and a :: "'b" assumes "b \<noteq> (0::'a)" shows "supp (b \<delta> a) = {a}"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r tab (tab\<^sub>0 r) (tab\<^sub>1 r)"
akra_bazzi_measure_decreases: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "akra_bazzi_measure (bs ! i * x + (hs ! i) x) < akra_bazzi_measure x"
cpq: shows "map_poly cor [:f, e, 0, 1:] = map_poly cor [:- x1, 1:] * map_poly cor q"
8: shows "sset (gtrace r v) \<inter> graph A w (Suc k) = {}"
ce_set_to_rel_srj: shows "surj ce_set_to_rel"
find_min_v_min: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "xs \<noteq> []"   and "normalized xs"   and "distinct (vals xs)"   and "distinct (prios xs)" shows "val (the (find_min xs)) = PQ.min (pqueue xs)"
set_of'_def: fixes x :: "float interval option" shows "set_of' x = (case x of None \<Rightarrow> UNIV | Some i \<Rightarrow> set_of (real_interval i))"
hermitian_proj_psi_minus_1: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat" assumes "grover_state_sig n f dims R K" shows "hermitian (2 \<cdot>\<^sub>m grover_state_sig.proj_psi n - 1\<^sub>m (grover_state.N n))"
val_eq_Some_sum_if_valid_neq_Leaf: fixes as :: "'b list"   and t :: "'b tree" assumes "valid as t"   and "t \<noteq> Leaf" shows "val t = Some (SWA.sum as (l t) (r t))"
emeasure_lim_sequence_emb: fixes x :: "'a"   and J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "x \<in> space M"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>j. M))" shows "emeasure (lim_sequence x) (prod_emb UNIV (\<lambda>_. M) J X) = emeasure (Ionescu_Tulcea.CI (K' x) (\<lambda>_. M) J) X"
simps_1: fixes \<xi> :: "nat \<Rightarrow> 'b set" shows "\<xi> \<Turnstile>\<^sub>r true\<^sub>r = True"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
rep_eq: fixes x :: "float"   and xa :: "float" shows "real_of_float (x + xa) = real_of_float x + real_of_float xa"
stabilizer_def: fixes m :: "'a set" shows "conjG.stabilizer m = {g \<in> carrier G. conjugation_action p g m = m}"
last_in_set: fixes as :: "'b list" assumes "as \<noteq> []" shows "last as \<in>. as"
rep_eq: fixes x :: "'b mpoly"   and xa :: "'b mpoly" shows "mapping_of (x + xa) = mapping_of x + mapping_of xa"
at_eq: shows "at t within {0..} = at t within {0..t + 1}"
NotAndD: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "\<not> [\<phi> \<^bold>& \<psi> in v]" shows "\<not> [\<phi> in v] \<or> \<not> [\<psi> in v]"
eq_resource_onI: fixes A :: "'a set"   and res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource" assumes "\<And>a. a \<in> A \<Longrightarrow> rel_spmf (rel_prod (=) (eq_resource_on A)) (run_resource res a) (run_resource res' a)" shows "A \<turnstile>\<^sub>R res \<sim> res'"
simps_1: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a fps" shows "radical r 0 a 0 = (1::'a)"
LT: shows "LT ars r ((s, [\<beta>_step]), (s, [\<alpha>_step]), \<gamma>1, \<gamma>2, \<gamma>3, \<delta>1, \<delta>2, \<delta>3)"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
incomplete_block_proper_subset: fixes bl :: "'a set" assumes "incomplete_block bl" shows "bl \<subset> \<V>"
intensional_simps_21: fixes P :: "'a \<Rightarrow> bool" shows "P \<and> \<not> P \<equiv> #False"
simps_5: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (SKIP;; c, s) = Some (c, s)"
singleton_image: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "f ` {a} = {f a}"
diff_add: fixes M :: "'k multiset"   and N :: "'k multiset"   and Q :: "'k multiset" shows "M - (N + Q) = M - N - Q"
n_narrow: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "P C1"   and "P C2"   and "C2 \<le> C1"   and "C1 \<triangle> C2 < C1" shows "n (C1 \<triangle> C2) < n C1"
a_star_redu_AndR_elim: fixes a :: "coname"   and M :: "trm"   and b :: "coname"   and N :: "trm"   and c :: "coname"   and R :: "trm" assumes "AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a* R" shows "\<exists>M' N'. R = AndR <a>.M' <b>.N' c \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N'"
BinOp\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>"
properPath_coind: fixes phi :: "'a stream \<Rightarrow> 'a treeFI \<Rightarrow> bool"   and as :: "'a stream"   and tr :: "'a treeFI" assumes "phi as tr"   and "\<And>as tr. phi as tr \<Longrightarrow> shd as = lab tr"   and "\<And>as tr. phi as tr \<Longrightarrow> \<exists>tr'\<in>set (sub tr). phi (stl as) tr'" shows "properPath as tr"
convergent: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "convergent X"   and "convergent Y" shows "convergent (\<lambda>n. prod (X n) (Y n))"
reduced_GB_unique: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite G"   and "is_reduced_GB G"   and "pmdl G = pmdl F" shows "reduced_GB F = G"
assms_2: shows "strip_all_single_var B = Some \<tau>"
finite_dimensional_vector_space_on_axioms: shows "finite_dimensional_vector_space_on (dest.tangent_space (f p)) (*\<^sub>R) basis"
alpha_Tree_FL_valid_Tree: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and t1 :: "('f, 'c, 'd, 'e) Tree"   and t2 :: "('f, 'c, 'd, 'e) Tree"   and P :: "'b" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "t1 =\<^sub>\<alpha> t2" shows "effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P t1 = effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P t2"
row_factor'_def: fixes r :: "nat" shows "row_factor' r = row_factor (digit_encode (nths (dims A) I) r)"
Diff_Int_distrib2: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
ordinal_mod_self: fixes x :: "ordinal" shows "x mod x = 0"
order_less_imp_le: fixes x :: "'e"   and y :: "'e" assumes "x < y" shows "x \<le> y"
cases: fixes x :: "'g list" obtains "x = []"   | xa :: "'g" where "x = [xa__]"   | xa :: "'g" and y :: "'g" and xs :: "'g list" where "x = xa # y # xs"
sdrop_shift: fixes i :: "nat"   and w :: "'c list"   and s :: "'c stream" shows "sdrop i (w @- s) = drop i w @- sdrop (i - length w) s"
Annos_change_is_lub: fixes wf_md :: "'b J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'b expr) mdecl \<Rightarrow> bool"   and P :: "'b J_prog"   and is_lub1 :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and is_lub2 :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and E :: "String.literal \<Rightarrow> ty option"   and es :: "'b expr list"   and es' :: "'b expr list" assumes "wf_prog wf_md P"   and "\<And>T1 T2 T3. \<lbrakk>is_lub1 T1 T2 T3; is_type P T1; is_type P T2\<rbrakk> \<Longrightarrow> is_lub2 T1 T2 T3"   and "\<And>T1 T2 T3. \<lbrakk>is_lub2 T1 T2 T3; is_type P T1; is_type P T2\<rbrakk> \<Longrightarrow> is_type P T3"   and "is_lub1,P,E \<turnstile> es [\<leadsto>] es'"   and "ran E \<union> set (blocks_types es) \<subseteq> types P" shows "is_lub2,P,E \<turnstile> es [\<leadsto>] es'"
approx_slp_outer: fixes p :: "nat"   and n :: "nat"   and slp :: "floatarith list"   and XS :: "(real \<times> real pdevs) list"   and RS :: "(real \<times> real pdevs) list"   and fas :: "floatarith list"   and xs :: "real list" assumes "approx_slp_outer p n slp XS = Some RS"   and "slp = slp_of_fas fas"   and "n = length fas"   and "xs \<in> Joints XS" shows "interpret_floatariths fas xs @ xs \<in> Joints (RS @ XS)"
complement_rep_number: fixes x :: "'a"   and r :: "int" assumes "x \<in> \<V>"   and "\<B> rep x = r" shows "\<B>\<^sup>C rep x = \<b> - r"
intros_1: fixes u :: "'d \<Rightarrow> real"   and x :: "'d"   and d :: "nat" assumes "u x = real d" shows "intv_elem x u (Const d)"
eval_poly_subst: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and q :: "'c \<Rightarrow> ('e monom \<times> 'd) list"   and p :: "('c monom \<times> 'd) list" assumes "\<And>w. f w = eval_poly g (q w)" shows "eval_poly f p = eval_poly g (poly_subst q p)"
simps_10: fixes v :: "tm" shows "qfree (Le v) = True"
arbitrary_union_of_relative_to: fixes P :: "'c set \<Rightarrow> bool"   and U :: "'c set" shows "arbitrary union_of P relative_to U = arbitrary union_of (P relative_to U)"
Cons: fixes bs :: "'b list" shows "\<forall>i<length (fst (filter2 P as bs)). P (fst (filter2 P as bs) ! i) (snd (filter2 P as bs) ! i)"
wls_fresh_subst: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and y :: "'var" assumes "wls s X"   and "wls (asSort ys) Y" shows "fresh zs z (X #[Y / y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> fresh zs z Y))"
emeasure_Sup_measure': fixes M :: "'b measure set"   and A :: "'b measure"   and X :: "'b set" assumes "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A"   and "X \<in> sets A"   and "M \<noteq> {}" shows "emeasure (Sup_measure' M) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M}. emeasure (comm_monoid_set.F sup bot id P) X)"
fls_conv_base_factor_shift_subdegree: fixes f :: "'b fls" shows "f = fls_shift (- fls_subdegree f) (fls_base_factor f)"
reflexive_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c" assumes "matrix_idempotent_semiring.reflexive f" shows "f (i, i) = top"
simps: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "(a \<in> support_flow f) = (\<exists>e. a = e \<and> 0 < f e)"
attractor_set_VVpstar: fixes v :: "'a"   and p :: "Player"   and W :: "'a set" assumes "\<not> G'.deadend v"   and "\<And>w. v \<rightarrow>\<^bsub>G'\<^esub> w \<Longrightarrow> w \<in> G'.attractor p W" shows "v \<in> G'.attractor p W"
subcls1I: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "D \<in> baseClasses Bs" shows "P \<turnstile> C \<prec>\<^sup>1 D"
option_relI_1: fixes R :: "('a \<times> 'b) set" shows "(None, None) \<in> \<langle>R\<rangle>option_rel"
between_imp_collinear: fixes a :: "'b"   and b :: "'b"   and x :: "'b" assumes "between (a, b) x" shows "collinear {a, x, b}"
vsubst_subst_compose1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "good Y" shows "X #[Y / y]_ys #[y1 // y]_ys = X #[Y #[y1 // y]_ys / y]_ys"
callee_invariant: fixes y :: "'b"   and s' :: "'s2"   and s :: "'s2"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee2 s x)"   and "I2 s"   and "x \<in> outs_\<I> \<I>" shows "I2 s'"
Limit_is_Ord: fixes \<xi> :: "V" assumes "Limit \<xi>" shows "Ord \<xi>"
assms_3: shows "\<forall>\<^sub>F x in F. f x \<in> {l x..u x}"
abs_eq: fixes xa :: "'state"   and x :: "('b, 'pred, 'act, 'effect) Tree" shows "FL_valid_Tree\<^sub>\<alpha> xa (abs_Tree\<^sub>\<alpha> x) = FL_valid_Tree xa x"
capacity_finite: fixes \<Delta> :: "('a, 'b) network_scheme"   and e :: "'a \<times> 'a" assumes "countable_network \<Delta>" shows "capacity \<Delta> e \<noteq> \<top>"
numeral_One: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
vsubset_vsingletonD: fixes A :: "V"   and x :: "V" assumes "A \<subseteq>\<^sub>\<circ> ZFC_in_HOL.set {x}" shows "A = []\<^sub>\<circ> \<or> A = ZFC_in_HOL.set {x}"
in_lists_iff_set: fixes xs :: "'a list"   and A :: "'a set" shows "(xs \<in> lists A) = (set xs \<subseteq> A)"
ord_resolve_11: shows "eligible As (D + negs (mset As))"
ord_le_eq_trans: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" assumes "subseq a b"   and "b = c" shows "subseq a c"
add_strict_increasing: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "{#} \<subset># a"   and "b \<subseteq># c" shows "b \<subset># a + c"
add_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
map2_val_compat'D1: fixes f :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'f) list \<times> 'o"   and zs :: "('a \<times> 'e) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "oalist_inv (f zs)"
Max_eq_length_takeWhile: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>m. P m \<Longrightarrow> m < n"   and "\<exists>m. P m" shows "Max {m. P m} = n - Suc (length (takeWhile (Not \<circ> P) (rev [0..<n])))"
elims: fixes x :: "ct list"   and xa :: "ct"   and xb :: "ct"   and y :: "checker_type" assumes "select_return_top x xa xb = y" obtains ts :: "ct list" and ct1 :: "ct" where "x = ts__" and "xa = ct1" and "xb = TAny" and "y = TopType (take (length ts__ - 3) ts__ @ [ct1])"   | ts :: "ct list" and v :: "t" where "x = tsa__" and "xa = TAny" and "xb = TSome v" and "y = TopType (take (length tsa__ - 3) tsa__ @ [TSome v])"   | ts :: "ct list" and t1 :: "t" and t2 :: "t" where "x = ts" and "xa = TSome t1" and "xb = TSome t2" and "y = (if t1 = t2 then TopType (take (length ts - 3) ts @ [TSome t1]) else Bot)"
retractionE: fixes g :: "'a" assumes "A.retraction g" obtains f :: "'a" where "A.ide (g \<cdot>\<^sub>A f)"
update_correct_1: fixes m :: "'a"   and k :: "'b"   and v :: "'c" assumes "invar m" shows "\<alpha> (update k v m) = \<alpha> m(k \<mapsto> v)"
post_meas0_is_state: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" assumes "state n v"   and "prob0 n v i \<noteq> 0" shows "state n (post_meas0 n v i)"
simps_2: fixes a :: "'c" shows "un_LCons1 (lconst a) = a"
interior_diff: fixes S :: "'b set"   and T :: "'b set" shows "interior (S - T) = interior S - closure T"
Sup_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
finite_chain3_betw: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.finite_chain_with3 betw a b c X" shows "betw a b c"
Collect_conj_eq: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
has_type_cl_sub: fixes A :: "type_scheme list"   and e :: "expr"   and t :: "typ" assumes "A |- e :: t" shows "\<forall>S. $ S A |- e :: $ S t"
arrI: fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and a :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and b :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" shows "S.arr f"
meet_related: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "x \<sqinter> y = x"
to_tree_order_get_root_node: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and ptr'' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to"   and "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"   and "ptr'' \<in> set to" shows "h \<turnstile> get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr"
foldr_ts_tuple_rel: fixes as :: "'a option list"   and P :: "nat \<Rightarrow> bool"   and auxlist :: "(nat \<times> 'a option list set) list" shows "(as \<in> foldr (\<union>) (concat (map (\<lambda>(t, rel). if P t then [rel] else []) auxlist)) {}) = (\<exists>t. (t, as) \<in> ts_tuple_rel (set auxlist) \<and> P t)"